Beginning of Section Conj_ZF_UPair_closed__1__1
(*** $I sig/Nov2021ConjPreamble.mgs ***)
L4
Variable x : set
(*** Conj_ZF_UPair_closed__1__1 TMRiWfwZrGZRqqLu4PMvcC1SGNsi55EAL7o bounty of about 25 bars ***)
L5
Variable y : set
L6
Variable z : set
L7
Hypothesis H0 : z ∈ UPair x y
L8
Theorem. (Conj_ZF_UPair_closed__1__1)
If_i (x ∈ Empty) x y ∈ Repl (𝒫 (𝒫 x)) (Ξ»w : set β‡’ If_i (x ∈ w) x y) β†’ z ∈ Repl (𝒫 (𝒫 x)) (Ξ»w : set β‡’ If_i (x ∈ w) x y)
In Proofgold the corresponding term root is f6cddc... and proposition id is d27324...
Proof:
Proof not loaded.
End of Section Conj_ZF_UPair_closed__1__1
Beginning of Section Conj_ZF_UPair_closed__5__1
L14
Variable x : set
(*** Conj_ZF_UPair_closed__5__1 TMaKuZgYoxjnJZ4Wuj7xF141Rxsgr8YPNL9 bounty of about 25 bars ***)
L15
Variable y : set
L16
Variable z : set
L17
Hypothesis H0 : ZF_closed x
L18
Hypothesis H2 : z ∈ x
L19
Theorem. (Conj_ZF_UPair_closed__5__1)
Repl (𝒫 (𝒫 y)) (Ξ»w : set β‡’ If_i (y ∈ w) y z) = UPair y z β†’ UPair y z ∈ x
In Proofgold the corresponding term root is d1689a... and proposition id is 736fe9...
Proof:
Proof not loaded.
End of Section Conj_ZF_UPair_closed__5__1
Beginning of Section Conj_ordinal_ordsucc_In_eq__1__1
L25
Variable x : set
(*** Conj_ordinal_ordsucc_In_eq__1__1 TMZCED7PCCQYHAZMjpTYoEJYVQDV8dHMvdh bounty of about 25 bars ***)
L26
Variable y : set
L27
Hypothesis H0 : ordinal x
L28
Theorem. (Conj_ordinal_ordsucc_In_eq__1__1)
ordinal (ordsucc y) β†’ ordsucc y ∈ x ∨ x = ordsucc y
In Proofgold the corresponding term root is 19571a... and proposition id is 572d45...
Proof:
Proof not loaded.
End of Section Conj_ordinal_ordsucc_In_eq__1__1
Beginning of Section Conj_ordinal_famunion__2__0
L34
Variable x : set
(*** Conj_ordinal_famunion__2__0 TMJimBpWJPHahokERpFVVLW5CYscT9BPAMs bounty of about 25 bars ***)
L35
Variable f : (set β†’ set)
L36
Variable y : set
L37
Variable z : set
L38
Hypothesis H1 : z ∈ x
L39
Hypothesis H2 : y ∈ f z
L40
Theorem. (Conj_ordinal_famunion__2__0)
ordinal (f z) β†’ TransSet y
In Proofgold the corresponding term root is 1cb0d5... and proposition id is 8a79f7...
Proof:
Proof not loaded.
End of Section Conj_ordinal_famunion__2__0
Beginning of Section Conj_KnasterTarski_set__3__0
L46
Variable x : set
(*** Conj_KnasterTarski_set__3__0 TMTN8gEBmZAHcVLDs3TPLxhWfwYgXqpCcwE bounty of about 25 bars ***)
L47
Variable f : (set β†’ set)
L48
Hypothesis H1 : Sep x (Ξ»y : set β‡’ βˆ€z : set, z ∈ 𝒫 x β†’ Subq (f z) z β†’ y ∈ z) ∈ 𝒫 x
L49
Hypothesis H2 : f (Sep x (Ξ»y : set β‡’ βˆ€z : set, z ∈ 𝒫 x β†’ Subq (f z) z β†’ y ∈ z)) ∈ 𝒫 x
L50
Hypothesis H3 : (βˆ€y : set, y ∈ 𝒫 x β†’ Subq (f y) y β†’ Subq (Sep x (Ξ»z : set β‡’ βˆ€w : set, w ∈ 𝒫 x β†’ Subq (f w) w β†’ z ∈ w)) y)
L51
Hypothesis H4 : Subq (f (Sep x (Ξ»y : set β‡’ βˆ€z : set, z ∈ 𝒫 x β†’ Subq (f z) z β†’ y ∈ z))) (Sep x (Ξ»y : set β‡’ βˆ€z : set, z ∈ 𝒫 x β†’ Subq (f z) z β†’ y ∈ z))
L52
Theorem. (Conj_KnasterTarski_set__3__0)
Subq (f (f (Sep x (Ξ»y : set β‡’ βˆ€z : set, z ∈ 𝒫 x β†’ Subq (f z) z β†’ y ∈ z)))) (f (Sep x (Ξ»y : set β‡’ βˆ€z : set, z ∈ 𝒫 x β†’ Subq (f z) z β†’ y ∈ z))) β†’ (βˆƒy : set, y ∈ 𝒫 x ∧ f y = y)
In Proofgold the corresponding term root is dfd671... and proposition id is 98ac0f...
Proof:
Proof not loaded.
End of Section Conj_KnasterTarski_set__3__0
Beginning of Section Conj_KnasterTarski_set__4__0
L58
Variable x : set
(*** Conj_KnasterTarski_set__4__0 TMLnrrDkmqrBbqmF3ezkji7rXkTCqVBhSXd bounty of about 25 bars ***)
L59
Variable f : (set β†’ set)
L60
Hypothesis H1 : Sep x (Ξ»y : set β‡’ βˆ€z : set, z ∈ 𝒫 x β†’ Subq (f z) z β†’ y ∈ z) ∈ 𝒫 x
L61
Hypothesis H2 : f (Sep x (Ξ»y : set β‡’ βˆ€z : set, z ∈ 𝒫 x β†’ Subq (f z) z β†’ y ∈ z)) ∈ 𝒫 x
L62
Hypothesis H3 : (βˆ€y : set, y ∈ 𝒫 x β†’ Subq (f y) y β†’ Subq (Sep x (Ξ»z : set β‡’ βˆ€w : set, w ∈ 𝒫 x β†’ Subq (f w) w β†’ z ∈ w)) y)
L63
Theorem. (Conj_KnasterTarski_set__4__0)
Subq (f (Sep x (Ξ»y : set β‡’ βˆ€z : set, z ∈ 𝒫 x β†’ Subq (f z) z β†’ y ∈ z))) (Sep x (Ξ»y : set β‡’ βˆ€z : set, z ∈ 𝒫 x β†’ Subq (f z) z β†’ y ∈ z)) β†’ (βˆƒy : set, y ∈ 𝒫 x ∧ f y = y)
In Proofgold the corresponding term root is 11da76... and proposition id is 16e542...
Proof:
Proof not loaded.
End of Section Conj_KnasterTarski_set__4__0
Beginning of Section Conj_SchroederBernstein__3__3
L69
Variable x : set
(*** Conj_SchroederBernstein__3__3 TMKiPykZFC3QousKfBFHUgTgsLHCrirmDkq bounty of about 25 bars ***)
L70
Variable y : set
L71
Variable f : (set β†’ set)
L72
Variable f2 : (set β†’ set)
L73
Variable z : set
L74
Variable w : set
L75
Variable u : set
L76
Variable v : set
L77
Hypothesis H0 : (βˆ€x2 : set, x2 ∈ y β†’ (βˆ€y2 : set, y2 ∈ y β†’ f2 x2 = f2 y2 β†’ x2 = y2))
L78
Hypothesis H1 : (Ξ»x2 : set β‡’ Repl (setminus y (Repl (setminus x x2) (Ξ»y2 : set β‡’ f y2))) (Ξ»y2 : set β‡’ f2 y2)) z = z
L79
Hypothesis H2 : w = f2 v
L80
Hypothesis H4 : u ∈ z
L81
Theorem. (Conj_SchroederBernstein__3__3)
u ∈ Repl (setminus y (Repl (setminus x z) f)) f2 β†’ inv y f2 w = inv y f2 u β†’ w = u
In Proofgold the corresponding term root is c39df2... and proposition id is 527b2f...
Proof:
Proof not loaded.
End of Section Conj_SchroederBernstein__3__3
Beginning of Section Conj_PigeonHole_nat__1__0
L87
Variable x : set
(*** Conj_PigeonHole_nat__1__0 TMY6v5YLkAD1d4QmaWZ9fBv1YhzzzzQ1XUJ bounty of about 25 bars ***)
L88
Variable f : (set β†’ set)
L89
Variable y : set
L90
Variable z : set
L91
Variable w : set
L92
Hypothesis H1 : z ∈ ordsucc (ordsucc x)
L93
Hypothesis H2 : ordsucc w ∈ ordsucc (ordsucc x)
L94
Hypothesis H3 : Β¬ Subq y z
L95
Hypothesis H4 : Subq y w
L96
Hypothesis H5 : f z = f (ordsucc w)
L97
Theorem. (Conj_PigeonHole_nat__1__0)
z β‰  ordsucc w
In Proofgold the corresponding term root is d0c8d7... and proposition id is 24d2cf...
Proof:
Proof not loaded.
End of Section Conj_PigeonHole_nat__1__0
Beginning of Section Conj_PigeonHole_nat__5__1
L103
Variable x : set
(*** Conj_PigeonHole_nat__5__1 TMVNArkhAjey8vvB2gCaVbmjaagwcyLiKYJ bounty of about 25 bars ***)
L104
Variable f : (set β†’ set)
L105
Variable y : set
L106
Variable z : set
L107
Variable w : set
L108
Hypothesis H0 : nat_p x
L109
Hypothesis H2 : z ∈ ordsucc x
L110
Hypothesis H3 : w ∈ ordsucc x
L111
Hypothesis H4 : z ∈ ordsucc (ordsucc x)
L112
Theorem. (Conj_PigeonHole_nat__5__1)
ordsucc z ∈ ordsucc (ordsucc x) β†’ If_i (Subq y z) (f (ordsucc z)) (f z) = If_i (Subq y w) (f (ordsucc w)) (f w) β†’ z = w
In Proofgold the corresponding term root is 69463d... and proposition id is a8682a...
Proof:
Proof not loaded.
End of Section Conj_PigeonHole_nat__5__1
Beginning of Section Conj_PigeonHole_nat_bij__2__2
L118
Variable x : set
(*** Conj_PigeonHole_nat_bij__2__2 TMRBHaXMizXz3nGHf4Ci1rJ5HnrVvffoWRX bounty of about 25 bars ***)
L119
Variable f : (set β†’ set)
L120
Variable y : set
L121
Variable z : set
L122
Variable w : set
L123
Hypothesis H0 : (βˆ€u : set, u ∈ x β†’ (βˆ€v : set, v ∈ x β†’ f u = f v β†’ u = v))
L124
Hypothesis H1 : Β¬ (βˆƒu : set, u ∈ x ∧ f u = y)
L125
Hypothesis H3 : w ∈ ordsucc x
L126
Theorem. (Conj_PigeonHole_nat_bij__2__2)
(z β‰  x β†’ z ∈ x) β†’ If_i (z = x) y (f z) = If_i (w = x) y (f w) β†’ z = w
In Proofgold the corresponding term root is de293d... and proposition id is 814b84...
Proof:
Proof not loaded.
End of Section Conj_PigeonHole_nat_bij__2__2
Beginning of Section Conj_finite_ind__2__4
L132
Variable p : (set β†’ prop)
(*** Conj_finite_ind__2__4 TMFUphzWSx8aKB1kaSnJBNNwYBgscuJiQRf bounty of about 25 bars ***)
L133
Variable x : set
L134
Variable y : set
L135
Variable f : (set β†’ set)
L136
Hypothesis H0 : (βˆ€z : set, βˆ€w : set, finite z β†’ nIn w z β†’ p z β†’ p (binunion z (Sing w)))
L137
Hypothesis H1 : nat_p x
L138
Hypothesis H2 : (βˆ€z : set, equip z x β†’ p z)
L139
Hypothesis H3 : (βˆ€z : set, z ∈ ordsucc x β†’ f z ∈ y)
L140
Hypothesis H5 : (βˆ€z : set, z ∈ y β†’ (βˆƒw : set, w ∈ ordsucc x ∧ f w = z))
L141
Theorem. (Conj_finite_ind__2__4)
y = binunion (Repl x f) (Sing (f x)) β†’ p y
In Proofgold the corresponding term root is 4ee808... and proposition id is 71cb45...
Proof:
Proof not loaded.
End of Section Conj_finite_ind__2__4
Beginning of Section Conj_Descr_Vo1_prop__1__1
L147
Variable P : ((set β†’ prop) β†’ prop)
(*** Conj_Descr_Vo1_prop__1__1 TMJEbBAX7t489skSyDBk87B3VJq3fD9y9y5 bounty of about 25 bars ***)
L148
Variable p : (set β†’ prop)
L149
Hypothesis H0 : (βˆ€q : set β†’ prop, βˆ€p2 : set β†’ prop, P q β†’ P p2 β†’ q = p2)
L150
Theorem. (Conj_Descr_Vo1_prop__1__1)
p = Descr_Vo1 P β†’ P (Descr_Vo1 P)
In Proofgold the corresponding term root is 5a51b6... and proposition id is 150974...
Proof:
Proof not loaded.
End of Section Conj_Descr_Vo1_prop__1__1
Beginning of Section Conj_nat_setsum1_ordsucc__1__0
L156
Variable x : set
(*** Conj_nat_setsum1_ordsucc__1__0 TMMkVv89B6dVBXNwMmBjyF8JAxVrG367z1V bounty of about 25 bars ***)
L157
Variable y : set
L158
Hypothesis H1 : x = ordsucc y
L159
Theorem. (Conj_nat_setsum1_ordsucc__1__0)
y ∈ x β†’ ordsucc y ∈ Inj1 x
In Proofgold the corresponding term root is 329db4... and proposition id is 01b082...
Proof:
Proof not loaded.
End of Section Conj_nat_setsum1_ordsucc__1__0
Beginning of Section Conj_PNoLt_trichotomy_or__6__2
L165
Variable x : set
(*** Conj_PNoLt_trichotomy_or__6__2 TMGdSYRypREbaCMjdgDuCTtKdaRoH3nfknc bounty of about 25 bars ***)
L166
Variable y : set
L167
Variable p : (set β†’ prop)
L168
Variable q : (set β†’ prop)
L169
Hypothesis H0 : TransSet y
L170
Hypothesis H1 : PNoEq_ (binintersect x y) p q
L171
Theorem. (Conj_PNoLt_trichotomy_or__6__2)
binintersect x y = x β†’ PNoLt x p y q ∨ x = y ∧ PNoEq_ x p q ∨ PNoLt y q x p
In Proofgold the corresponding term root is 5fbb6c... and proposition id is 21717c...
Proof:
Proof not loaded.
End of Section Conj_PNoLt_trichotomy_or__6__2
Beginning of Section Conj_PNoLt_trichotomy_or__7__2
L177
Variable x : set
(*** Conj_PNoLt_trichotomy_or__7__2 TMGAPyaWQWbvFHmMnj2nVubcV9WdEQYbgEg bounty of about 25 bars ***)
L178
Variable y : set
L179
Variable p : (set β†’ prop)
L180
Variable q : (set β†’ prop)
L181
Hypothesis H0 : ordinal x
L182
Hypothesis H1 : ordinal y
L183
Hypothesis H3 : TransSet y
L184
Theorem. (Conj_PNoLt_trichotomy_or__7__2)
ordinal (binintersect x y) β†’ PNoLt x p y q ∨ x = y ∧ PNoEq_ x p q ∨ PNoLt y q x p
In Proofgold the corresponding term root is 1bc7a3... and proposition id is 070102...
Proof:
Proof not loaded.
End of Section Conj_PNoLt_trichotomy_or__7__2
Beginning of Section Conj_PNoLt_tra__1__0
L190
Variable x : set
(*** Conj_PNoLt_tra__1__0 TMP2S2f7eN72k7Ft9eqBGUwGr685mV2iun8 bounty of about 25 bars ***)
L191
Variable y : set
L192
Variable z : set
L193
Variable p : (set β†’ prop)
L194
Variable q : (set β†’ prop)
L195
Variable p2 : (set β†’ prop)
L196
Variable w : set
L197
Hypothesis H1 : ordinal y
L198
Hypothesis H2 : TransSet z
L199
Hypothesis H3 : PNoEq_ x p q
L200
Hypothesis H4 : q x
L201
Hypothesis H5 : w ∈ y
L202
Hypothesis H6 : w ∈ z
L203
Hypothesis H7 : PNoEq_ w q p2
L204
Hypothesis H8 : Β¬ q w
L205
Hypothesis H9 : p2 w
L206
Theorem. (Conj_PNoLt_tra__1__0)
ordinal w β†’ PNoLt x p z p2
In Proofgold the corresponding term root is 64613e... and proposition id is 2ffa11...
Proof:
Proof not loaded.
End of Section Conj_PNoLt_tra__1__0
Beginning of Section Conj_PNoLt_tra__2__12
L212
Variable x : set
(*** Conj_PNoLt_tra__2__12 TMbsKqES3JZ25sb39E5FLWXfezbnLFGPCif bounty of about 25 bars ***)
L213
Variable y : set
L214
Variable z : set
L215
Variable p : (set β†’ prop)
L216
Variable q : (set β†’ prop)
L217
Variable p2 : (set β†’ prop)
L218
Variable w : set
L219
Variable u : set
L220
Hypothesis H0 : ordinal y
L221
Hypothesis H1 : TransSet x
L222
Hypothesis H2 : TransSet z
L223
Hypothesis H3 : w ∈ x
L224
Hypothesis H4 : PNoEq_ w p q
L225
Hypothesis H5 : Β¬ p w
L226
Hypothesis H6 : q w
L227
Hypothesis H7 : ordinal w
L228
Hypothesis H8 : u ∈ y
L229
Hypothesis H9 : u ∈ z
L230
Hypothesis H10 : PNoEq_ u q p2
L231
Hypothesis H11 : Β¬ q u
L232
Theorem. (Conj_PNoLt_tra__2__12)
ordinal u β†’ PNoLt x p z p2
In Proofgold the corresponding term root is 068007... and proposition id is cced19...
Proof:
Proof not loaded.
End of Section Conj_PNoLt_tra__2__12
Beginning of Section Conj_PNoLe_tra__1__0
L238
Variable x : set
(*** Conj_PNoLe_tra__1__0 TMZo1t6fSzg6H75vQLBhtHhzmjjdKUHaJa8 bounty of about 25 bars ***)
L239
Variable y : set
L240
Variable z : set
L241
Variable p : (set β†’ prop)
L242
Variable q : (set β†’ prop)
L243
Variable p2 : (set β†’ prop)
L244
Hypothesis H1 : ordinal z
L245
Hypothesis H2 : PNoLe y q z p2
L246
Hypothesis H3 : x = y
L247
Hypothesis H4 : PNoEq_ x p q
L248
Theorem. (Conj_PNoLe_tra__1__0)
PNoEq_ y p q β†’ PNoLe y p z p2
In Proofgold the corresponding term root is 6e0b08... and proposition id is 7337bf...
Proof:
Proof not loaded.
End of Section Conj_PNoLe_tra__1__0
Beginning of Section Conj_PNo_rel_strict_upperbd_antimon__6__1
L254
Variable P : (set β†’ ((set β†’ prop) β†’ prop))
(*** Conj_PNo_rel_strict_upperbd_antimon__6__1 TMcET1NKGkLLHZkqeXZ6wrGiUNehBwPcHQV bounty of about 25 bars ***)
L255
Variable x : set
L256
Variable p : (set β†’ prop)
L257
Variable y : set
L258
Hypothesis H0 : ordinal x
L259
Hypothesis H2 : TransSet x
L260
Theorem. (Conj_PNo_rel_strict_upperbd_antimon__6__1)
ordinal y β†’ (βˆ€z : set, z ∈ x β†’ (βˆ€q : set β†’ prop, PNo_downc P z q β†’ PNoLt z q x p)) β†’ (βˆ€z : set, z ∈ y β†’ (βˆ€q : set β†’ prop, PNo_downc P z q β†’ PNoLt z q y p))
In Proofgold the corresponding term root is 8f263d... and proposition id is 3d21a9...
Proof:
Proof not loaded.
End of Section Conj_PNo_rel_strict_upperbd_antimon__6__1
Beginning of Section Conj_PNo_rel_strict_upperbd_antimon__6__2
L266
Variable P : (set β†’ ((set β†’ prop) β†’ prop))
(*** Conj_PNo_rel_strict_upperbd_antimon__6__2 TMJywbFerFTHWRLM6wsuE5PWHEc8X7X8P6z bounty of about 25 bars ***)
L267
Variable x : set
L268
Variable p : (set β†’ prop)
L269
Variable y : set
L270
Hypothesis H0 : ordinal x
L271
Hypothesis H1 : y ∈ x
L272
Theorem. (Conj_PNo_rel_strict_upperbd_antimon__6__2)
ordinal y β†’ (βˆ€z : set, z ∈ x β†’ (βˆ€q : set β†’ prop, PNo_downc P z q β†’ PNoLt z q x p)) β†’ (βˆ€z : set, z ∈ y β†’ (βˆ€q : set β†’ prop, PNo_downc P z q β†’ PNoLt z q y p))
In Proofgold the corresponding term root is 6b0317... and proposition id is 4772a3...
Proof:
Proof not loaded.
End of Section Conj_PNo_rel_strict_upperbd_antimon__6__2
Beginning of Section Conj_PNo_rel_strict_lowerbd_antimon__4__0
L278
Variable P : (set β†’ ((set β†’ prop) β†’ prop))
(*** Conj_PNo_rel_strict_lowerbd_antimon__4__0 TMZeqeGZKBwUwRbq92ceL6K1Vmtb5irrQst bounty of about 25 bars ***)
L279
Variable x : set
L280
Variable p : (set β†’ prop)
L281
Variable y : set
L282
Variable z : set
L283
Variable q : (set β†’ prop)
L284
Hypothesis H1 : TransSet x
L285
Hypothesis H2 : TransSet y
L286
Hypothesis H3 : (βˆ€w : set, w ∈ x β†’ (βˆ€p2 : set β†’ prop, PNo_upc P w p2 β†’ PNoLt x p w p2))
L287
Hypothesis H4 : z ∈ y
L288
Hypothesis H5 : PNo_upc P z q
L289
Theorem. (Conj_PNo_rel_strict_lowerbd_antimon__4__0)
z ∈ x β†’ PNoLt y p z q
In Proofgold the corresponding term root is b0eccf... and proposition id is deba05...
Proof:
Proof not loaded.
End of Section Conj_PNo_rel_strict_lowerbd_antimon__4__0
Beginning of Section Conj_PNo_rel_imv_ex__4__9
L295
Variable P : (set β†’ ((set β†’ prop) β†’ prop))
(*** Conj_PNo_rel_imv_ex__4__9 TMPkhVJkFy9Ms7gkgDoFpsQKxepoVTDKUxj bounty of about 25 bars ***)
L296
Variable x : set
L297
Variable p : (set β†’ prop)
L298
Variable y : set
L299
Variable q : (set β†’ prop)
L300
Variable z : set
L301
Hypothesis H0 : ordinal x
L302
Hypothesis H1 : (βˆ€w : set, w ∈ x β†’ (βˆ€p2 : set β†’ prop, PNo_upc P w p2 β†’ PNoLt x p w p2))
L303
Hypothesis H2 : PNoEq_ x p (Ξ»w : set β‡’ p w ∨ w = x)
L304
Hypothesis H3 : PNo_upc P y q
L305
Hypothesis H4 : ordinal y
L306
Hypothesis H5 : y = x
L307
Hypothesis H6 : z ∈ x
L308
Hypothesis H7 : PNoEq_ z q (Ξ»w : set β‡’ p w ∨ w = x)
L309
Hypothesis H8 : Β¬ q z
L310
Theorem. (Conj_PNo_rel_imv_ex__4__9)
ordinal z β†’ PNoLt (ordsucc x) (Ξ»w : set β‡’ p w ∨ w = x) x q
In Proofgold the corresponding term root is 5ab3cf... and proposition id is 24c2fe...
Proof:
Proof not loaded.
End of Section Conj_PNo_rel_imv_ex__4__9
Beginning of Section Conj_PNo_rel_imv_ex__7__0
L316
Variable x : set
(*** Conj_PNo_rel_imv_ex__7__0 TMUqnU2Yz3CfJbQ6v5hbWEn8MujYPvVExsP bounty of about 25 bars ***)
L317
Variable p : (set β†’ prop)
L318
Variable q : (set β†’ prop)
L319
Variable y : set
L320
Hypothesis H1 : PNoEq_ x p (Ξ»z : set β‡’ p z ∨ z = x)
L321
Hypothesis H2 : y ∈ x
L322
Hypothesis H3 : PNoEq_ y (Ξ»z : set β‡’ p z ∨ z = x) q
L323
Hypothesis H4 : ¬ (p y ∨ y = x)
L324
Hypothesis H5 : ordinal y
L325
Hypothesis H6 : PNoLt y q x p
L326
Theorem. (Conj_PNo_rel_imv_ex__7__0)
Β¬ PNoLt x p y q
In Proofgold the corresponding term root is d9bff3... and proposition id is eb8531...
Proof:
Proof not loaded.
End of Section Conj_PNo_rel_imv_ex__7__0
Beginning of Section Conj_PNo_rel_imv_ex__7__3
L332
Variable x : set
(*** Conj_PNo_rel_imv_ex__7__3 TMMECdQEe62jktMZnRd6z9ZFpoDkuuVcofU bounty of about 25 bars ***)
L333
Variable p : (set β†’ prop)
L334
Variable q : (set β†’ prop)
L335
Variable y : set
L336
Hypothesis H0 : ordinal x
L337
Hypothesis H1 : PNoEq_ x p (Ξ»z : set β‡’ p z ∨ z = x)
L338
Hypothesis H2 : y ∈ x
L339
Hypothesis H4 : ¬ (p y ∨ y = x)
L340
Hypothesis H5 : ordinal y
L341
Hypothesis H6 : PNoLt y q x p
L342
Theorem. (Conj_PNo_rel_imv_ex__7__3)
Β¬ PNoLt x p y q
In Proofgold the corresponding term root is cb3490... and proposition id is e94bd7...
Proof:
Proof not loaded.
End of Section Conj_PNo_rel_imv_ex__7__3
Beginning of Section Conj_PNo_rel_imv_ex__7__4
L348
Variable x : set
(*** Conj_PNo_rel_imv_ex__7__4 TMQfmyfL9TveWvprJ64AKc7umWBsurN1bf3 bounty of about 25 bars ***)
L349
Variable p : (set β†’ prop)
L350
Variable q : (set β†’ prop)
L351
Variable y : set
L352
Hypothesis H0 : ordinal x
L353
Hypothesis H1 : PNoEq_ x p (Ξ»z : set β‡’ p z ∨ z = x)
L354
Hypothesis H2 : y ∈ x
L355
Hypothesis H3 : PNoEq_ y (Ξ»z : set β‡’ p z ∨ z = x) q
L356
Hypothesis H5 : ordinal y
L357
Hypothesis H6 : PNoLt y q x p
L358
Theorem. (Conj_PNo_rel_imv_ex__7__4)
Β¬ PNoLt x p y q
In Proofgold the corresponding term root is 9eefe4... and proposition id is e1d832...
Proof:
Proof not loaded.
End of Section Conj_PNo_rel_imv_ex__7__4
Beginning of Section Conj_PNo_rel_imv_ex__15__3
L364
Variable P : (set β†’ ((set β†’ prop) β†’ prop))
(*** Conj_PNo_rel_imv_ex__15__3 TMdaA3PkxB8s49rta6HPT6QuMuNfzfAR39H bounty of about 25 bars ***)
L365
Variable Q : (set β†’ ((set β†’ prop) β†’ prop))
L366
Variable x : set
L367
Variable y : set
L368
Variable p : (set β†’ prop)
L369
Hypothesis H0 : Β¬ (βˆƒq : set β†’ prop, PNo_rel_strict_uniq_imv P Q x q)
L370
Hypothesis H1 : x = ordsucc y
L371
Hypothesis H2 : ordinal y
L372
Hypothesis H4 : binintersect y (ordsucc y) = y
L373
Hypothesis H5 : binintersect (ordsucc y) y = y
L374
Hypothesis H6 : (βˆ€z : set, z ∈ y β†’ (βˆ€q : set β†’ prop, PNo_downc P z q β†’ PNoLt z q y p))
L375
Hypothesis H7 : (βˆ€z : set, z ∈ y β†’ (βˆ€q : set β†’ prop, PNo_upc Q z q β†’ PNoLt y p z q))
L376
Hypothesis H8 : (βˆ€q : set β†’ prop, PNo_rel_strict_imv P Q y q β†’ PNoEq_ y p q)
L377
Hypothesis H9 : PNoEq_ y p (Ξ»z : set β‡’ p z ∨ z = y)
L378
Hypothesis H10 : PNoLt y p (ordsucc y) (Ξ»z : set β‡’ p z ∨ z = y)
L379
Hypothesis H11 : Β¬ (PNo_rel_strict_imv P Q x (Ξ»z : set β‡’ p z ∧ z β‰  y) ∧ PNo_rel_strict_imv P Q x (Ξ»z : set β‡’ p z ∨ z = y))
L380
Hypothesis H12 : (βˆ€q : set β†’ prop, PNo_upc Q y q β†’ Β¬ PNoEq_ y p q)
L381
Theorem. (Conj_PNo_rel_imv_ex__15__3)
Β¬ PNo_rel_strict_imv P Q (ordsucc y) (Ξ»z : set β‡’ p z ∨ z = y)
In Proofgold the corresponding term root is a231b3... and proposition id is eef5ef...
Proof:
Proof not loaded.
End of Section Conj_PNo_rel_imv_ex__15__3
Beginning of Section Conj_PNo_rel_imv_ex__15__11
L387
Variable P : (set β†’ ((set β†’ prop) β†’ prop))
(*** Conj_PNo_rel_imv_ex__15__11 TMcFGbZa7n4xC3Kk7XJMkVTM2TQtc9TDmND bounty of about 25 bars ***)
L388
Variable Q : (set β†’ ((set β†’ prop) β†’ prop))
L389
Variable x : set
L390
Variable y : set
L391
Variable p : (set β†’ prop)
L392
Hypothesis H0 : Β¬ (βˆƒq : set β†’ prop, PNo_rel_strict_uniq_imv P Q x q)
L393
Hypothesis H1 : x = ordsucc y
L394
Hypothesis H2 : ordinal y
L395
Hypothesis H3 : ordinal (ordsucc y)
L396
Hypothesis H4 : binintersect y (ordsucc y) = y
L397
Hypothesis H5 : binintersect (ordsucc y) y = y
L398
Hypothesis H6 : (βˆ€z : set, z ∈ y β†’ (βˆ€q : set β†’ prop, PNo_downc P z q β†’ PNoLt z q y p))
L399
Hypothesis H7 : (βˆ€z : set, z ∈ y β†’ (βˆ€q : set β†’ prop, PNo_upc Q z q β†’ PNoLt y p z q))
L400
Hypothesis H8 : (βˆ€q : set β†’ prop, PNo_rel_strict_imv P Q y q β†’ PNoEq_ y p q)
L401
Hypothesis H9 : PNoEq_ y p (Ξ»z : set β‡’ p z ∨ z = y)
L402
Hypothesis H10 : PNoLt y p (ordsucc y) (Ξ»z : set β‡’ p z ∨ z = y)
L403
Hypothesis H12 : (βˆ€q : set β†’ prop, PNo_upc Q y q β†’ Β¬ PNoEq_ y p q)
L404
Theorem. (Conj_PNo_rel_imv_ex__15__11)
Β¬ PNo_rel_strict_imv P Q (ordsucc y) (Ξ»z : set β‡’ p z ∨ z = y)
In Proofgold the corresponding term root is a9afe8... and proposition id is b8cd09...
Proof:
Proof not loaded.
End of Section Conj_PNo_rel_imv_ex__15__11
Beginning of Section Conj_PNo_rel_imv_ex__16__2
L410
Variable x : set
(*** Conj_PNo_rel_imv_ex__16__2 TMP1ZTggZrEcrc8QArKuhb2NVS3udxnaFfT bounty of about 25 bars ***)
L411
Variable p : (set β†’ prop)
L412
Variable q : (set β†’ prop)
L413
Variable y : set
L414
Hypothesis H0 : ordinal x
L415
Hypothesis H1 : PNoEq_ x (Ξ»z : set β‡’ p z ∧ z β‰  x) p
L416
Hypothesis H3 : PNoEq_ y q (Ξ»z : set β‡’ p z ∧ z β‰  x)
L417
Hypothesis H4 : p y ∧ y β‰  x
L418
Hypothesis H5 : ordinal y
L419
Hypothesis H6 : PNoLt x p y q
L420
Theorem. (Conj_PNo_rel_imv_ex__16__2)
Β¬ PNoLt y q x p
In Proofgold the corresponding term root is bb97e9... and proposition id is ec2d3f...
Proof:
Proof not loaded.
End of Section Conj_PNo_rel_imv_ex__16__2
Beginning of Section Conj_PNo_rel_imv_ex__17__0
L426
Variable P : (set β†’ ((set β†’ prop) β†’ prop))
(*** Conj_PNo_rel_imv_ex__17__0 TMJHXAcqP2jrpeSxezB2zfZpacvRoMAStkT bounty of about 25 bars ***)
L427
Variable x : set
L428
Variable p : (set β†’ prop)
L429
Variable y : set
L430
Variable q : (set β†’ prop)
L431
Variable z : set
L432
Hypothesis H1 : (βˆ€w : set, w ∈ x β†’ (βˆ€p2 : set β†’ prop, PNo_upc P w p2 β†’ PNoLt x p w p2))
L433
Hypothesis H2 : PNoEq_ x (Ξ»w : set β‡’ p w ∧ w β‰  x) p
L434
Hypothesis H3 : PNo_upc P y q
L435
Hypothesis H4 : ordinal y
L436
Hypothesis H5 : y = x
L437
Hypothesis H6 : z ∈ x
L438
Hypothesis H7 : PNoEq_ z q (Ξ»w : set β‡’ p w ∧ w β‰  x)
L439
Hypothesis H8 : Β¬ q z
L440
Hypothesis H9 : p z ∧ z β‰  x
L441
Hypothesis H10 : ordinal z
L442
Theorem. (Conj_PNo_rel_imv_ex__17__0)
Β¬ PNoLt x p z q
In Proofgold the corresponding term root is b0e8ab... and proposition id is 5a9ced...
Proof:
Proof not loaded.
End of Section Conj_PNo_rel_imv_ex__17__0
Beginning of Section Conj_PNo_rel_imv_ex__19__3
L448
Variable P : (set β†’ ((set β†’ prop) β†’ prop))
(*** Conj_PNo_rel_imv_ex__19__3 TMZ4Krnt5Syu2fEJSsAsgwz1hfSqaGNNDUW bounty of about 25 bars ***)
L449
Variable x : set
L450
Variable p : (set β†’ prop)
L451
Variable y : set
L452
Variable q : (set β†’ prop)
L453
Hypothesis H0 : ordinal x
L454
Hypothesis H1 : ordinal (ordsucc x)
L455
Hypothesis H2 : binintersect x (ordsucc x) = x
L456
Hypothesis H4 : PNoEq_ x (Ξ»z : set β‡’ p z ∧ z β‰  x) p
L457
Hypothesis H5 : PNoLt (ordsucc x) (Ξ»z : set β‡’ p z ∧ z β‰  x) x p
L458
Hypothesis H6 : y ∈ ordsucc x
L459
Hypothesis H7 : PNo_upc P y q
L460
Theorem. (Conj_PNo_rel_imv_ex__19__3)
ordinal y β†’ PNoLt (ordsucc x) (Ξ»z : set β‡’ p z ∧ z β‰  x) y q
In Proofgold the corresponding term root is 233b53... and proposition id is 8d1209...
Proof:
Proof not loaded.
End of Section Conj_PNo_rel_imv_ex__19__3
Beginning of Section Conj_PNo_rel_imv_ex__20__5
L466
Variable P : (set β†’ ((set β†’ prop) β†’ prop))
(*** Conj_PNo_rel_imv_ex__20__5 TMUDoiz2H8EVUYnshq47r2ZTj1j3fmGo66s bounty of about 25 bars ***)
L467
Variable x : set
L468
Variable p : (set β†’ prop)
L469
Variable y : set
L470
Variable q : (set β†’ prop)
L471
Variable z : set
L472
Hypothesis H0 : ordinal x
L473
Hypothesis H1 : (βˆ€w : set, w ∈ x β†’ (βˆ€p2 : set β†’ prop, PNo_downc P w p2 β†’ PNoLt w p2 x p))
L474
Hypothesis H2 : PNo_downc P y q
L475
Hypothesis H3 : ordinal y
L476
Hypothesis H4 : y = x
L477
Hypothesis H6 : q z
L478
Hypothesis H7 : ordinal z
L479
Hypothesis H8 : PNoLt x p z q
L480
Theorem. (Conj_PNo_rel_imv_ex__20__5)
PNoLt z q x p β†’ PNoLt x q (ordsucc x) (Ξ»w : set β‡’ p w ∧ w β‰  x)
In Proofgold the corresponding term root is 23c728... and proposition id is b05661...
Proof:
Proof not loaded.
End of Section Conj_PNo_rel_imv_ex__20__5
Beginning of Section Conj_PNo_rel_imv_ex__22__6
L486
Variable P : (set β†’ ((set β†’ prop) β†’ prop))
(*** Conj_PNo_rel_imv_ex__22__6 TMYo2KqyE14yVRUbzyppxd9PeSZ4hWGzH4k bounty of about 25 bars ***)
L487
Variable x : set
L488
Variable p : (set β†’ prop)
L489
Variable y : set
L490
Variable q : (set β†’ prop)
L491
Variable z : set
L492
Hypothesis H0 : ordinal x
L493
Hypothesis H1 : (βˆ€w : set, w ∈ x β†’ (βˆ€p2 : set β†’ prop, PNo_downc P w p2 β†’ PNoLt w p2 x p))
L494
Hypothesis H2 : PNoEq_ x (Ξ»w : set β‡’ p w ∧ w β‰  x) p
L495
Hypothesis H3 : PNo_downc P y q
L496
Hypothesis H4 : ordinal y
L497
Hypothesis H5 : y = x
L498
Hypothesis H7 : PNoEq_ z (Ξ»w : set β‡’ p w ∧ w β‰  x) q
L499
Hypothesis H8 : Β¬ (p z ∧ z β‰  x)
L500
Hypothesis H9 : q z
L501
Theorem. (Conj_PNo_rel_imv_ex__22__6)
ordinal z β†’ PNoLt x q (ordsucc x) (Ξ»w : set β‡’ p w ∧ w β‰  x)
In Proofgold the corresponding term root is e3f4d2... and proposition id is 0ef788...
Proof:
Proof not loaded.
End of Section Conj_PNo_rel_imv_ex__22__6
Beginning of Section Conj_PNo_rel_imv_ex__29__1
L507
Variable P : (set β†’ ((set β†’ prop) β†’ prop))
(*** Conj_PNo_rel_imv_ex__29__1 TMRjXrmeJcsW4Zp9nEgAZkMWqz8uDRWNfEi bounty of about 25 bars ***)
L508
Variable Q : (set β†’ ((set β†’ prop) β†’ prop))
L509
Variable x : set
L510
Variable y : set
L511
Variable p : (set β†’ prop)
L512
Hypothesis H0 : Β¬ (βˆƒq : set β†’ prop, PNo_rel_strict_uniq_imv P Q x q)
L513
Hypothesis H2 : ordinal y
L514
Hypothesis H3 : ordinal (ordsucc y)
L515
Hypothesis H4 : binintersect y (ordsucc y) = y
L516
Hypothesis H5 : binintersect (ordsucc y) y = y
L517
Hypothesis H6 : (βˆ€z : set, z ∈ y β†’ (βˆ€q : set β†’ prop, PNo_downc P z q β†’ PNoLt z q y p))
L518
Hypothesis H7 : (βˆ€z : set, z ∈ y β†’ (βˆ€q : set β†’ prop, PNo_upc Q z q β†’ PNoLt y p z q))
L519
Hypothesis H8 : (βˆ€q : set β†’ prop, PNo_rel_strict_imv P Q y q β†’ PNoEq_ y p q)
L520
Hypothesis H9 : PNoEq_ y (Ξ»z : set β‡’ p z ∧ z β‰  y) p
L521
Hypothesis H10 : PNoLt (ordsucc y) (Ξ»z : set β‡’ p z ∧ z β‰  y) y p
L522
Hypothesis H11 : Β¬ (PNo_rel_strict_imv P Q x (Ξ»z : set β‡’ p z ∧ z β‰  y) ∧ PNo_rel_strict_imv P Q x (Ξ»z : set β‡’ p z ∨ z = y))
L523
Hypothesis H12 : (βˆ€q : set β†’ prop, PNo_downc P y q β†’ Β¬ PNoEq_ y p q)
L524
Theorem. (Conj_PNo_rel_imv_ex__29__1)
Β¬ PNo_rel_strict_imv P Q (ordsucc y) (Ξ»z : set β‡’ p z ∧ z β‰  y)
In Proofgold the corresponding term root is 8cba16... and proposition id is e00566...
Proof:
Proof not loaded.
End of Section Conj_PNo_rel_imv_ex__29__1
Beginning of Section Conj_PNo_rel_imv_ex__32__9
L530
Variable P : (set β†’ ((set β†’ prop) β†’ prop))
(*** Conj_PNo_rel_imv_ex__32__9 TMSa3THJvaCmqeT9cm6ArY2DGZB2Njhc7v3 bounty of about 25 bars ***)
L531
Variable Q : (set β†’ ((set β†’ prop) β†’ prop))
L532
Variable x : set
L533
Variable y : set
L534
Variable p : (set β†’ prop)
L535
Hypothesis H0 : PNoLt_pwise (PNo_downc P) (PNo_upc Q)
L536
Hypothesis H1 : Β¬ (βˆƒq : set β†’ prop, PNo_rel_strict_uniq_imv P Q x q)
L537
Hypothesis H2 : Β¬ (βˆƒz : set, z ∈ x ∧ (βˆƒq : set β†’ prop, PNo_rel_strict_split_imv P Q z q))
L538
Hypothesis H3 : y ∈ x
L539
Hypothesis H4 : x = ordsucc y
L540
Hypothesis H5 : ordinal y
L541
Hypothesis H6 : ordinal (ordsucc y)
L542
Hypothesis H7 : binintersect y (ordsucc y) = y
L543
Hypothesis H8 : binintersect (ordsucc y) y = y
L544
Hypothesis H10 : (βˆ€z : set, z ∈ y β†’ (βˆ€q : set β†’ prop, PNo_upc Q z q β†’ PNoLt y p z q))
L545
Hypothesis H11 : (βˆ€q : set β†’ prop, PNo_rel_strict_imv P Q y q β†’ PNoEq_ y p q)
L546
Hypothesis H12 : PNoEq_ y (Ξ»z : set β‡’ p z ∧ z β‰  y) p
L547
Hypothesis H13 : PNoLt (ordsucc y) (Ξ»z : set β‡’ p z ∧ z β‰  y) y p
L548
Hypothesis H14 : PNoEq_ y p (Ξ»z : set β‡’ p z ∨ z = y)
L549
Hypothesis H15 : p y ∨ y = y
L550
Theorem. (Conj_PNo_rel_imv_ex__32__9)
Β¬ PNoLt y p (ordsucc y) (Ξ»z : set β‡’ p z ∨ z = y)
In Proofgold the corresponding term root is b4e1a0... and proposition id is 9bc844...
Proof:
Proof not loaded.
End of Section Conj_PNo_rel_imv_ex__32__9
Beginning of Section Conj_PNo_rel_imv_ex__32__11
L556
Variable P : (set β†’ ((set β†’ prop) β†’ prop))
(*** Conj_PNo_rel_imv_ex__32__11 TMbTncLNNpNfkERbUjx8kv5pSWrwxxdAwYV bounty of about 25 bars ***)
L557
Variable Q : (set β†’ ((set β†’ prop) β†’ prop))
L558
Variable x : set
L559
Variable y : set
L560
Variable p : (set β†’ prop)
L561
Hypothesis H0 : PNoLt_pwise (PNo_downc P) (PNo_upc Q)
L562
Hypothesis H1 : Β¬ (βˆƒq : set β†’ prop, PNo_rel_strict_uniq_imv P Q x q)
L563
Hypothesis H2 : Β¬ (βˆƒz : set, z ∈ x ∧ (βˆƒq : set β†’ prop, PNo_rel_strict_split_imv P Q z q))
L564
Hypothesis H3 : y ∈ x
L565
Hypothesis H4 : x = ordsucc y
L566
Hypothesis H5 : ordinal y
L567
Hypothesis H6 : ordinal (ordsucc y)
L568
Hypothesis H7 : binintersect y (ordsucc y) = y
L569
Hypothesis H8 : binintersect (ordsucc y) y = y
L570
Hypothesis H9 : (βˆ€z : set, z ∈ y β†’ (βˆ€q : set β†’ prop, PNo_downc P z q β†’ PNoLt z q y p))
L571
Hypothesis H10 : (βˆ€z : set, z ∈ y β†’ (βˆ€q : set β†’ prop, PNo_upc Q z q β†’ PNoLt y p z q))
L572
Hypothesis H12 : PNoEq_ y (Ξ»z : set β‡’ p z ∧ z β‰  y) p
L573
Hypothesis H13 : PNoLt (ordsucc y) (Ξ»z : set β‡’ p z ∧ z β‰  y) y p
L574
Hypothesis H14 : PNoEq_ y p (Ξ»z : set β‡’ p z ∨ z = y)
L575
Hypothesis H15 : p y ∨ y = y
L576
Theorem. (Conj_PNo_rel_imv_ex__32__11)
Β¬ PNoLt y p (ordsucc y) (Ξ»z : set β‡’ p z ∨ z = y)
In Proofgold the corresponding term root is bf03ed... and proposition id is 414ac4...
Proof:
Proof not loaded.
End of Section Conj_PNo_rel_imv_ex__32__11
Beginning of Section Conj_PNo_rel_imv_ex__37__7
L582
Variable P : (set β†’ ((set β†’ prop) β†’ prop))
(*** Conj_PNo_rel_imv_ex__37__7 TMcJMYMAiAXtcxRJSe51e3P5atj4uZXxDxB bounty of about 25 bars ***)
L583
Variable Q : (set β†’ ((set β†’ prop) β†’ prop))
L584
Variable x : set
L585
Variable y : set
L586
Variable p : (set β†’ prop)
L587
Hypothesis H0 : PNoLt_pwise (PNo_downc P) (PNo_upc Q)
L588
Hypothesis H1 : Β¬ (βˆƒq : set β†’ prop, PNo_rel_strict_uniq_imv P Q x q)
L589
Hypothesis H2 : Β¬ (βˆƒz : set, z ∈ x ∧ (βˆƒq : set β†’ prop, PNo_rel_strict_split_imv P Q z q))
L590
Hypothesis H3 : y ∈ x
L591
Hypothesis H4 : x = ordsucc y
L592
Hypothesis H5 : ordinal y
L593
Hypothesis H6 : ordinal (ordsucc y)
L594
Hypothesis H8 : binintersect (ordsucc y) y = y
L595
Hypothesis H9 : (βˆ€z : set, z ∈ y β†’ (βˆ€q : set β†’ prop, PNo_downc P z q β†’ PNoLt z q y p))
L596
Hypothesis H10 : (βˆ€z : set, z ∈ y β†’ (βˆ€q : set β†’ prop, PNo_upc Q z q β†’ PNoLt y p z q))
L597
Hypothesis H11 : (βˆ€q : set β†’ prop, PNo_rel_strict_imv P Q y q β†’ PNoEq_ y p q)
L598
Theorem. (Conj_PNo_rel_imv_ex__37__7)
Β¬ PNoEq_ y (Ξ»z : set β‡’ p z ∧ z β‰  y) p
In Proofgold the corresponding term root is 0fe916... and proposition id is 905ede...
Proof:
Proof not loaded.
End of Section Conj_PNo_rel_imv_ex__37__7
Beginning of Section Conj_PNo_rel_imv_ex__38__8
L604
Variable P : (set β†’ ((set β†’ prop) β†’ prop))
(*** Conj_PNo_rel_imv_ex__38__8 TMbM6LC8i8uFLu3U3VFDbweyiHuAPXPBKzH bounty of about 25 bars ***)
L605
Variable Q : (set β†’ ((set β†’ prop) β†’ prop))
L606
Variable x : set
L607
Variable y : set
L608
Hypothesis H0 : PNoLt_pwise (PNo_downc P) (PNo_upc Q)
L609
Hypothesis H1 : Β¬ (βˆƒp : set β†’ prop, PNo_rel_strict_uniq_imv P Q x p)
L610
Hypothesis H2 : Β¬ (βˆƒz : set, z ∈ x ∧ (βˆƒp : set β†’ prop, PNo_rel_strict_split_imv P Q z p))
L611
Hypothesis H3 : (βˆ€z : set, z ∈ x β†’ (βˆƒp : set β†’ prop, PNo_rel_strict_uniq_imv P Q z p))
L612
Hypothesis H4 : y ∈ x
L613
Hypothesis H5 : x = ordsucc y
L614
Hypothesis H6 : ordinal y
L615
Hypothesis H7 : ordinal (ordsucc y)
L616
Theorem. (Conj_PNo_rel_imv_ex__38__8)
binintersect (ordsucc y) y β‰  y
In Proofgold the corresponding term root is da8890... and proposition id is d9c8fe...
Proof:
Proof not loaded.
End of Section Conj_PNo_rel_imv_ex__38__8
Beginning of Section Conj_PNo_rel_imv_ex__39__1
L622
Variable P : (set β†’ ((set β†’ prop) β†’ prop))
(*** Conj_PNo_rel_imv_ex__39__1 TMaJTxrE2RP1uZqiXFTk3KPKGrdXfrJ6Lp7 bounty of about 25 bars ***)
L623
Variable Q : (set β†’ ((set β†’ prop) β†’ prop))
L624
Variable x : set
L625
Variable y : set
L626
Hypothesis H0 : PNoLt_pwise (PNo_downc P) (PNo_upc Q)
L627
Hypothesis H2 : Β¬ (βˆƒz : set, z ∈ x ∧ (βˆƒp : set β†’ prop, PNo_rel_strict_split_imv P Q z p))
L628
Hypothesis H3 : (βˆ€z : set, z ∈ x β†’ (βˆƒp : set β†’ prop, PNo_rel_strict_uniq_imv P Q z p))
L629
Hypothesis H4 : y ∈ x
L630
Hypothesis H5 : x = ordsucc y
L631
Hypothesis H6 : ordinal y
L632
Hypothesis H7 : ordinal (ordsucc y)
L633
Theorem. (Conj_PNo_rel_imv_ex__39__1)
binintersect y (ordsucc y) β‰  y
In Proofgold the corresponding term root is b70e25... and proposition id is b66192...
Proof:
Proof not loaded.
End of Section Conj_PNo_rel_imv_ex__39__1
Beginning of Section Conj_PNo_rel_imv_ex__40__5
L639
Variable P : (set β†’ ((set β†’ prop) β†’ prop))
(*** Conj_PNo_rel_imv_ex__40__5 TMQVSK5chGYgFcx6Ybq4xeT6zLbkzBzTAcj bounty of about 25 bars ***)
L640
Variable Q : (set β†’ ((set β†’ prop) β†’ prop))
L641
Variable x : set
L642
Variable y : set
L643
Hypothesis H0 : PNoLt_pwise (PNo_downc P) (PNo_upc Q)
L644
Hypothesis H1 : ordinal x
L645
Hypothesis H2 : Β¬ (βˆƒp : set β†’ prop, PNo_rel_strict_uniq_imv P Q x p)
L646
Hypothesis H3 : Β¬ (βˆƒz : set, z ∈ x ∧ (βˆƒp : set β†’ prop, PNo_rel_strict_split_imv P Q z p))
L647
Hypothesis H4 : (βˆ€z : set, z ∈ x β†’ (βˆƒp : set β†’ prop, PNo_rel_strict_uniq_imv P Q z p))
L648
Hypothesis H6 : x = ordsucc y
L649
Hypothesis H7 : ordinal y
L650
Theorem. (Conj_PNo_rel_imv_ex__40__5)
Β¬ ordinal (ordsucc y)
In Proofgold the corresponding term root is 335ad9... and proposition id is 51980f...
Proof:
Proof not loaded.
End of Section Conj_PNo_rel_imv_ex__40__5
Beginning of Section Conj_PNo_rel_imv_ex__45__7
L656
Variable P : (set β†’ ((set β†’ prop) β†’ prop))
(*** Conj_PNo_rel_imv_ex__45__7 TMSMeR1ffX7wZjXdhctDSLiu8PCv4wcJV7Q bounty of about 25 bars ***)
L657
Variable Q : (set β†’ ((set β†’ prop) β†’ prop))
L658
Variable x : set
L659
Variable y : set
L660
Variable p : (set β†’ prop)
L661
Hypothesis H0 : TransSet x
L662
Hypothesis H1 : ordinal y
L663
Hypothesis H2 : (βˆ€z : set, z ∈ y β†’ z ∈ x β†’ PNo_rel_strict_uniq_imv P Q z (Ξ»w : set β‡’ βˆ€q : set β†’ prop, PNo_rel_strict_imv P Q (ordsucc w) q β†’ q w))
L664
Hypothesis H3 : y ∈ x
L665
Hypothesis H4 : PNo_rel_strict_imv P Q y p
L666
Hypothesis H5 : PNo_rel_strict_upperbd P y p
L667
Hypothesis H6 : PNo_rel_strict_lowerbd Q y p
L668
Theorem. (Conj_PNo_rel_imv_ex__45__7)
PNoEq_ y (Ξ»z : set β‡’ βˆ€q : set β†’ prop, PNo_rel_strict_imv P Q (ordsucc z) q β†’ q z) p β†’ PNo_rel_strict_imv P Q y (Ξ»z : set β‡’ βˆ€q : set β†’ prop, PNo_rel_strict_imv P Q (ordsucc z) q β†’ q z) ∧ (βˆ€q : set β†’ prop, PNo_rel_strict_imv P Q y q β†’ PNoEq_ y (Ξ»z : set β‡’ βˆ€p2 : set β†’ prop, PNo_rel_strict_imv P Q (ordsucc z) p2 β†’ p2 z) q)
In Proofgold the corresponding term root is 24ce4f... and proposition id is 8a635d...
Proof:
Proof not loaded.
End of Section Conj_PNo_rel_imv_ex__45__7
Beginning of Section Conj_PNo_rel_imv_ex__49__2
L674
Variable P : (set β†’ ((set β†’ prop) β†’ prop))
(*** Conj_PNo_rel_imv_ex__49__2 TMdG6zUPs33QZ9w9NuB9kM5zMdx1CT9RFEG bounty of about 25 bars ***)
L675
Variable Q : (set β†’ ((set β†’ prop) β†’ prop))
L676
Variable x : set
L677
Variable y : set
L678
Variable p : (set β†’ prop)
L679
Hypothesis H0 : PNo_upc Q y p
L680
Hypothesis H1 : ordinal y
L681
Hypothesis H3 : PNoEq_ y p (Ξ»z : set β‡’ βˆ€q : set β†’ prop, PNo_rel_strict_imv P Q (ordsucc z) q β†’ q z)
L682
Hypothesis H4 : (βˆ€q : set β†’ prop, PNo_rel_strict_imv P Q (ordsucc y) q β†’ q y)
L683
Hypothesis H5 : PNo_rel_strict_lowerbd Q (ordsucc y) (Ξ»z : set β‡’ βˆ€q : set β†’ prop, PNo_rel_strict_imv P Q (ordsucc z) q β†’ q z)
L684
Theorem. (Conj_PNo_rel_imv_ex__49__2)
PNoLt (ordsucc y) (Ξ»z : set β‡’ βˆ€q : set β†’ prop, PNo_rel_strict_imv P Q (ordsucc z) q β†’ q z) y p β†’ PNoLt x (Ξ»z : set β‡’ βˆ€q : set β†’ prop, PNo_rel_strict_imv P Q (ordsucc z) q β†’ q z) y p
In Proofgold the corresponding term root is e39ce0... and proposition id is 4434ae...
Proof:
Proof not loaded.
End of Section Conj_PNo_rel_imv_ex__49__2
Beginning of Section Conj_PNo_rel_imv_ex__54__4
L690
Variable P : (set β†’ ((set β†’ prop) β†’ prop))
(*** Conj_PNo_rel_imv_ex__54__4 TMLjymvpMEP9p77j9wc3pY7XjBEiBhjdWyp bounty of about 25 bars ***)
L691
Variable Q : (set β†’ ((set β†’ prop) β†’ prop))
L692
Variable x : set
L693
Variable y : set
L694
Variable p : (set β†’ prop)
L695
Variable z : set
L696
Hypothesis H0 : (βˆ€w : set, w ∈ x β†’ ordsucc w ∈ x)
L697
Hypothesis H1 : (βˆ€w : set, w ∈ x β†’ PNo_rel_strict_uniq_imv P Q w (Ξ»u : set β‡’ βˆ€q : set β†’ prop, PNo_rel_strict_imv P Q (ordsucc u) q β†’ q u))
L698
Hypothesis H2 : PNo_upc Q y p
L699
Hypothesis H3 : ordinal y
L700
Hypothesis H5 : z ∈ x
L701
Hypothesis H6 : PNoEq_ z p (Ξ»w : set β‡’ βˆ€q : set β†’ prop, PNo_rel_strict_imv P Q (ordsucc w) q β†’ q w)
L702
Hypothesis H7 : Β¬ p z
L703
Hypothesis H8 : (βˆ€q : set β†’ prop, PNo_rel_strict_imv P Q (ordsucc z) q β†’ q z)
L704
Theorem. (Conj_PNo_rel_imv_ex__54__4)
ordinal z β†’ PNoLt x (Ξ»w : set β‡’ βˆ€q : set β†’ prop, PNo_rel_strict_imv P Q (ordsucc w) q β†’ q w) y p
In Proofgold the corresponding term root is 812c2a... and proposition id is c8cd71...
Proof:
Proof not loaded.
End of Section Conj_PNo_rel_imv_ex__54__4
Beginning of Section Conj_PNo_rel_imv_ex__58__3
L710
Variable P : (set β†’ ((set β†’ prop) β†’ prop))
(*** Conj_PNo_rel_imv_ex__58__3 TMXsABnafy262JXapKahHoNdMf1hQwMdh71 bounty of about 25 bars ***)
L711
Variable Q : (set β†’ ((set β†’ prop) β†’ prop))
L712
Variable x : set
L713
Variable p : (set β†’ prop)
L714
Variable y : set
L715
Hypothesis H0 : ordinal x
L716
Hypothesis H1 : ordinal (ordsucc x)
L717
Hypothesis H2 : PNo_rel_strict_imv P Q (ordsucc x) p
L718
Hypothesis H4 : p y
L719
Hypothesis H5 : (βˆ€q : set β†’ prop, PNo_rel_strict_imv P Q (ordsucc y) q β†’ PNoEq_ (ordsucc y) (Ξ»z : set β‡’ βˆ€p2 : set β†’ prop, PNo_rel_strict_imv P Q (ordsucc z) p2 β†’ p2 z) q)
L720
Theorem. (Conj_PNo_rel_imv_ex__58__3)
PNoEq_ (ordsucc y) (Ξ»z : set β‡’ βˆ€q : set β†’ prop, PNo_rel_strict_imv P Q (ordsucc z) q β†’ q z) p β†’ (βˆ€q : set β†’ prop, PNo_rel_strict_imv P Q (ordsucc y) q β†’ q y)
In Proofgold the corresponding term root is 8c96ea... and proposition id is 8b768a...
Proof:
Proof not loaded.
End of Section Conj_PNo_rel_imv_ex__58__3
Beginning of Section Conj_PNo_rel_imv_ex__62__6
L726
Variable P : (set β†’ ((set β†’ prop) β†’ prop))
(*** Conj_PNo_rel_imv_ex__62__6 TMZ1CZUr9oZDmUpi9P6Y1JMLDXLAtHAN1sq bounty of about 25 bars ***)
L727
Variable Q : (set β†’ ((set β†’ prop) β†’ prop))
L728
Variable x : set
L729
Variable y : set
L730
Variable p : (set β†’ prop)
L731
Variable q : (set β†’ prop)
L732
Variable z : set
L733
Hypothesis H0 : TransSet x
L734
Hypothesis H1 : (βˆ€w : set, w ∈ x β†’ ordsucc w ∈ x)
L735
Hypothesis H2 : (βˆ€w : set, w ∈ x β†’ PNo_rel_strict_uniq_imv P Q w (Ξ»u : set β‡’ βˆ€p2 : set β†’ prop, PNo_rel_strict_imv P Q (ordsucc u) p2 β†’ p2 u))
L736
Hypothesis H3 : y ∈ x
L737
Hypothesis H4 : ordinal y
L738
Hypothesis H5 : ordinal (ordsucc y)
L739
Hypothesis H7 : PNo_rel_strict_imv P Q (ordsucc y) q
L740
Hypothesis H8 : z ∈ y
L741
Hypothesis H9 : Β¬ p z
L742
Hypothesis H10 : q z
L743
Theorem. (Conj_PNo_rel_imv_ex__62__6)
z ∈ x β†’ q y
In Proofgold the corresponding term root is 9d3f53... and proposition id is d094cd...
Proof:
Proof not loaded.
End of Section Conj_PNo_rel_imv_ex__62__6
Beginning of Section Conj_PNo_rel_imv_ex__64__8
L749
Variable P : (set β†’ ((set β†’ prop) β†’ prop))
(*** Conj_PNo_rel_imv_ex__64__8 TMVSk81mccW1NNMSJtNsBrXmaBcmnqAuQvW bounty of about 25 bars ***)
L750
Variable Q : (set β†’ ((set β†’ prop) β†’ prop))
L751
Variable x : set
L752
Variable y : set
L753
Variable p : (set β†’ prop)
L754
Variable q : (set β†’ prop)
L755
Hypothesis H0 : TransSet x
L756
Hypothesis H1 : (βˆ€z : set, z ∈ x β†’ ordsucc z ∈ x)
L757
Hypothesis H2 : (βˆ€z : set, z ∈ x β†’ PNo_rel_strict_uniq_imv P Q z (Ξ»w : set β‡’ βˆ€p2 : set β†’ prop, PNo_rel_strict_imv P Q (ordsucc w) p2 β†’ p2 w))
L758
Hypothesis H3 : y ∈ x
L759
Hypothesis H4 : PNo_downc P y p
L760
Hypothesis H5 : ordinal y
L761
Hypothesis H6 : ordinal (ordsucc y)
L762
Hypothesis H7 : PNoEq_ y (Ξ»z : set β‡’ βˆ€p2 : set β†’ prop, PNo_rel_strict_imv P Q (ordsucc z) p2 β†’ p2 z) p
L763
Hypothesis H9 : PNo_rel_strict_upperbd P (ordsucc y) q
L764
Theorem. (Conj_PNo_rel_imv_ex__64__8)
PNoLt y p (ordsucc y) q β†’ q y
In Proofgold the corresponding term root is b474bb... and proposition id is d28067...
Proof:
Proof not loaded.
End of Section Conj_PNo_rel_imv_ex__64__8
Beginning of Section Conj_PNo_rel_imv_ex__65__3
L770
Variable P : (set β†’ ((set β†’ prop) β†’ prop))
(*** Conj_PNo_rel_imv_ex__65__3 TMXsABnafy262JXapKahHoNdMf1hQwMdh71 bounty of about 25 bars ***)
L771
Variable Q : (set β†’ ((set β†’ prop) β†’ prop))
L772
Variable x : set
L773
Variable p : (set β†’ prop)
L774
Variable y : set
L775
Hypothesis H0 : ordinal x
L776
Hypothesis H1 : ordinal (ordsucc x)
L777
Hypothesis H2 : PNo_rel_strict_imv P Q (ordsucc x) p
L778
Hypothesis H4 : p y
L779
Hypothesis H5 : (βˆ€q : set β†’ prop, PNo_rel_strict_imv P Q (ordsucc y) q β†’ PNoEq_ (ordsucc y) (Ξ»z : set β‡’ βˆ€p2 : set β†’ prop, PNo_rel_strict_imv P Q (ordsucc z) p2 β†’ p2 z) q)
L780
Theorem. (Conj_PNo_rel_imv_ex__65__3)
PNoEq_ (ordsucc y) (Ξ»z : set β‡’ βˆ€q : set β†’ prop, PNo_rel_strict_imv P Q (ordsucc z) q β†’ q z) p β†’ (βˆ€q : set β†’ prop, PNo_rel_strict_imv P Q (ordsucc y) q β†’ q y)
In Proofgold the corresponding term root is 8c96ea... and proposition id is 8b768a...
Proof:
Proof not loaded.
End of Section Conj_PNo_rel_imv_ex__65__3
Beginning of Section Conj_PNo_rel_imv_ex__68__8
L786
Variable P : (set β†’ ((set β†’ prop) β†’ prop))
(*** Conj_PNo_rel_imv_ex__68__8 TMNr25UtU3bGKrNRzZ9a1G9dTQcmv7z44ah bounty of about 25 bars ***)
L787
Variable Q : (set β†’ ((set β†’ prop) β†’ prop))
L788
Variable x : set
L789
Variable p : (set β†’ prop)
L790
Variable y : set
L791
Variable q : (set β†’ prop)
L792
Variable z : set
L793
Hypothesis H0 : (βˆ€w : set, w ∈ x β†’ ordsucc w ∈ x)
L794
Hypothesis H1 : (βˆ€w : set, w ∈ x β†’ PNo_rel_strict_uniq_imv P Q w (Ξ»u : set β‡’ βˆ€p2 : set β†’ prop, PNo_rel_strict_imv P Q (ordsucc u) p2 β†’ p2 u))
L795
Hypothesis H2 : PNoEq_ y (Ξ»w : set β‡’ βˆ€p2 : set β†’ prop, PNo_rel_strict_imv P Q (ordsucc w) p2 β†’ p2 w) p
L796
Hypothesis H3 : ordinal y
L797
Hypothesis H4 : ordinal (ordsucc y)
L798
Hypothesis H5 : PNo_rel_strict_imv P Q (ordsucc y) q
L799
Hypothesis H6 : z ∈ y
L800
Hypothesis H7 : Β¬ p z
L801
Hypothesis H9 : z ∈ x
L802
Theorem. (Conj_PNo_rel_imv_ex__68__8)
ordsucc z ∈ x β†’ q y
In Proofgold the corresponding term root is ec127a... and proposition id is d9314d...
Proof:
Proof not loaded.
End of Section Conj_PNo_rel_imv_ex__68__8
Beginning of Section Conj_PNo_rel_imv_ex__71__8
L808
Variable P : (set β†’ ((set β†’ prop) β†’ prop))
(*** Conj_PNo_rel_imv_ex__71__8 TMcyfwW5fwja3EhvFF2VMY3PzBkQqpzC9re bounty of about 25 bars ***)
L809
Variable Q : (set β†’ ((set β†’ prop) β†’ prop))
L810
Variable x : set
L811
Variable y : set
L812
Variable p : (set β†’ prop)
L813
Variable z : set
L814
Variable q : (set β†’ prop)
L815
Hypothesis H0 : TransSet x
L816
Hypothesis H1 : (βˆ€w : set, w ∈ x β†’ ordsucc w ∈ x)
L817
Hypothesis H2 : (βˆ€w : set, w ∈ x β†’ PNo_rel_strict_uniq_imv P Q w (Ξ»u : set β‡’ βˆ€p2 : set β†’ prop, PNo_rel_strict_imv P Q (ordsucc u) p2 β†’ p2 u))
L818
Hypothesis H3 : PNo_downc P y p
L819
Hypothesis H4 : ordinal y
L820
Hypothesis H5 : z ∈ x
L821
Hypothesis H6 : z ∈ y
L822
Hypothesis H7 : PNoEq_ z (Ξ»w : set β‡’ βˆ€p2 : set β†’ prop, PNo_rel_strict_imv P Q (ordsucc w) p2 β†’ p2 w) p
L823
Hypothesis H9 : ordinal z
L824
Hypothesis H10 : ordinal (ordsucc z)
L825
Hypothesis H11 : PNo_rel_strict_imv P Q (ordsucc z) q
L826
Hypothesis H12 : PNo_rel_strict_upperbd P (ordsucc z) q
L827
Theorem. (Conj_PNo_rel_imv_ex__71__8)
PNoLt z p (ordsucc z) q β†’ q z
In Proofgold the corresponding term root is 1875f2... and proposition id is 1b79cd...
Proof:
Proof not loaded.
End of Section Conj_PNo_rel_imv_ex__71__8
Beginning of Section Conj_PNo_rel_imv_ex__73__4
L833
Variable P : (set β†’ ((set β†’ prop) β†’ prop))
(*** Conj_PNo_rel_imv_ex__73__4 TMTZNeDzxBf2Paf1Ak1kw7jGphtHhn12HUc bounty of about 25 bars ***)
L834
Variable Q : (set β†’ ((set β†’ prop) β†’ prop))
L835
Variable x : set
L836
Variable y : set
L837
Variable p : (set β†’ prop)
L838
Variable z : set
L839
Hypothesis H0 : TransSet x
L840
Hypothesis H1 : (βˆ€w : set, w ∈ x β†’ ordsucc w ∈ x)
L841
Hypothesis H2 : (βˆ€w : set, w ∈ x β†’ PNo_rel_strict_uniq_imv P Q w (Ξ»u : set β‡’ βˆ€q : set β†’ prop, PNo_rel_strict_imv P Q (ordsucc u) q β†’ q u))
L842
Hypothesis H3 : PNo_downc P y p
L843
Hypothesis H5 : z ∈ x
L844
Hypothesis H6 : z ∈ y
L845
Hypothesis H7 : PNoEq_ z (Ξ»w : set β‡’ βˆ€q : set β†’ prop, PNo_rel_strict_imv P Q (ordsucc w) q β†’ q w) p
L846
Hypothesis H8 : Β¬ (βˆ€q : set β†’ prop, PNo_rel_strict_imv P Q (ordsucc z) q β†’ q z)
L847
Hypothesis H9 : p z
L848
Theorem. (Conj_PNo_rel_imv_ex__73__4)
ordinal z β†’ PNoLt y p x (Ξ»w : set β‡’ βˆ€q : set β†’ prop, PNo_rel_strict_imv P Q (ordsucc w) q β†’ q w)
In Proofgold the corresponding term root is 1de5d2... and proposition id is feb97a...
Proof:
Proof not loaded.
End of Section Conj_PNo_rel_imv_ex__73__4
Beginning of Section Conj_PNo_rel_imv_ex__75__0
L854
Variable P : (set β†’ ((set β†’ prop) β†’ prop))
(*** Conj_PNo_rel_imv_ex__75__0 TMcYEpLRNq6iyc94Ea8qin2z3nWN7gRtU4E bounty of about 25 bars ***)
L855
Variable Q : (set β†’ ((set β†’ prop) β†’ prop))
L856
Variable x : set
L857
Variable y : set
L858
Variable p : (set β†’ prop)
L859
Hypothesis H1 : TransSet x
L860
Hypothesis H2 : (βˆ€z : set, z ∈ x β†’ ordsucc z ∈ x)
L861
Hypothesis H3 : (βˆ€z : set, z ∈ x β†’ PNo_rel_strict_uniq_imv P Q z (Ξ»w : set β‡’ βˆ€q : set β†’ prop, PNo_rel_strict_imv P Q (ordsucc w) q β†’ q w))
L862
Hypothesis H4 : y ∈ x
L863
Hypothesis H5 : PNo_downc P y p
L864
Theorem. (Conj_PNo_rel_imv_ex__75__0)
ordinal y β†’ PNoLt y p x (Ξ»z : set β‡’ βˆ€q : set β†’ prop, PNo_rel_strict_imv P Q (ordsucc z) q β†’ q z)
In Proofgold the corresponding term root is 43c9c7... and proposition id is 5f8026...
Proof:
Proof not loaded.
End of Section Conj_PNo_rel_imv_ex__75__0
Beginning of Section Conj_PNo_rel_imv_ex__77__2
L870
Variable P : (set β†’ ((set β†’ prop) β†’ prop))
(*** Conj_PNo_rel_imv_ex__77__2 TMRhqLyJ5fYxZwYa4CqwpZhTBg3fFGZvY2n bounty of about 25 bars ***)
L871
Variable Q : (set β†’ ((set β†’ prop) β†’ prop))
L872
Variable x : set
L873
Hypothesis H0 : ordinal x
L874
Hypothesis H1 : TransSet x
L875
Hypothesis H3 : (βˆ€y : set, y ∈ x β†’ (βˆƒp : set β†’ prop, PNo_rel_strict_uniq_imv P Q y p))
L876
Hypothesis H4 : (βˆ€y : set, y ∈ x β†’ ordsucc y ∈ x)
L877
Theorem. (Conj_PNo_rel_imv_ex__77__2)
Β¬ (βˆ€y : set, ordinal y β†’ y ∈ x β†’ PNo_rel_strict_uniq_imv P Q y (Ξ»z : set β‡’ βˆ€p : set β†’ prop, PNo_rel_strict_imv P Q (ordsucc z) p β†’ p z))
In Proofgold the corresponding term root is d1dcc5... and proposition id is 773d2d...
Proof:
Proof not loaded.
End of Section Conj_PNo_rel_imv_ex__77__2
Beginning of Section Conj_PNo_lenbdd_strict_imv_extend0__3__3
L883
Variable P : (set β†’ ((set β†’ prop) β†’ prop))
(*** Conj_PNo_lenbdd_strict_imv_extend0__3__3 TMHFSDDLMoqRPaTU6EkA9DePhT8cdJWQ6H4 bounty of about 25 bars ***)
L884
Variable x : set
L885
Variable p : (set β†’ prop)
L886
Variable y : set
L887
Variable q : (set β†’ prop)
L888
Hypothesis H0 : ordinal x
L889
Hypothesis H1 : PNo_lenbdd x P
L890
Hypothesis H2 : PNo_rel_strict_lowerbd P x p
L891
Hypothesis H4 : PNoEq_ x p (Ξ»z : set β‡’ p z ∧ z β‰  x)
L892
Hypothesis H5 : y ∈ ordsucc x
L893
Hypothesis H6 : PNo_upc P y q
L894
Theorem. (Conj_PNo_lenbdd_strict_imv_extend0__3__3)
ordinal y β†’ PNoLt (ordsucc x) (Ξ»z : set β‡’ p z ∧ z β‰  x) y q
In Proofgold the corresponding term root is 47e1a3... and proposition id is c6c301...
Proof:
Proof not loaded.
End of Section Conj_PNo_lenbdd_strict_imv_extend0__3__3
Beginning of Section Conj_PNo_lenbdd_strict_imv_extend0__4__2
L900
Variable x : set
(*** Conj_PNo_lenbdd_strict_imv_extend0__4__2 TMKoYkwGP2qYik5hxqWGwirkimZ9JarWCPu bounty of about 25 bars ***)
L901
Variable p : (set β†’ prop)
L902
Variable y : set
L903
Variable q : (set β†’ prop)
L904
Variable z : set
L905
Hypothesis H0 : ordinal x
L906
Hypothesis H1 : TransSet x
L907
Hypothesis H3 : y ∈ x
L908
Hypothesis H4 : z ∈ y
L909
Theorem. (Conj_PNo_lenbdd_strict_imv_extend0__4__2)
z ∈ x β†’ PNoEq_ z (Ξ»w : set β‡’ p w ∧ w β‰  x) q ∧ Β¬ (p z ∧ z β‰  x) ∧ q z β†’ PNoLt x p y q
In Proofgold the corresponding term root is 4b7934... and proposition id is 2db1e1...
Proof:
Proof not loaded.
End of Section Conj_PNo_lenbdd_strict_imv_extend0__4__2
Beginning of Section Conj_PNo_lenbdd_strict_imv_extend0__5__0
L915
Variable P : (set β†’ ((set β†’ prop) β†’ prop))
(*** Conj_PNo_lenbdd_strict_imv_extend0__5__0 TMJjZGcGRQQ2Ce13nNH5rvihU47HrNvukAc bounty of about 25 bars ***)
L916
Variable x : set
L917
Variable p : (set β†’ prop)
L918
Variable y : set
L919
Variable q : (set β†’ prop)
L920
Hypothesis H1 : TransSet x
L921
Hypothesis H2 : PNo_rel_strict_upperbd P x p
L922
Hypothesis H3 : ordinal (ordsucc x)
L923
Hypothesis H4 : PNoEq_ x p (Ξ»z : set β‡’ p z ∧ z β‰  x)
L924
Hypothesis H5 : ordinal y
L925
Hypothesis H6 : y ∈ x
L926
Hypothesis H7 : y ∈ ordsucc x
L927
Hypothesis H8 : PNo_downc P y q
L928
Theorem. (Conj_PNo_lenbdd_strict_imv_extend0__5__0)
PNoLt y q x p β†’ PNoLt y q (ordsucc x) (Ξ»z : set β‡’ p z ∧ z β‰  x)
In Proofgold the corresponding term root is db129c... and proposition id is e737e1...
Proof:
Proof not loaded.
End of Section Conj_PNo_lenbdd_strict_imv_extend0__5__0
Beginning of Section Conj_PNo_lenbdd_strict_imv_extend0__6__7
L934
Variable P : (set β†’ ((set β†’ prop) β†’ prop))
(*** Conj_PNo_lenbdd_strict_imv_extend0__6__7 TMUBiU1chGGXj36SUYxwP6wggWDbLFNXhKn bounty of about 25 bars ***)
L935
Variable x : set
L936
Variable p : (set β†’ prop)
L937
Variable y : set
L938
Variable q : (set β†’ prop)
L939
Hypothesis H0 : ordinal x
L940
Hypothesis H1 : TransSet x
L941
Hypothesis H2 : PNo_rel_strict_upperbd P x p
L942
Hypothesis H3 : ordinal (ordsucc x)
L943
Hypothesis H4 : PNoEq_ x p (Ξ»z : set β‡’ p z ∧ z β‰  x)
L944
Hypothesis H5 : ordinal y
L945
Hypothesis H6 : P y q
L946
Hypothesis H8 : y ∈ ordsucc x
L947
Theorem. (Conj_PNo_lenbdd_strict_imv_extend0__6__7)
PNo_downc P y q β†’ PNoLt y q (ordsucc x) (Ξ»z : set β‡’ p z ∧ z β‰  x)
In Proofgold the corresponding term root is 6007a4... and proposition id is f7371a...
Proof:
Proof not loaded.
End of Section Conj_PNo_lenbdd_strict_imv_extend0__6__7
Beginning of Section Conj_PNo_lenbdd_strict_imv_extend0__9__0
L953
Variable P : (set β†’ ((set β†’ prop) β†’ prop))
(*** Conj_PNo_lenbdd_strict_imv_extend0__9__0 TMdEFEo47d5rFP7v9XENqvQhExHMXarWWA6 bounty of about 25 bars ***)
L954
Variable x : set
L955
Variable p : (set β†’ prop)
L956
Variable y : set
L957
Variable q : (set β†’ prop)
L958
Hypothesis H1 : TransSet x
L959
Hypothesis H2 : PNo_lenbdd x P
L960
Hypothesis H3 : PNo_rel_strict_upperbd P x p
L961
Hypothesis H4 : ordinal (ordsucc x)
L962
Hypothesis H5 : PNoEq_ x p (Ξ»z : set β‡’ p z ∧ z β‰  x)
L963
Hypothesis H6 : y ∈ ordsucc x
L964
Hypothesis H7 : PNo_downc P y q
L965
Theorem. (Conj_PNo_lenbdd_strict_imv_extend0__9__0)
ordinal y β†’ PNoLt y q (ordsucc x) (Ξ»z : set β‡’ p z ∧ z β‰  x)
In Proofgold the corresponding term root is c6e08d... and proposition id is ab4fa2...
Proof:
Proof not loaded.
End of Section Conj_PNo_lenbdd_strict_imv_extend0__9__0
Beginning of Section Conj_PNo_lenbdd_strict_imv_extend0__10__0
L971
Variable P : (set β†’ ((set β†’ prop) β†’ prop))
(*** Conj_PNo_lenbdd_strict_imv_extend0__10__0 TMN8mYxu4LJAyaWvaGFtVRLA9uv1trJQGam bounty of about 25 bars ***)
L972
Variable Q : (set β†’ ((set β†’ prop) β†’ prop))
L973
Variable x : set
L974
Variable p : (set β†’ prop)
L975
Hypothesis H1 : TransSet x
L976
Hypothesis H2 : PNo_lenbdd x P
L977
Hypothesis H3 : PNo_lenbdd x Q
L978
Hypothesis H4 : PNo_rel_strict_upperbd P x p
L979
Hypothesis H5 : PNo_rel_strict_lowerbd Q x p
L980
Hypothesis H6 : ordinal (ordsucc x)
L981
Theorem. (Conj_PNo_lenbdd_strict_imv_extend0__10__0)
PNoEq_ x p (Ξ»y : set β‡’ p y ∧ y β‰  x) β†’ PNo_rel_strict_upperbd P (ordsucc x) (Ξ»y : set β‡’ p y ∧ y β‰  x) ∧ PNo_rel_strict_lowerbd Q (ordsucc x) (Ξ»y : set β‡’ p y ∧ y β‰  x)
In Proofgold the corresponding term root is ea39d4... and proposition id is fa1af6...
Proof:
Proof not loaded.
End of Section Conj_PNo_lenbdd_strict_imv_extend0__10__0
Beginning of Section Conj_PNo_lenbdd_strict_imv_extend1__2__0
L987
Variable P : (set β†’ ((set β†’ prop) β†’ prop))
(*** Conj_PNo_lenbdd_strict_imv_extend1__2__0 TMSZSeFY78qUQwvLBeQLwxMVzvgWzZb3Vim bounty of about 25 bars ***)
L988
Variable x : set
L989
Variable p : (set β†’ prop)
L990
Variable y : set
L991
Variable q : (set β†’ prop)
L992
Hypothesis H1 : TransSet x
L993
Hypothesis H2 : PNo_rel_strict_lowerbd P x p
L994
Hypothesis H3 : ordinal (ordsucc x)
L995
Hypothesis H4 : PNoEq_ x p (Ξ»z : set β‡’ p z ∨ z = x)
L996
Hypothesis H5 : ordinal y
L997
Hypothesis H6 : y ∈ x
L998
Hypothesis H7 : y ∈ ordsucc x
L999
Hypothesis H8 : PNo_upc P y q
L1000
Theorem. (Conj_PNo_lenbdd_strict_imv_extend1__2__0)
PNoLt x p y q β†’ PNoLt (ordsucc x) (Ξ»z : set β‡’ p z ∨ z = x) y q
In Proofgold the corresponding term root is a48c87... and proposition id is acf5b6...
Proof:
Proof not loaded.
End of Section Conj_PNo_lenbdd_strict_imv_extend1__2__0
Beginning of Section Conj_PNo_lenbdd_strict_imv_extend1__2__1
L1006
Variable P : (set β†’ ((set β†’ prop) β†’ prop))
(*** Conj_PNo_lenbdd_strict_imv_extend1__2__1 TMHPVZsv2MoYfbFNYANEqTw8XNS7UGjworc bounty of about 25 bars ***)
L1007
Variable x : set
L1008
Variable p : (set β†’ prop)
L1009
Variable y : set
L1010
Variable q : (set β†’ prop)
L1011
Hypothesis H0 : ordinal x
L1012
Hypothesis H2 : PNo_rel_strict_lowerbd P x p
L1013
Hypothesis H3 : ordinal (ordsucc x)
L1014
Hypothesis H4 : PNoEq_ x p (Ξ»z : set β‡’ p z ∨ z = x)
L1015
Hypothesis H5 : ordinal y
L1016
Hypothesis H6 : y ∈ x
L1017
Hypothesis H7 : y ∈ ordsucc x
L1018
Hypothesis H8 : PNo_upc P y q
L1019
Theorem. (Conj_PNo_lenbdd_strict_imv_extend1__2__1)
PNoLt x p y q β†’ PNoLt (ordsucc x) (Ξ»z : set β‡’ p z ∨ z = x) y q
In Proofgold the corresponding term root is f5faa9... and proposition id is f4b34f...
Proof:
Proof not loaded.
End of Section Conj_PNo_lenbdd_strict_imv_extend1__2__1
Beginning of Section Conj_PNo_lenbdd_strict_imv_extend1__2__5
L1025
Variable P : (set β†’ ((set β†’ prop) β†’ prop))
(*** Conj_PNo_lenbdd_strict_imv_extend1__2__5 TMULkU7fQWJBbHSXaGeNqtRGiJQAEj9LyvD bounty of about 25 bars ***)
L1026
Variable x : set
L1027
Variable p : (set β†’ prop)
L1028
Variable y : set
L1029
Variable q : (set β†’ prop)
L1030
Hypothesis H0 : ordinal x
L1031
Hypothesis H1 : TransSet x
L1032
Hypothesis H2 : PNo_rel_strict_lowerbd P x p
L1033
Hypothesis H3 : ordinal (ordsucc x)
L1034
Hypothesis H4 : PNoEq_ x p (Ξ»z : set β‡’ p z ∨ z = x)
L1035
Hypothesis H6 : y ∈ x
L1036
Hypothesis H7 : y ∈ ordsucc x
L1037
Hypothesis H8 : PNo_upc P y q
L1038
Theorem. (Conj_PNo_lenbdd_strict_imv_extend1__2__5)
PNoLt x p y q β†’ PNoLt (ordsucc x) (Ξ»z : set β‡’ p z ∨ z = x) y q
In Proofgold the corresponding term root is 98bd96... and proposition id is 8008ed...
Proof:
Proof not loaded.
End of Section Conj_PNo_lenbdd_strict_imv_extend1__2__5
Beginning of Section Conj_PNo_lenbdd_strict_imv_extend1__4__6
L1044
Variable P : (set β†’ ((set β†’ prop) β†’ prop))
(*** Conj_PNo_lenbdd_strict_imv_extend1__4__6 TMSw5T2HCVFdjpAYPLQTbmnrSAcevxwLM9f bounty of about 25 bars ***)
L1045
Variable x : set
L1046
Variable p : (set β†’ prop)
L1047
Variable y : set
L1048
Variable q : (set β†’ prop)
L1049
Hypothesis H0 : ordinal x
L1050
Hypothesis H1 : TransSet x
L1051
Hypothesis H2 : PNo_rel_strict_lowerbd P x p
L1052
Hypothesis H3 : ordinal (ordsucc x)
L1053
Hypothesis H4 : PNoEq_ x p (Ξ»z : set β‡’ p z ∨ z = x)
L1054
Hypothesis H5 : ordinal y
L1055
Hypothesis H7 : y ∈ x
L1056
Theorem. (Conj_PNo_lenbdd_strict_imv_extend1__4__6)
y ∈ ordsucc x β†’ PNoLt (ordsucc x) (Ξ»z : set β‡’ p z ∨ z = x) y q
In Proofgold the corresponding term root is 464827... and proposition id is 5626a9...
Proof:
Proof not loaded.
End of Section Conj_PNo_lenbdd_strict_imv_extend1__4__6
Beginning of Section Conj_PNo_lenbdd_strict_imv_extend1__7__6
L1062
Variable P : (set β†’ ((set β†’ prop) β†’ prop))
(*** Conj_PNo_lenbdd_strict_imv_extend1__7__6 TMGBcjtjfJvCNPbgY63jS4MXjmAh2nCw51Y bounty of about 25 bars ***)
L1063
Variable x : set
L1064
Variable p : (set β†’ prop)
L1065
Variable y : set
L1066
Variable q : (set β†’ prop)
L1067
Hypothesis H0 : ordinal x
L1068
Hypothesis H1 : TransSet x
L1069
Hypothesis H2 : PNo_lenbdd x P
L1070
Hypothesis H3 : PNo_rel_strict_lowerbd P x p
L1071
Hypothesis H4 : ordinal (ordsucc x)
L1072
Hypothesis H5 : PNoEq_ x p (Ξ»z : set β‡’ p z ∨ z = x)
L1073
Hypothesis H7 : PNo_upc P y q
L1074
Theorem. (Conj_PNo_lenbdd_strict_imv_extend1__7__6)
ordinal y β†’ PNoLt (ordsucc x) (Ξ»z : set β‡’ p z ∨ z = x) y q
In Proofgold the corresponding term root is d948b1... and proposition id is 05de45...
Proof:
Proof not loaded.
End of Section Conj_PNo_lenbdd_strict_imv_extend1__7__6
Beginning of Section Conj_PNo_strict_upperbd_imp_rel_strict_upperbd__1__5
L1080
Variable x : set
(*** Conj_PNo_strict_upperbd_imp_rel_strict_upperbd__1__5 TMTyWYggM9BDvitg8kQK4KBANdTcrEhebZp bounty of about 25 bars ***)
L1081
Variable y : set
L1082
Variable p : (set β†’ prop)
L1083
Variable z : set
L1084
Variable q : (set β†’ prop)
L1085
Hypothesis H0 : ordinal x
L1086
Hypothesis H1 : z ∈ y
L1087
Hypothesis H2 : TransSet x
L1088
Hypothesis H3 : ordinal y
L1089
Hypothesis H4 : ordinal z
L1090
Hypothesis H6 : PNoLt z q x p
L1091
Hypothesis H7 : z ∈ x
L1092
Theorem. (Conj_PNo_strict_upperbd_imp_rel_strict_upperbd__1__5)
Subq z x β†’ PNoLt z q y p
In Proofgold the corresponding term root is 19af79... and proposition id is e8dc29...
Proof:
Proof not loaded.
End of Section Conj_PNo_strict_upperbd_imp_rel_strict_upperbd__1__5
Beginning of Section Conj_PNo_strict_upperbd_imp_rel_strict_upperbd__3__9
L1098
Variable x : set
(*** Conj_PNo_strict_upperbd_imp_rel_strict_upperbd__3__9 TMasZ9DTBvLQ4CXZa8ALpPeTfMwvWvRQsnR bounty of about 25 bars ***)
L1099
Variable y : set
L1100
Variable p : (set β†’ prop)
L1101
Variable z : set
L1102
Variable q : (set β†’ prop)
L1103
Variable w : set
L1104
Variable p2 : (set β†’ prop)
L1105
Hypothesis H0 : ordinal x
L1106
Hypothesis H1 : y ∈ ordsucc x
L1107
Hypothesis H2 : z ∈ y
L1108
Hypothesis H3 : TransSet x
L1109
Hypothesis H4 : ordinal y
L1110
Hypothesis H5 : ordinal z
L1111
Hypothesis H6 : Subq z y
L1112
Hypothesis H7 : ordinal w
L1113
Hypothesis H8 : PNoLe z q w p2
L1114
Theorem. (Conj_PNo_strict_upperbd_imp_rel_strict_upperbd__3__9)
PNoLt z q x p β†’ PNoLt z q y p
In Proofgold the corresponding term root is 57a9c8... and proposition id is 3a5b6c...
Proof:
Proof not loaded.
End of Section Conj_PNo_strict_upperbd_imp_rel_strict_upperbd__3__9
Beginning of Section Conj_PNo_strict_upperbd_imp_rel_strict_upperbd__4__0
L1120
Variable P : (set β†’ ((set β†’ prop) β†’ prop))
(*** Conj_PNo_strict_upperbd_imp_rel_strict_upperbd__4__0 TMEh2ZQdZ49iaw8LqRZuDkfDA6pa4wSmZ6a bounty of about 25 bars ***)
L1121
Variable x : set
L1122
Variable y : set
L1123
Variable p : (set β†’ prop)
L1124
Variable z : set
L1125
Variable q : (set β†’ prop)
L1126
Variable w : set
L1127
Variable p2 : (set β†’ prop)
L1128
Hypothesis H1 : y ∈ ordsucc x
L1129
Hypothesis H2 : PNo_strict_upperbd P x p
L1130
Hypothesis H3 : z ∈ y
L1131
Hypothesis H4 : TransSet x
L1132
Hypothesis H5 : ordinal y
L1133
Hypothesis H6 : ordinal z
L1134
Hypothesis H7 : Subq z y
L1135
Hypothesis H8 : ordinal w
L1136
Hypothesis H9 : P w p2
L1137
Hypothesis H10 : PNoLe z q w p2
L1138
Theorem. (Conj_PNo_strict_upperbd_imp_rel_strict_upperbd__4__0)
PNoLt w p2 x p β†’ PNoLt z q y p
In Proofgold the corresponding term root is 6be64c... and proposition id is 3d4191...
Proof:
Proof not loaded.
End of Section Conj_PNo_strict_upperbd_imp_rel_strict_upperbd__4__0
Beginning of Section Conj_PNo_strict_upperbd_imp_rel_strict_upperbd__7__4
L1144
Variable P : (set β†’ ((set β†’ prop) β†’ prop))
(*** Conj_PNo_strict_upperbd_imp_rel_strict_upperbd__7__4 TMRegrq8pGGwApkFnhLQM3YWz51XuJxNi7R bounty of about 25 bars ***)
L1145
Variable x : set
L1146
Variable y : set
L1147
Variable p : (set β†’ prop)
L1148
Variable z : set
L1149
Variable q : (set β†’ prop)
L1150
Hypothesis H0 : ordinal x
L1151
Hypothesis H1 : y ∈ ordsucc x
L1152
Hypothesis H2 : PNo_strict_upperbd P x p
L1153
Hypothesis H3 : z ∈ y
L1154
Hypothesis H5 : TransSet x
L1155
Hypothesis H6 : ordinal y
L1156
Theorem. (Conj_PNo_strict_upperbd_imp_rel_strict_upperbd__7__4)
TransSet y β†’ PNoLt z q y p
In Proofgold the corresponding term root is 0504d8... and proposition id is b18346...
Proof:
Proof not loaded.
End of Section Conj_PNo_strict_upperbd_imp_rel_strict_upperbd__7__4
Beginning of Section Conj_PNo_strict_upperbd_imp_rel_strict_upperbd__9__2
L1162
Variable P : (set β†’ ((set β†’ prop) β†’ prop))
(*** Conj_PNo_strict_upperbd_imp_rel_strict_upperbd__9__2 TMUu9yndFHGkhCkWCG8EGHLuB2xjSi1V65E bounty of about 25 bars ***)
L1163
Variable x : set
L1164
Variable y : set
L1165
Variable p : (set β†’ prop)
L1166
Variable z : set
L1167
Variable q : (set β†’ prop)
L1168
Hypothesis H0 : ordinal x
L1169
Hypothesis H1 : y ∈ ordsucc x
L1170
Hypothesis H3 : z ∈ y
L1171
Hypothesis H4 : PNo_downc P z q
L1172
Hypothesis H5 : TransSet x
L1173
Theorem. (Conj_PNo_strict_upperbd_imp_rel_strict_upperbd__9__2)
ordinal (ordsucc x) β†’ PNoLt z q y p
In Proofgold the corresponding term root is d165a1... and proposition id is 973e14...
Proof:
Proof not loaded.
End of Section Conj_PNo_strict_upperbd_imp_rel_strict_upperbd__9__2
Beginning of Section Conj_PNo_strict_lowerbd_imp_rel_strict_lowerbd__4__8
L1179
Variable P : (set β†’ ((set β†’ prop) β†’ prop))
(*** Conj_PNo_strict_lowerbd_imp_rel_strict_lowerbd__4__8 TMMyqikEgkE3KEQcbUPAPGYcGC6o9TiBxeb bounty of about 25 bars ***)
L1180
Variable x : set
L1181
Variable y : set
L1182
Variable p : (set β†’ prop)
L1183
Variable z : set
L1184
Variable q : (set β†’ prop)
L1185
Variable w : set
L1186
Variable p2 : (set β†’ prop)
L1187
Hypothesis H0 : ordinal x
L1188
Hypothesis H1 : y ∈ ordsucc x
L1189
Hypothesis H2 : PNo_strict_lowerbd P x p
L1190
Hypothesis H3 : z ∈ y
L1191
Hypothesis H4 : TransSet x
L1192
Hypothesis H5 : ordinal y
L1193
Hypothesis H6 : ordinal z
L1194
Hypothesis H7 : Subq z y
L1195
Hypothesis H9 : P w p2
L1196
Hypothesis H10 : PNoLe w p2 z q
L1197
Theorem. (Conj_PNo_strict_lowerbd_imp_rel_strict_lowerbd__4__8)
PNoLt x p w p2 β†’ PNoLt y p z q
In Proofgold the corresponding term root is 3eff33... and proposition id is 0009bf...
Proof:
Proof not loaded.
End of Section Conj_PNo_strict_lowerbd_imp_rel_strict_lowerbd__4__8
Beginning of Section Conj_PNo_strict_lowerbd_imp_rel_strict_lowerbd__5__5
L1203
Variable P : (set β†’ ((set β†’ prop) β†’ prop))
(*** Conj_PNo_strict_lowerbd_imp_rel_strict_lowerbd__5__5 TMS99xStnfrm5WUUEpTTeQkD81zKpQpFf5X bounty of about 25 bars ***)
L1204
Variable x : set
L1205
Variable y : set
L1206
Variable p : (set β†’ prop)
L1207
Variable z : set
L1208
Variable q : (set β†’ prop)
L1209
Hypothesis H0 : ordinal x
L1210
Hypothesis H1 : y ∈ ordsucc x
L1211
Hypothesis H2 : PNo_strict_lowerbd P x p
L1212
Hypothesis H3 : z ∈ y
L1213
Hypothesis H4 : PNo_upc P z q
L1214
Hypothesis H6 : ordinal y
L1215
Hypothesis H7 : TransSet y
L1216
Hypothesis H8 : ordinal z
L1217
Theorem. (Conj_PNo_strict_lowerbd_imp_rel_strict_lowerbd__5__5)
Subq z y β†’ PNoLt y p z q
In Proofgold the corresponding term root is 98f54d... and proposition id is ed1ac3...
Proof:
Proof not loaded.
End of Section Conj_PNo_strict_lowerbd_imp_rel_strict_lowerbd__5__5
Beginning of Section Conj_PNo_strict_lowerbd_imp_rel_strict_lowerbd__7__2
L1223
Variable P : (set β†’ ((set β†’ prop) β†’ prop))
(*** Conj_PNo_strict_lowerbd_imp_rel_strict_lowerbd__7__2 TMWcaUDiL2YQAZvFqthnj1zU3twtmTtDoVi bounty of about 25 bars ***)
L1224
Variable x : set
L1225
Variable y : set
L1226
Variable p : (set β†’ prop)
L1227
Variable z : set
L1228
Variable q : (set β†’ prop)
L1229
Hypothesis H0 : ordinal x
L1230
Hypothesis H1 : y ∈ ordsucc x
L1231
Hypothesis H3 : z ∈ y
L1232
Hypothesis H4 : PNo_upc P z q
L1233
Hypothesis H5 : TransSet x
L1234
Hypothesis H6 : ordinal y
L1235
Theorem. (Conj_PNo_strict_lowerbd_imp_rel_strict_lowerbd__7__2)
TransSet y β†’ PNoLt y p z q
In Proofgold the corresponding term root is 3620ac... and proposition id is 76f790...
Proof:
Proof not loaded.
End of Section Conj_PNo_strict_lowerbd_imp_rel_strict_lowerbd__7__2
Beginning of Section Conj_PNo_rel_split_imv_imp_strict_imv__3__0
L1241
Variable P : (set β†’ ((set β†’ prop) β†’ prop))
(*** Conj_PNo_rel_split_imv_imp_strict_imv__3__0 TMGGCEt3CDedULNRck7Nu1mJwFb4VNuDtR6 bounty of about 25 bars ***)
L1242
Variable x : set
L1243
Variable p : (set β†’ prop)
L1244
Variable y : set
L1245
Variable q : (set β†’ prop)
L1246
Variable z : set
L1247
Hypothesis H1 : ordinal (ordsucc x)
L1248
Hypothesis H2 : PNo_rel_strict_lowerbd P (ordsucc x) (Ξ»w : set β‡’ p w ∨ w = x)
L1249
Hypothesis H3 : ordinal y
L1250
Hypothesis H4 : P y q
L1251
Hypothesis H5 : z ∈ ordsucc x
L1252
Hypothesis H6 : PNoEq_ z q p
L1253
Hypothesis H7 : p z ∨ z = x
L1254
Hypothesis H8 : ordinal z
L1255
Hypothesis H9 : PNoLt y q z q
L1256
Theorem. (Conj_PNo_rel_split_imv_imp_strict_imv__3__0)
Β¬ PNo_upc P z q
In Proofgold the corresponding term root is e36d58... and proposition id is 730d7b...
Proof:
Proof not loaded.
End of Section Conj_PNo_rel_split_imv_imp_strict_imv__3__0
Beginning of Section Conj_PNo_rel_split_imv_imp_strict_imv__11__7
L1262
Variable P : (set β†’ ((set β†’ prop) β†’ prop))
(*** Conj_PNo_rel_split_imv_imp_strict_imv__11__7 TMSD52yYnoeDoRvRq2hdSf2SqgfYbvqV3bz bounty of about 25 bars ***)
L1263
Variable x : set
L1264
Variable p : (set β†’ prop)
L1265
Variable y : set
L1266
Variable q : (set β†’ prop)
L1267
Variable z : set
L1268
Hypothesis H0 : ordinal x
L1269
Hypothesis H1 : ordinal (ordsucc x)
L1270
Hypothesis H2 : PNo_rel_strict_upperbd P (ordsucc x) (Ξ»w : set β‡’ p w ∧ w β‰  x)
L1271
Hypothesis H3 : ordinal y
L1272
Hypothesis H4 : P y q
L1273
Hypothesis H5 : z ∈ ordsucc x
L1274
Hypothesis H6 : PNoEq_ z p q
L1275
Hypothesis H8 : ordinal z
L1276
Hypothesis H9 : PNoLt z q y q
L1277
Theorem. (Conj_PNo_rel_split_imv_imp_strict_imv__11__7)
Β¬ PNo_downc P z q
In Proofgold the corresponding term root is 9002a6... and proposition id is f680b2...
Proof:
Proof not loaded.
End of Section Conj_PNo_rel_split_imv_imp_strict_imv__11__7
Beginning of Section Conj_PNo_rel_split_imv_imp_strict_imv__14__6
L1283
Variable P : (set β†’ ((set β†’ prop) β†’ prop))
(*** Conj_PNo_rel_split_imv_imp_strict_imv__14__6 TMEybc8mW9dUVZFgriZ2xToEKv1pkzpxKGD bounty of about 25 bars ***)
L1284
Variable x : set
L1285
Variable p : (set β†’ prop)
L1286
Variable y : set
L1287
Variable q : (set β†’ prop)
L1288
Hypothesis H0 : ordinal x
L1289
Hypothesis H1 : ordinal (ordsucc x)
L1290
Hypothesis H2 : PNo_rel_strict_upperbd P (ordsucc x) (Ξ»z : set β‡’ p z ∧ z β‰  x)
L1291
Hypothesis H3 : Β¬ (p x ∧ x β‰  x)
L1292
Hypothesis H4 : ordinal y
L1293
Hypothesis H5 : P y q
L1294
Theorem. (Conj_PNo_rel_split_imv_imp_strict_imv__14__6)
(βˆ€z : set, z ∈ ordsucc x β†’ z ∈ y β†’ PNoEq_ z p q β†’ q z β†’ p z ∧ z β‰  x) β†’ PNoLt y q x p
In Proofgold the corresponding term root is 43ba98... and proposition id is 42ce31...
Proof:
Proof not loaded.
End of Section Conj_PNo_rel_split_imv_imp_strict_imv__14__6
Beginning of Section Conj_PNo_rel_split_imv_imp_strict_imv__15__2
L1300
Variable P : (set β†’ ((set β†’ prop) β†’ prop))
(*** Conj_PNo_rel_split_imv_imp_strict_imv__15__2 TMbfvvkKvVMX9j8Y3tH6RGeVJUTv9tPThhQ bounty of about 25 bars ***)
L1301
Variable x : set
L1302
Variable p : (set β†’ prop)
L1303
Variable y : set
L1304
Variable q : (set β†’ prop)
L1305
Hypothesis H0 : ordinal x
L1306
Hypothesis H1 : ordinal (ordsucc x)
L1307
Hypothesis H3 : Β¬ (p x ∧ x β‰  x)
L1308
Hypothesis H4 : PNoLt (ordsucc x) (Ξ»z : set β‡’ p z ∧ z β‰  x) x p
L1309
Hypothesis H5 : ordinal y
L1310
Hypothesis H6 : P y q
L1311
Hypothesis H7 : PNo_downc P y q
L1312
Theorem. (Conj_PNo_rel_split_imv_imp_strict_imv__15__2)
(y ∈ ordsucc x β†’ PNoLt y q x p) β†’ PNoLt y q x p
In Proofgold the corresponding term root is 0b205f... and proposition id is 8f0e92...
Proof:
Proof not loaded.
End of Section Conj_PNo_rel_split_imv_imp_strict_imv__15__2
Beginning of Section Conj_PNo_rel_split_imv_imp_strict_imv__15__6
L1318
Variable P : (set β†’ ((set β†’ prop) β†’ prop))
(*** Conj_PNo_rel_split_imv_imp_strict_imv__15__6 TMK1xNLseFgMdUN3vMWZzqAQdY4enZMUv1p bounty of about 25 bars ***)
L1319
Variable x : set
L1320
Variable p : (set β†’ prop)
L1321
Variable y : set
L1322
Variable q : (set β†’ prop)
L1323
Hypothesis H0 : ordinal x
L1324
Hypothesis H1 : ordinal (ordsucc x)
L1325
Hypothesis H2 : PNo_rel_strict_upperbd P (ordsucc x) (Ξ»z : set β‡’ p z ∧ z β‰  x)
L1326
Hypothesis H3 : Β¬ (p x ∧ x β‰  x)
L1327
Hypothesis H4 : PNoLt (ordsucc x) (Ξ»z : set β‡’ p z ∧ z β‰  x) x p
L1328
Hypothesis H5 : ordinal y
L1329
Hypothesis H7 : PNo_downc P y q
L1330
Theorem. (Conj_PNo_rel_split_imv_imp_strict_imv__15__6)
(y ∈ ordsucc x β†’ PNoLt y q x p) β†’ PNoLt y q x p
In Proofgold the corresponding term root is fbb0de... and proposition id is 03512a...
Proof:
Proof not loaded.
End of Section Conj_PNo_rel_split_imv_imp_strict_imv__15__6
Beginning of Section Conj_PNo_rel_split_imv_imp_strict_imv__19__4
L1336
Variable P : (set β†’ ((set β†’ prop) β†’ prop))
(*** Conj_PNo_rel_split_imv_imp_strict_imv__19__4 TMazkeAkLcpM35R9Mvf1d6AWB8zmEK1KoBZ bounty of about 25 bars ***)
L1337
Variable Q : (set β†’ ((set β†’ prop) β†’ prop))
L1338
Variable x : set
L1339
Variable p : (set β†’ prop)
L1340
Hypothesis H0 : ordinal x
L1341
Hypothesis H1 : ordinal (ordsucc x)
L1342
Hypothesis H2 : PNo_rel_strict_upperbd P (ordsucc x) (Ξ»y : set β‡’ p y ∧ y β‰  x)
L1343
Hypothesis H3 : PNo_rel_strict_lowerbd Q (ordsucc x) (Ξ»y : set β‡’ p y ∨ y = x)
L1344
Theorem. (Conj_PNo_rel_split_imv_imp_strict_imv__19__4)
p x ∨ x = x β†’ PNo_strict_upperbd P x p ∧ PNo_strict_lowerbd Q x p
In Proofgold the corresponding term root is 209657... and proposition id is efcf62...
Proof:
Proof not loaded.
End of Section Conj_PNo_rel_split_imv_imp_strict_imv__19__4
Beginning of Section Conj_PNo_strict_imv_pred_eq__3__7
L1350
Variable P : (set β†’ ((set β†’ prop) β†’ prop))
(*** Conj_PNo_strict_imv_pred_eq__3__7 TMcwTwpZaM7Zax4fqkzuER6wGJn4kjRkjwC bounty of about 25 bars ***)
L1351
Variable Q : (set β†’ ((set β†’ prop) β†’ prop))
L1352
Variable x : set
L1353
Variable p : (set β†’ prop)
L1354
Variable q : (set β†’ prop)
L1355
Variable y : set
L1356
Hypothesis H0 : ordinal x
L1357
Hypothesis H1 : (βˆ€z : set, z ∈ x β†’ (βˆ€p2 : set β†’ prop, Β¬ PNo_strict_imv P Q z p2))
L1358
Hypothesis H2 : PNo_strict_lowerbd Q x p
L1359
Hypothesis H3 : PNo_strict_upperbd P x q
L1360
Hypothesis H4 : ordinal y
L1361
Hypothesis H5 : y ∈ x
L1362
Hypothesis H6 : Β¬ q y
L1363
Theorem. (Conj_PNo_strict_imv_pred_eq__3__7)
Β¬ PNoLt x q y q
In Proofgold the corresponding term root is b6294d... and proposition id is f2e677...
Proof:
Proof not loaded.
End of Section Conj_PNo_strict_imv_pred_eq__3__7
Beginning of Section Conj_PNo_strict_imv_pred_eq__6__3
L1369
Variable P : (set β†’ ((set β†’ prop) β†’ prop))
(*** Conj_PNo_strict_imv_pred_eq__6__3 TMTmARE13etB6hE9K6uH4uXvLhPh8BePC2V bounty of about 25 bars ***)
L1370
Variable Q : (set β†’ ((set β†’ prop) β†’ prop))
L1371
Variable x : set
L1372
Variable p : (set β†’ prop)
L1373
Variable q : (set β†’ prop)
L1374
Hypothesis H0 : ordinal x
L1375
Hypothesis H1 : TransSet x
L1376
Hypothesis H2 : (βˆ€y : set, y ∈ x β†’ (βˆ€p2 : set β†’ prop, Β¬ PNo_strict_imv P Q y p2))
L1377
Hypothesis H4 : PNo_strict_lowerbd Q x p
L1378
Hypothesis H5 : PNo_strict_upperbd P x q
L1379
Hypothesis H6 : PNo_strict_lowerbd Q x q
L1380
Theorem. (Conj_PNo_strict_imv_pred_eq__6__3)
(βˆ€y : set, ordinal y β†’ y ∈ x β†’ (p y ↔ q y)) β†’ (βˆ€y : set, y ∈ x β†’ (p y ↔ q y))
In Proofgold the corresponding term root is 740615... and proposition id is d71fbf...
Proof:
Proof not loaded.
End of Section Conj_PNo_strict_imv_pred_eq__6__3
Beginning of Section Conj_PNo_bd_In__1__3
L1386
Variable P : (set β†’ ((set β†’ prop) β†’ prop))
(*** Conj_PNo_bd_In__1__3 TMSNNhhLCkfPpfVWqdZMxaxp1kwpbknsyAq bounty of about 25 bars ***)
L1387
Variable Q : (set β†’ ((set β†’ prop) β†’ prop))
L1388
Variable x : set
L1389
Variable y : set
L1390
Variable p : (set β†’ prop)
L1391
Hypothesis H0 : (βˆ€z : set, z ∈ PNo_bd P Q β†’ (βˆ€q : set β†’ prop, Β¬ PNo_strict_imv P Q z q))
L1392
Hypothesis H1 : y ∈ ordsucc x
L1393
Hypothesis H2 : PNo_strict_imv P Q y p
L1394
Theorem. (Conj_PNo_bd_In__1__3)
¬ y ∈ PNo_bd P Q
In Proofgold the corresponding term root is 75a884... and proposition id is 33d612...
Proof:
Proof not loaded.
End of Section Conj_PNo_bd_In__1__3
Beginning of Section Conj_SNoLtE__1__3
L1400
Variable x : set
(*** Conj_SNoLtE__1__3 TMdQPqnVozoS1DBSku1D8KJe6vQLp5cUVa1 bounty of about 25 bars ***)
L1401
Variable y : set
L1402
Variable P : prop
L1403
Variable z : set
L1404
Hypothesis H0 : (βˆ€w : set, SNo w β†’ SNoLev w ∈ binintersect (SNoLev x) (SNoLev y) β†’ SNoEq_ (SNoLev w) w x β†’ SNoEq_ (SNoLev w) w y β†’ x < w β†’ w < y β†’ nIn (SNoLev w) x β†’ SNoLev w ∈ y β†’ P)
L1405
Hypothesis H1 : z ∈ binintersect (SNoLev x) (SNoLev y)
L1406
Hypothesis H2 : PNoEq_ z (Ξ»w : set β‡’ w ∈ x) (Ξ»w : set β‡’ w ∈ y)
L1407
Hypothesis H4 : z ∈ y
L1408
Hypothesis H5 : z ∈ SNoLev x
L1409
Hypothesis H6 : z ∈ SNoLev y
L1410
Hypothesis H7 : SNo (PSNo z (Ξ»w : set β‡’ w ∈ x))
L1411
Hypothesis H8 : SNoLev (PSNo z (Ξ»w : set β‡’ w ∈ x)) = z
L1412
Hypothesis H9 : SNoEq_ z (PSNo z (Ξ»w : set β‡’ w ∈ x)) x
L1413
Theorem. (Conj_SNoLtE__1__3)
SNoEq_ z (PSNo z (Ξ»w : set β‡’ w ∈ x)) y β†’ P
In Proofgold the corresponding term root is 445d91... and proposition id is 267f4b...
Proof:
Proof not loaded.
End of Section Conj_SNoLtE__1__3
Beginning of Section Conj_SNoLtE__1__4
L1419
Variable x : set
(*** Conj_SNoLtE__1__4 TMJgjL45h6ziFXBioksUVwe7ZeE7AvUXWAC bounty of about 25 bars ***)
L1420
Variable y : set
L1421
Variable P : prop
L1422
Variable z : set
L1423
Hypothesis H0 : (βˆ€w : set, SNo w β†’ SNoLev w ∈ binintersect (SNoLev x) (SNoLev y) β†’ SNoEq_ (SNoLev w) w x β†’ SNoEq_ (SNoLev w) w y β†’ x < w β†’ w < y β†’ nIn (SNoLev w) x β†’ SNoLev w ∈ y β†’ P)
L1424
Hypothesis H1 : z ∈ binintersect (SNoLev x) (SNoLev y)
L1425
Hypothesis H2 : PNoEq_ z (Ξ»w : set β‡’ w ∈ x) (Ξ»w : set β‡’ w ∈ y)
L1426
Hypothesis H3 : ¬ z ∈ x
L1427
Hypothesis H5 : z ∈ SNoLev x
L1428
Hypothesis H6 : z ∈ SNoLev y
L1429
Hypothesis H7 : SNo (PSNo z (Ξ»w : set β‡’ w ∈ x))
L1430
Hypothesis H8 : SNoLev (PSNo z (Ξ»w : set β‡’ w ∈ x)) = z
L1431
Hypothesis H9 : SNoEq_ z (PSNo z (Ξ»w : set β‡’ w ∈ x)) x
L1432
Theorem. (Conj_SNoLtE__1__4)
SNoEq_ z (PSNo z (Ξ»w : set β‡’ w ∈ x)) y β†’ P
In Proofgold the corresponding term root is 2f0432... and proposition id is 2719d2...
Proof:
Proof not loaded.
End of Section Conj_SNoLtE__1__4
Beginning of Section Conj_SNoLtE__1__5
L1438
Variable x : set
(*** Conj_SNoLtE__1__5 TMKJPxmBAFbH7XXj2fJ5pWNasTPbKWbMb7G bounty of about 25 bars ***)
L1439
Variable y : set
L1440
Variable P : prop
L1441
Variable z : set
L1442
Hypothesis H0 : (βˆ€w : set, SNo w β†’ SNoLev w ∈ binintersect (SNoLev x) (SNoLev y) β†’ SNoEq_ (SNoLev w) w x β†’ SNoEq_ (SNoLev w) w y β†’ x < w β†’ w < y β†’ nIn (SNoLev w) x β†’ SNoLev w ∈ y β†’ P)
L1443
Hypothesis H1 : z ∈ binintersect (SNoLev x) (SNoLev y)
L1444
Hypothesis H2 : PNoEq_ z (Ξ»w : set β‡’ w ∈ x) (Ξ»w : set β‡’ w ∈ y)
L1445
Hypothesis H3 : ¬ z ∈ x
L1446
Hypothesis H4 : z ∈ y
L1447
Hypothesis H6 : z ∈ SNoLev y
L1448
Hypothesis H7 : SNo (PSNo z (Ξ»w : set β‡’ w ∈ x))
L1449
Hypothesis H8 : SNoLev (PSNo z (Ξ»w : set β‡’ w ∈ x)) = z
L1450
Hypothesis H9 : SNoEq_ z (PSNo z (Ξ»w : set β‡’ w ∈ x)) x
L1451
Theorem. (Conj_SNoLtE__1__5)
SNoEq_ z (PSNo z (Ξ»w : set β‡’ w ∈ x)) y β†’ P
In Proofgold the corresponding term root is f86dba... and proposition id is f28ab9...
Proof:
Proof not loaded.
End of Section Conj_SNoLtE__1__5
Beginning of Section Conj_SNoLtE__1__8
L1457
Variable x : set
(*** Conj_SNoLtE__1__8 TMMKbr6xKkKmuxkFx4vUPZ1ESib4K5C3dTA bounty of about 25 bars ***)
L1458
Variable y : set
L1459
Variable P : prop
L1460
Variable z : set
L1461
Hypothesis H0 : (βˆ€w : set, SNo w β†’ SNoLev w ∈ binintersect (SNoLev x) (SNoLev y) β†’ SNoEq_ (SNoLev w) w x β†’ SNoEq_ (SNoLev w) w y β†’ x < w β†’ w < y β†’ nIn (SNoLev w) x β†’ SNoLev w ∈ y β†’ P)
L1462
Hypothesis H1 : z ∈ binintersect (SNoLev x) (SNoLev y)
L1463
Hypothesis H2 : PNoEq_ z (Ξ»w : set β‡’ w ∈ x) (Ξ»w : set β‡’ w ∈ y)
L1464
Hypothesis H3 : ¬ z ∈ x
L1465
Hypothesis H4 : z ∈ y
L1466
Hypothesis H5 : z ∈ SNoLev x
L1467
Hypothesis H6 : z ∈ SNoLev y
L1468
Hypothesis H7 : SNo (PSNo z (Ξ»w : set β‡’ w ∈ x))
L1469
Hypothesis H9 : SNoEq_ z (PSNo z (Ξ»w : set β‡’ w ∈ x)) x
L1470
Theorem. (Conj_SNoLtE__1__8)
SNoEq_ z (PSNo z (Ξ»w : set β‡’ w ∈ x)) y β†’ P
In Proofgold the corresponding term root is c01103... and proposition id is c3520c...
Proof:
Proof not loaded.
End of Section Conj_SNoLtE__1__8
Beginning of Section Conj_SNoLtE__1__9
L1476
Variable x : set
(*** Conj_SNoLtE__1__9 TMaJh4YBa7noJ5vCMzbZiPyPmw7oqYkr2W1 bounty of about 25 bars ***)
L1477
Variable y : set
L1478
Variable P : prop
L1479
Variable z : set
L1480
Hypothesis H0 : (βˆ€w : set, SNo w β†’ SNoLev w ∈ binintersect (SNoLev x) (SNoLev y) β†’ SNoEq_ (SNoLev w) w x β†’ SNoEq_ (SNoLev w) w y β†’ x < w β†’ w < y β†’ nIn (SNoLev w) x β†’ SNoLev w ∈ y β†’ P)
L1481
Hypothesis H1 : z ∈ binintersect (SNoLev x) (SNoLev y)
L1482
Hypothesis H2 : PNoEq_ z (Ξ»w : set β‡’ w ∈ x) (Ξ»w : set β‡’ w ∈ y)
L1483
Hypothesis H3 : ¬ z ∈ x
L1484
Hypothesis H4 : z ∈ y
L1485
Hypothesis H5 : z ∈ SNoLev x
L1486
Hypothesis H6 : z ∈ SNoLev y
L1487
Hypothesis H7 : SNo (PSNo z (Ξ»w : set β‡’ w ∈ x))
L1488
Hypothesis H8 : SNoLev (PSNo z (Ξ»w : set β‡’ w ∈ x)) = z
L1489
Theorem. (Conj_SNoLtE__1__9)
SNoEq_ z (PSNo z (Ξ»w : set β‡’ w ∈ x)) y β†’ P
In Proofgold the corresponding term root is ddbb2e... and proposition id is 8612ba...
Proof:
Proof not loaded.
End of Section Conj_SNoLtE__1__9
Beginning of Section Conj_SNoLtE__6__5
L1495
Variable x : set
(*** Conj_SNoLtE__6__5 TMGP9P8sY3CFbBbXaB82XSjAVKrqfT6dEqA bounty of about 25 bars ***)
L1496
Variable y : set
L1497
Variable P : prop
L1498
Variable z : set
L1499
Hypothesis H0 : (βˆ€w : set, SNo w β†’ SNoLev w ∈ binintersect (SNoLev x) (SNoLev y) β†’ SNoEq_ (SNoLev w) w x β†’ SNoEq_ (SNoLev w) w y β†’ x < w β†’ w < y β†’ nIn (SNoLev w) x β†’ SNoLev w ∈ y β†’ P)
L1500
Hypothesis H1 : ordinal (SNoLev x)
L1501
Hypothesis H2 : z ∈ binintersect (SNoLev x) (SNoLev y)
L1502
Hypothesis H3 : PNoEq_ z (Ξ»w : set β‡’ w ∈ x) (Ξ»w : set β‡’ w ∈ y)
L1503
Hypothesis H4 : ¬ z ∈ x
L1504
Hypothesis H6 : z ∈ SNoLev x
L1505
Hypothesis H7 : z ∈ SNoLev y
L1506
Theorem. (Conj_SNoLtE__6__5)
ordinal z β†’ P
In Proofgold the corresponding term root is 4f4229... and proposition id is 98564e...
Proof:
Proof not loaded.
End of Section Conj_SNoLtE__6__5
Beginning of Section Conj_SNoLtE__8__3
L1512
Variable x : set
(*** Conj_SNoLtE__8__3 TMVrqu7AeB67D8CtBz4oWD87dGHLjfaeKZN bounty of about 25 bars ***)
L1513
Variable y : set
L1514
Variable P : prop
L1515
Hypothesis H0 : SNo x
L1516
Hypothesis H1 : SNo y
L1517
Hypothesis H2 : x < y
L1518
Hypothesis H4 : SNoLev x ∈ SNoLev y β†’ SNoEq_ (SNoLev x) x y β†’ SNoLev x ∈ y β†’ P
L1519
Hypothesis H5 : SNoLev y ∈ SNoLev x β†’ SNoEq_ (SNoLev y) x y β†’ nIn (SNoLev y) x β†’ P
L1520
Theorem. (Conj_SNoLtE__8__3)
ordinal (SNoLev x) β†’ P
In Proofgold the corresponding term root is 9e8f80... and proposition id is 009cab...
Proof:
Proof not loaded.
End of Section Conj_SNoLtE__8__3
Beginning of Section Conj_SNoCutP_SNoCut__1__4
L1526
Variable x : set
(*** Conj_SNoCutP_SNoCut__1__4 TMFfd7YezCcxhJoZLfd2pQayMKkmji9UvjZ bounty of about 25 bars ***)
L1527
Variable y : set
L1528
Variable z : set
L1529
Hypothesis H0 : ordinal (PNo_bd (Ξ»w : set β‡’ Ξ»p : set β†’ prop β‡’ ordinal w ∧ PSNo w p ∈ x) (Ξ»w : set β‡’ Ξ»p : set β†’ prop β‡’ ordinal w ∧ PSNo w p ∈ y))
L1530
Hypothesis H1 : PNo_strict_upperbd (Ξ»w : set β‡’ Ξ»p : set β†’ prop β‡’ ordinal w ∧ PSNo w p ∈ x) (PNo_bd (Ξ»w : set β‡’ Ξ»p : set β†’ prop β‡’ ordinal w ∧ PSNo w p ∈ x) (Ξ»w : set β‡’ Ξ»p : set β†’ prop β‡’ ordinal w ∧ PSNo w p ∈ y)) (PNo_pred (Ξ»w : set β‡’ Ξ»p : set β†’ prop β‡’ ordinal w ∧ PSNo w p ∈ x) (Ξ»w : set β‡’ Ξ»p : set β†’ prop β‡’ ordinal w ∧ PSNo w p ∈ y))
L1531
Hypothesis H2 : z ∈ x
L1532
Hypothesis H3 : ordinal (SNoLev z)
L1533
Theorem. (Conj_SNoCutP_SNoCut__1__4)
ordinal (SNoLev z) ∧ PSNo (SNoLev z) (Ξ»w : set β‡’ w ∈ z) ∈ x β†’ z < PSNo (PNo_bd (Ξ»w : set β‡’ Ξ»p : set β†’ prop β‡’ ordinal w ∧ PSNo w p ∈ x) (Ξ»w : set β‡’ Ξ»p : set β†’ prop β‡’ ordinal w ∧ PSNo w p ∈ y)) (PNo_pred (Ξ»w : set β‡’ Ξ»p : set β†’ prop β‡’ ordinal w ∧ PSNo w p ∈ x) (Ξ»w : set β‡’ Ξ»p : set β†’ prop β‡’ ordinal w ∧ PSNo w p ∈ y))
In Proofgold the corresponding term root is 47ac26... and proposition id is d9ef37...
Proof:
Proof not loaded.
End of Section Conj_SNoCutP_SNoCut__1__4
Beginning of Section Conj_SNoCutP_SNoCut__9__3
L1539
Variable x : set
(*** Conj_SNoCutP_SNoCut__9__3 TMPhgADvu7SX3HqSNXWvtZAtQ5aRGDAnpzg bounty of about 25 bars ***)
L1540
Variable y : set
L1541
Variable z : set
L1542
Variable w : set
L1543
Hypothesis H0 : ordinal (PNo_bd (Ξ»u : set β‡’ Ξ»p : set β†’ prop β‡’ ordinal u ∧ PSNo u p ∈ x) (Ξ»u : set β‡’ Ξ»p : set β†’ prop β‡’ ordinal u ∧ PSNo u p ∈ y))
L1544
Hypothesis H1 : PNo_strict_imv (Ξ»u : set β‡’ Ξ»p : set β†’ prop β‡’ ordinal u ∧ PSNo u p ∈ x) (Ξ»u : set β‡’ Ξ»p : set β†’ prop β‡’ ordinal u ∧ PSNo u p ∈ y) (PNo_bd (Ξ»u : set β‡’ Ξ»p : set β†’ prop β‡’ ordinal u ∧ PSNo u p ∈ x) (Ξ»u : set β‡’ Ξ»p : set β†’ prop β‡’ ordinal u ∧ PSNo u p ∈ y)) (PNo_pred (Ξ»u : set β‡’ Ξ»p : set β†’ prop β‡’ ordinal u ∧ PSNo u p ∈ x) (Ξ»u : set β‡’ Ξ»p : set β†’ prop β‡’ ordinal u ∧ PSNo u p ∈ y))
L1545
Hypothesis H2 : (βˆ€u : set, u ∈ PNo_bd (Ξ»v : set β‡’ Ξ»p : set β†’ prop β‡’ ordinal v ∧ PSNo v p ∈ x) (Ξ»v : set β‡’ Ξ»p : set β†’ prop β‡’ ordinal v ∧ PSNo v p ∈ y) β†’ (βˆ€p : set β†’ prop, Β¬ PNo_strict_imv (Ξ»v : set β‡’ Ξ»q : set β†’ prop β‡’ ordinal v ∧ PSNo v q ∈ x) (Ξ»v : set β‡’ Ξ»q : set β†’ prop β‡’ ordinal v ∧ PSNo v q ∈ y) u p))
L1546
Hypothesis H4 : PNo_strict_imv (Ξ»u : set β‡’ Ξ»p : set β†’ prop β‡’ ordinal u ∧ PSNo u p ∈ x) (Ξ»u : set β‡’ Ξ»p : set β†’ prop β‡’ ordinal u ∧ PSNo u p ∈ y) (SNoLev z) (Ξ»u : set β‡’ u ∈ z)
L1547
Hypothesis H5 : Subq (PNo_bd (Ξ»u : set β‡’ Ξ»p : set β†’ prop β‡’ ordinal u ∧ PSNo u p ∈ x) (Ξ»u : set β‡’ Ξ»p : set β†’ prop β‡’ ordinal u ∧ PSNo u p ∈ y)) (SNoLev z)
L1548
Hypothesis H6 : w ∈ PNo_bd (Ξ»u : set β‡’ Ξ»p : set β†’ prop β‡’ ordinal u ∧ PSNo u p ∈ x) (Ξ»u : set β‡’ Ξ»p : set β†’ prop β‡’ ordinal u ∧ PSNo u p ∈ y)
L1549
Hypothesis H7 : PNoEq_ w (Ξ»u : set β‡’ u ∈ z) (PNo_pred (Ξ»u : set β‡’ Ξ»p : set β†’ prop β‡’ ordinal u ∧ PSNo u p ∈ x) (Ξ»u : set β‡’ Ξ»p : set β†’ prop β‡’ ordinal u ∧ PSNo u p ∈ y))
L1550
Hypothesis H8 : nIn w z
L1551
Hypothesis H9 : PNo_pred (Ξ»u : set β‡’ Ξ»p : set β†’ prop β‡’ ordinal u ∧ PSNo u p ∈ x) (Ξ»u : set β‡’ Ξ»p : set β†’ prop β‡’ ordinal u ∧ PSNo u p ∈ y) w
L1552
Hypothesis H10 : ordinal w
L1553
Hypothesis H11 : ordinal (ordsucc w)
L1554
Hypothesis H12 : Β¬ (w ∈ z ∧ w β‰  w)
L1555
Theorem. (Conj_SNoCutP_SNoCut__9__3)
¬ (w ∈ z ∨ w = w)
In Proofgold the corresponding term root is 16dec8... and proposition id is 039f5e...
Proof:
Proof not loaded.
End of Section Conj_SNoCutP_SNoCut__9__3
Beginning of Section Conj_SNoCutP_SNoCut__9__11
L1561
Variable x : set
(*** Conj_SNoCutP_SNoCut__9__11 TMLBTvhZReSf3EJGVFXvM3XKu8uFfwBVHMG bounty of about 25 bars ***)
L1562
Variable y : set
L1563
Variable z : set
L1564
Variable w : set
L1565
Hypothesis H0 : ordinal (PNo_bd (Ξ»u : set β‡’ Ξ»p : set β†’ prop β‡’ ordinal u ∧ PSNo u p ∈ x) (Ξ»u : set β‡’ Ξ»p : set β†’ prop β‡’ ordinal u ∧ PSNo u p ∈ y))
L1566
Hypothesis H1 : PNo_strict_imv (Ξ»u : set β‡’ Ξ»p : set β†’ prop β‡’ ordinal u ∧ PSNo u p ∈ x) (Ξ»u : set β‡’ Ξ»p : set β†’ prop β‡’ ordinal u ∧ PSNo u p ∈ y) (PNo_bd (Ξ»u : set β‡’ Ξ»p : set β†’ prop β‡’ ordinal u ∧ PSNo u p ∈ x) (Ξ»u : set β‡’ Ξ»p : set β†’ prop β‡’ ordinal u ∧ PSNo u p ∈ y)) (PNo_pred (Ξ»u : set β‡’ Ξ»p : set β†’ prop β‡’ ordinal u ∧ PSNo u p ∈ x) (Ξ»u : set β‡’ Ξ»p : set β†’ prop β‡’ ordinal u ∧ PSNo u p ∈ y))
L1567
Hypothesis H2 : (βˆ€u : set, u ∈ PNo_bd (Ξ»v : set β‡’ Ξ»p : set β†’ prop β‡’ ordinal v ∧ PSNo v p ∈ x) (Ξ»v : set β‡’ Ξ»p : set β†’ prop β‡’ ordinal v ∧ PSNo v p ∈ y) β†’ (βˆ€p : set β†’ prop, Β¬ PNo_strict_imv (Ξ»v : set β‡’ Ξ»q : set β†’ prop β‡’ ordinal v ∧ PSNo v q ∈ x) (Ξ»v : set β‡’ Ξ»q : set β†’ prop β‡’ ordinal v ∧ PSNo v q ∈ y) u p))
L1568
Hypothesis H3 : ordinal (SNoLev z)
L1569
Hypothesis H4 : PNo_strict_imv (Ξ»u : set β‡’ Ξ»p : set β†’ prop β‡’ ordinal u ∧ PSNo u p ∈ x) (Ξ»u : set β‡’ Ξ»p : set β†’ prop β‡’ ordinal u ∧ PSNo u p ∈ y) (SNoLev z) (Ξ»u : set β‡’ u ∈ z)
L1570
Hypothesis H5 : Subq (PNo_bd (Ξ»u : set β‡’ Ξ»p : set β†’ prop β‡’ ordinal u ∧ PSNo u p ∈ x) (Ξ»u : set β‡’ Ξ»p : set β†’ prop β‡’ ordinal u ∧ PSNo u p ∈ y)) (SNoLev z)
L1571
Hypothesis H6 : w ∈ PNo_bd (Ξ»u : set β‡’ Ξ»p : set β†’ prop β‡’ ordinal u ∧ PSNo u p ∈ x) (Ξ»u : set β‡’ Ξ»p : set β†’ prop β‡’ ordinal u ∧ PSNo u p ∈ y)
L1572
Hypothesis H7 : PNoEq_ w (Ξ»u : set β‡’ u ∈ z) (PNo_pred (Ξ»u : set β‡’ Ξ»p : set β†’ prop β‡’ ordinal u ∧ PSNo u p ∈ x) (Ξ»u : set β‡’ Ξ»p : set β†’ prop β‡’ ordinal u ∧ PSNo u p ∈ y))
L1573
Hypothesis H8 : nIn w z
L1574
Hypothesis H9 : PNo_pred (Ξ»u : set β‡’ Ξ»p : set β†’ prop β‡’ ordinal u ∧ PSNo u p ∈ x) (Ξ»u : set β‡’ Ξ»p : set β†’ prop β‡’ ordinal u ∧ PSNo u p ∈ y) w
L1575
Hypothesis H10 : ordinal w
L1576
Hypothesis H12 : Β¬ (w ∈ z ∧ w β‰  w)
L1577
Theorem. (Conj_SNoCutP_SNoCut__9__11)
¬ (w ∈ z ∨ w = w)
In Proofgold the corresponding term root is dcdfad... and proposition id is 8140c3...
Proof:
Proof not loaded.
End of Section Conj_SNoCutP_SNoCut__9__11
Beginning of Section Conj_SNoCutP_SNoCut__10__0
L1583
Variable x : set
(*** Conj_SNoCutP_SNoCut__10__0 TMJdq8XLSQYPaMSsGxLGNYwJwRYeGHJPhds bounty of about 25 bars ***)
L1584
Variable y : set
L1585
Variable z : set
L1586
Variable w : set
L1587
Hypothesis H1 : PNo_strict_imv (Ξ»u : set β‡’ Ξ»p : set β†’ prop β‡’ ordinal u ∧ PSNo u p ∈ x) (Ξ»u : set β‡’ Ξ»p : set β†’ prop β‡’ ordinal u ∧ PSNo u p ∈ y) (PNo_bd (Ξ»u : set β‡’ Ξ»p : set β†’ prop β‡’ ordinal u ∧ PSNo u p ∈ x) (Ξ»u : set β‡’ Ξ»p : set β†’ prop β‡’ ordinal u ∧ PSNo u p ∈ y)) (PNo_pred (Ξ»u : set β‡’ Ξ»p : set β†’ prop β‡’ ordinal u ∧ PSNo u p ∈ x) (Ξ»u : set β‡’ Ξ»p : set β†’ prop β‡’ ordinal u ∧ PSNo u p ∈ y))
L1588
Hypothesis H2 : (βˆ€u : set, u ∈ PNo_bd (Ξ»v : set β‡’ Ξ»p : set β†’ prop β‡’ ordinal v ∧ PSNo v p ∈ x) (Ξ»v : set β‡’ Ξ»p : set β†’ prop β‡’ ordinal v ∧ PSNo v p ∈ y) β†’ (βˆ€p : set β†’ prop, Β¬ PNo_strict_imv (Ξ»v : set β‡’ Ξ»q : set β†’ prop β‡’ ordinal v ∧ PSNo v q ∈ x) (Ξ»v : set β‡’ Ξ»q : set β†’ prop β‡’ ordinal v ∧ PSNo v q ∈ y) u p))
L1589
Hypothesis H3 : ordinal (SNoLev z)
L1590
Hypothesis H4 : PNo_strict_imv (Ξ»u : set β‡’ Ξ»p : set β†’ prop β‡’ ordinal u ∧ PSNo u p ∈ x) (Ξ»u : set β‡’ Ξ»p : set β†’ prop β‡’ ordinal u ∧ PSNo u p ∈ y) (SNoLev z) (Ξ»u : set β‡’ u ∈ z)
L1591
Hypothesis H5 : Subq (PNo_bd (Ξ»u : set β‡’ Ξ»p : set β†’ prop β‡’ ordinal u ∧ PSNo u p ∈ x) (Ξ»u : set β‡’ Ξ»p : set β†’ prop β‡’ ordinal u ∧ PSNo u p ∈ y)) (SNoLev z)
L1592
Hypothesis H6 : w ∈ PNo_bd (Ξ»u : set β‡’ Ξ»p : set β†’ prop β‡’ ordinal u ∧ PSNo u p ∈ x) (Ξ»u : set β‡’ Ξ»p : set β†’ prop β‡’ ordinal u ∧ PSNo u p ∈ y)
L1593
Hypothesis H7 : PNoEq_ w (Ξ»u : set β‡’ u ∈ z) (PNo_pred (Ξ»u : set β‡’ Ξ»p : set β†’ prop β‡’ ordinal u ∧ PSNo u p ∈ x) (Ξ»u : set β‡’ Ξ»p : set β†’ prop β‡’ ordinal u ∧ PSNo u p ∈ y))
L1594
Hypothesis H8 : nIn w z
L1595
Hypothesis H9 : PNo_pred (Ξ»u : set β‡’ Ξ»p : set β†’ prop β‡’ ordinal u ∧ PSNo u p ∈ x) (Ξ»u : set β‡’ Ξ»p : set β†’ prop β‡’ ordinal u ∧ PSNo u p ∈ y) w
L1596
Hypothesis H10 : ordinal w
L1597
Hypothesis H11 : ordinal (ordsucc w)
L1598
Theorem. (Conj_SNoCutP_SNoCut__10__0)
Β¬ Β¬ (w ∈ z ∧ w β‰  w)
In Proofgold the corresponding term root is 6b8630... and proposition id is 219cfc...
Proof:
Proof not loaded.
End of Section Conj_SNoCutP_SNoCut__10__0
Beginning of Section Conj_SNoCutP_SNoCut__12__7
L1604
Variable x : set
(*** Conj_SNoCutP_SNoCut__12__7 TMEvgoKq2HNz7B21Y3F9MjnqevRyE15e6sp bounty of about 25 bars ***)
L1605
Variable y : set
L1606
Variable z : set
L1607
Variable w : set
L1608
Hypothesis H0 : ordinal (PNo_bd (Ξ»u : set β‡’ Ξ»p : set β†’ prop β‡’ ordinal u ∧ PSNo u p ∈ x) (Ξ»u : set β‡’ Ξ»p : set β†’ prop β‡’ ordinal u ∧ PSNo u p ∈ y))
L1609
Hypothesis H1 : PNo_strict_imv (Ξ»u : set β‡’ Ξ»p : set β†’ prop β‡’ ordinal u ∧ PSNo u p ∈ x) (Ξ»u : set β‡’ Ξ»p : set β†’ prop β‡’ ordinal u ∧ PSNo u p ∈ y) (PNo_bd (Ξ»u : set β‡’ Ξ»p : set β†’ prop β‡’ ordinal u ∧ PSNo u p ∈ x) (Ξ»u : set β‡’ Ξ»p : set β†’ prop β‡’ ordinal u ∧ PSNo u p ∈ y)) (PNo_pred (Ξ»u : set β‡’ Ξ»p : set β†’ prop β‡’ ordinal u ∧ PSNo u p ∈ x) (Ξ»u : set β‡’ Ξ»p : set β†’ prop β‡’ ordinal u ∧ PSNo u p ∈ y))
L1610
Hypothesis H2 : (βˆ€u : set, u ∈ PNo_bd (Ξ»v : set β‡’ Ξ»p : set β†’ prop β‡’ ordinal v ∧ PSNo v p ∈ x) (Ξ»v : set β‡’ Ξ»p : set β†’ prop β‡’ ordinal v ∧ PSNo v p ∈ y) β†’ (βˆ€p : set β†’ prop, Β¬ PNo_strict_imv (Ξ»v : set β‡’ Ξ»q : set β†’ prop β‡’ ordinal v ∧ PSNo v q ∈ x) (Ξ»v : set β‡’ Ξ»q : set β†’ prop β‡’ ordinal v ∧ PSNo v q ∈ y) u p))
L1611
Hypothesis H3 : ordinal (SNoLev z)
L1612
Hypothesis H4 : PNo_strict_imv (Ξ»u : set β‡’ Ξ»p : set β†’ prop β‡’ ordinal u ∧ PSNo u p ∈ x) (Ξ»u : set β‡’ Ξ»p : set β†’ prop β‡’ ordinal u ∧ PSNo u p ∈ y) (SNoLev z) (Ξ»u : set β‡’ u ∈ z)
L1613
Hypothesis H5 : Subq (PNo_bd (Ξ»u : set β‡’ Ξ»p : set β†’ prop β‡’ ordinal u ∧ PSNo u p ∈ x) (Ξ»u : set β‡’ Ξ»p : set β†’ prop β‡’ ordinal u ∧ PSNo u p ∈ y)) (SNoLev z)
L1614
Hypothesis H6 : w ∈ PNo_bd (Ξ»u : set β‡’ Ξ»p : set β†’ prop β‡’ ordinal u ∧ PSNo u p ∈ x) (Ξ»u : set β‡’ Ξ»p : set β†’ prop β‡’ ordinal u ∧ PSNo u p ∈ y)
L1615
Hypothesis H8 : nIn w z
L1616
Hypothesis H9 : PNo_pred (Ξ»u : set β‡’ Ξ»p : set β†’ prop β‡’ ordinal u ∧ PSNo u p ∈ x) (Ξ»u : set β‡’ Ξ»p : set β†’ prop β‡’ ordinal u ∧ PSNo u p ∈ y) w
L1617
Theorem. (Conj_SNoCutP_SNoCut__12__7)
Β¬ ordinal w
In Proofgold the corresponding term root is 61b482... and proposition id is f8c08b...
Proof:
Proof not loaded.
End of Section Conj_SNoCutP_SNoCut__12__7
Beginning of Section Conj_SNoCutP_SNoCut__14__7
L1623
Variable x : set
(*** Conj_SNoCutP_SNoCut__14__7 TMPSDC4rZZxZELQSdd3JKvLq7z9R2KaXV6g bounty of about 25 bars ***)
L1624
Variable y : set
L1625
Variable z : set
L1626
Variable w : set
L1627
Hypothesis H0 : ordinal (PNo_bd (Ξ»u : set β‡’ Ξ»p : set β†’ prop β‡’ ordinal u ∧ PSNo u p ∈ x) (Ξ»u : set β‡’ Ξ»p : set β†’ prop β‡’ ordinal u ∧ PSNo u p ∈ y))
L1628
Hypothesis H1 : PNo_strict_imv (Ξ»u : set β‡’ Ξ»p : set β†’ prop β‡’ ordinal u ∧ PSNo u p ∈ x) (Ξ»u : set β‡’ Ξ»p : set β†’ prop β‡’ ordinal u ∧ PSNo u p ∈ y) (PNo_bd (Ξ»u : set β‡’ Ξ»p : set β†’ prop β‡’ ordinal u ∧ PSNo u p ∈ x) (Ξ»u : set β‡’ Ξ»p : set β†’ prop β‡’ ordinal u ∧ PSNo u p ∈ y)) (PNo_pred (Ξ»u : set β‡’ Ξ»p : set β†’ prop β‡’ ordinal u ∧ PSNo u p ∈ x) (Ξ»u : set β‡’ Ξ»p : set β†’ prop β‡’ ordinal u ∧ PSNo u p ∈ y))
L1629
Hypothesis H2 : (βˆ€u : set, u ∈ PNo_bd (Ξ»v : set β‡’ Ξ»p : set β†’ prop β‡’ ordinal v ∧ PSNo v p ∈ x) (Ξ»v : set β‡’ Ξ»p : set β†’ prop β‡’ ordinal v ∧ PSNo v p ∈ y) β†’ (βˆ€p : set β†’ prop, Β¬ PNo_strict_imv (Ξ»v : set β‡’ Ξ»q : set β†’ prop β‡’ ordinal v ∧ PSNo v q ∈ x) (Ξ»v : set β‡’ Ξ»q : set β†’ prop β‡’ ordinal v ∧ PSNo v q ∈ y) u p))
L1630
Hypothesis H3 : ordinal (SNoLev z)
L1631
Hypothesis H4 : PNo_strict_imv (Ξ»u : set β‡’ Ξ»p : set β†’ prop β‡’ ordinal u ∧ PSNo u p ∈ x) (Ξ»u : set β‡’ Ξ»p : set β†’ prop β‡’ ordinal u ∧ PSNo u p ∈ y) (SNoLev z) (Ξ»u : set β‡’ u ∈ z)
L1632
Hypothesis H5 : Subq (PNo_bd (Ξ»u : set β‡’ Ξ»p : set β†’ prop β‡’ ordinal u ∧ PSNo u p ∈ x) (Ξ»u : set β‡’ Ξ»p : set β†’ prop β‡’ ordinal u ∧ PSNo u p ∈ y)) (SNoLev z)
L1633
Hypothesis H6 : w ∈ PNo_bd (Ξ»u : set β‡’ Ξ»p : set β†’ prop β‡’ ordinal u ∧ PSNo u p ∈ x) (Ξ»u : set β‡’ Ξ»p : set β†’ prop β‡’ ordinal u ∧ PSNo u p ∈ y)
L1634
Hypothesis H8 : Β¬ PNo_pred (Ξ»u : set β‡’ Ξ»p : set β†’ prop β‡’ ordinal u ∧ PSNo u p ∈ x) (Ξ»u : set β‡’ Ξ»p : set β†’ prop β‡’ ordinal u ∧ PSNo u p ∈ y) w
L1635
Hypothesis H9 : w ∈ z
L1636
Hypothesis H10 : ordinal w
L1637
Hypothesis H11 : ordinal (ordsucc w)
L1638
Theorem. (Conj_SNoCutP_SNoCut__14__7)
Β¬ Β¬ (w ∈ z ∧ w β‰  w)
In Proofgold the corresponding term root is 344de8... and proposition id is 10e359...
Proof:
Proof not loaded.
End of Section Conj_SNoCutP_SNoCut__14__7
Beginning of Section Conj_SNoCutP_SNoCut__15__3
L1644
Variable x : set
(*** Conj_SNoCutP_SNoCut__15__3 TMPE5WnELYkkzrvYw2YJcsm3JmDnBS7fnEw bounty of about 25 bars ***)
L1645
Variable y : set
L1646
Variable z : set
L1647
Variable w : set
L1648
Hypothesis H0 : ordinal (PNo_bd (Ξ»u : set β‡’ Ξ»p : set β†’ prop β‡’ ordinal u ∧ PSNo u p ∈ x) (Ξ»u : set β‡’ Ξ»p : set β†’ prop β‡’ ordinal u ∧ PSNo u p ∈ y))
L1649
Hypothesis H1 : PNo_strict_imv (Ξ»u : set β‡’ Ξ»p : set β†’ prop β‡’ ordinal u ∧ PSNo u p ∈ x) (Ξ»u : set β‡’ Ξ»p : set β†’ prop β‡’ ordinal u ∧ PSNo u p ∈ y) (PNo_bd (Ξ»u : set β‡’ Ξ»p : set β†’ prop β‡’ ordinal u ∧ PSNo u p ∈ x) (Ξ»u : set β‡’ Ξ»p : set β†’ prop β‡’ ordinal u ∧ PSNo u p ∈ y)) (PNo_pred (Ξ»u : set β‡’ Ξ»p : set β†’ prop β‡’ ordinal u ∧ PSNo u p ∈ x) (Ξ»u : set β‡’ Ξ»p : set β†’ prop β‡’ ordinal u ∧ PSNo u p ∈ y))
L1650
Hypothesis H2 : (βˆ€u : set, u ∈ PNo_bd (Ξ»v : set β‡’ Ξ»p : set β†’ prop β‡’ ordinal v ∧ PSNo v p ∈ x) (Ξ»v : set β‡’ Ξ»p : set β†’ prop β‡’ ordinal v ∧ PSNo v p ∈ y) β†’ (βˆ€p : set β†’ prop, Β¬ PNo_strict_imv (Ξ»v : set β‡’ Ξ»q : set β†’ prop β‡’ ordinal v ∧ PSNo v q ∈ x) (Ξ»v : set β‡’ Ξ»q : set β†’ prop β‡’ ordinal v ∧ PSNo v q ∈ y) u p))
L1651
Hypothesis H4 : PNo_strict_imv (Ξ»u : set β‡’ Ξ»p : set β†’ prop β‡’ ordinal u ∧ PSNo u p ∈ x) (Ξ»u : set β‡’ Ξ»p : set β†’ prop β‡’ ordinal u ∧ PSNo u p ∈ y) (SNoLev z) (Ξ»u : set β‡’ u ∈ z)
L1652
Hypothesis H5 : Subq (PNo_bd (Ξ»u : set β‡’ Ξ»p : set β†’ prop β‡’ ordinal u ∧ PSNo u p ∈ x) (Ξ»u : set β‡’ Ξ»p : set β†’ prop β‡’ ordinal u ∧ PSNo u p ∈ y)) (SNoLev z)
L1653
Hypothesis H6 : w ∈ PNo_bd (Ξ»u : set β‡’ Ξ»p : set β†’ prop β‡’ ordinal u ∧ PSNo u p ∈ x) (Ξ»u : set β‡’ Ξ»p : set β†’ prop β‡’ ordinal u ∧ PSNo u p ∈ y)
L1654
Hypothesis H7 : PNoEq_ w (PNo_pred (Ξ»u : set β‡’ Ξ»p : set β†’ prop β‡’ ordinal u ∧ PSNo u p ∈ x) (Ξ»u : set β‡’ Ξ»p : set β†’ prop β‡’ ordinal u ∧ PSNo u p ∈ y)) (Ξ»u : set β‡’ u ∈ z)
L1655
Hypothesis H8 : Β¬ PNo_pred (Ξ»u : set β‡’ Ξ»p : set β†’ prop β‡’ ordinal u ∧ PSNo u p ∈ x) (Ξ»u : set β‡’ Ξ»p : set β†’ prop β‡’ ordinal u ∧ PSNo u p ∈ y) w
L1656
Hypothesis H9 : w ∈ z
L1657
Hypothesis H10 : ordinal w
L1658
Theorem. (Conj_SNoCutP_SNoCut__15__3)
Β¬ ordinal (ordsucc w)
In Proofgold the corresponding term root is 1334f6... and proposition id is 350c53...
Proof:
Proof not loaded.
End of Section Conj_SNoCutP_SNoCut__15__3
Beginning of Section Conj_SNoCutP_SNoCut__20__2
L1664
Variable x : set
(*** Conj_SNoCutP_SNoCut__20__2 TMMYndscEdRQjjCuK4nkZGLXyE1tDkBy9gd bounty of about 25 bars ***)
L1665
Variable y : set
L1666
Variable z : set
L1667
Hypothesis H0 : ordinal (PNo_bd (Ξ»w : set β‡’ Ξ»p : set β†’ prop β‡’ ordinal w ∧ PSNo w p ∈ x) (Ξ»w : set β‡’ Ξ»p : set β†’ prop β‡’ ordinal w ∧ PSNo w p ∈ y))
L1668
Hypothesis H1 : PNo_strict_imv (Ξ»w : set β‡’ Ξ»p : set β†’ prop β‡’ ordinal w ∧ PSNo w p ∈ x) (Ξ»w : set β‡’ Ξ»p : set β†’ prop β‡’ ordinal w ∧ PSNo w p ∈ y) (PNo_bd (Ξ»w : set β‡’ Ξ»p : set β†’ prop β‡’ ordinal w ∧ PSNo w p ∈ x) (Ξ»w : set β‡’ Ξ»p : set β†’ prop β‡’ ordinal w ∧ PSNo w p ∈ y)) (PNo_pred (Ξ»w : set β‡’ Ξ»p : set β†’ prop β‡’ ordinal w ∧ PSNo w p ∈ x) (Ξ»w : set β‡’ Ξ»p : set β†’ prop β‡’ ordinal w ∧ PSNo w p ∈ y))
L1669
Hypothesis H3 : SNoLev (SNoCut x y) = PNo_bd (Ξ»w : set β‡’ Ξ»p : set β†’ prop β‡’ ordinal w ∧ PSNo w p ∈ x) (Ξ»w : set β‡’ Ξ»p : set β†’ prop β‡’ ordinal w ∧ PSNo w p ∈ y)
L1670
Hypothesis H4 : PNoEq_ (PNo_bd (Ξ»w : set β‡’ Ξ»p : set β†’ prop β‡’ ordinal w ∧ PSNo w p ∈ x) (Ξ»w : set β‡’ Ξ»p : set β†’ prop β‡’ ordinal w ∧ PSNo w p ∈ y)) (Ξ»w : set β‡’ w ∈ SNoCut x y) (PNo_pred (Ξ»w : set β‡’ Ξ»p : set β†’ prop β‡’ ordinal w ∧ PSNo w p ∈ x) (Ξ»w : set β‡’ Ξ»p : set β†’ prop β‡’ ordinal w ∧ PSNo w p ∈ y))
L1671
Hypothesis H5 : SNo z
L1672
Hypothesis H6 : (βˆ€w : set, w ∈ x β†’ w < z)
L1673
Hypothesis H7 : (βˆ€w : set, w ∈ y β†’ z < w)
L1674
Theorem. (Conj_SNoCutP_SNoCut__20__2)
ordinal (SNoLev z) β†’ Subq (SNoLev (SNoCut x y)) (SNoLev z) ∧ PNoEq_ (SNoLev (SNoCut x y)) (Ξ»w : set β‡’ w ∈ SNoCut x y) (Ξ»w : set β‡’ w ∈ z)
In Proofgold the corresponding term root is 46700e... and proposition id is b40f33...
Proof:
Proof not loaded.
End of Section Conj_SNoCutP_SNoCut__20__2
Beginning of Section Conj_SNoCutP_SNoCut__21__7
L1680
Variable x : set
(*** Conj_SNoCutP_SNoCut__21__7 TMVjJiy6dQ4D55oC9yfHEYM8R3GXkUHgkDQ bounty of about 25 bars ***)
L1681
Variable y : set
L1682
Hypothesis H0 : (βˆ€z : set, z ∈ y β†’ SNo z)
L1683
Hypothesis H1 : ordinal (PNo_bd (Ξ»z : set β‡’ Ξ»p : set β†’ prop β‡’ ordinal z ∧ PSNo z p ∈ x) (Ξ»z : set β‡’ Ξ»p : set β†’ prop β‡’ ordinal z ∧ PSNo z p ∈ y))
L1684
Hypothesis H2 : PNo_strict_imv (Ξ»z : set β‡’ Ξ»p : set β†’ prop β‡’ ordinal z ∧ PSNo z p ∈ x) (Ξ»z : set β‡’ Ξ»p : set β†’ prop β‡’ ordinal z ∧ PSNo z p ∈ y) (PNo_bd (Ξ»z : set β‡’ Ξ»p : set β†’ prop β‡’ ordinal z ∧ PSNo z p ∈ x) (Ξ»z : set β‡’ Ξ»p : set β†’ prop β‡’ ordinal z ∧ PSNo z p ∈ y)) (PNo_pred (Ξ»z : set β‡’ Ξ»p : set β†’ prop β‡’ ordinal z ∧ PSNo z p ∈ x) (Ξ»z : set β‡’ Ξ»p : set β†’ prop β‡’ ordinal z ∧ PSNo z p ∈ y))
L1685
Hypothesis H3 : (βˆ€z : set, z ∈ PNo_bd (Ξ»w : set β‡’ Ξ»p : set β†’ prop β‡’ ordinal w ∧ PSNo w p ∈ x) (Ξ»w : set β‡’ Ξ»p : set β†’ prop β‡’ ordinal w ∧ PSNo w p ∈ y) β†’ (βˆ€p : set β†’ prop, Β¬ PNo_strict_imv (Ξ»w : set β‡’ Ξ»q : set β†’ prop β‡’ ordinal w ∧ PSNo w q ∈ x) (Ξ»w : set β‡’ Ξ»q : set β†’ prop β‡’ ordinal w ∧ PSNo w q ∈ y) z p))
L1686
Hypothesis H4 : PNo_strict_lowerbd (Ξ»z : set β‡’ Ξ»p : set β†’ prop β‡’ ordinal z ∧ PSNo z p ∈ y) (PNo_bd (Ξ»z : set β‡’ Ξ»p : set β†’ prop β‡’ ordinal z ∧ PSNo z p ∈ x) (Ξ»z : set β‡’ Ξ»p : set β†’ prop β‡’ ordinal z ∧ PSNo z p ∈ y)) (PNo_pred (Ξ»z : set β‡’ Ξ»p : set β†’ prop β‡’ ordinal z ∧ PSNo z p ∈ x) (Ξ»z : set β‡’ Ξ»p : set β†’ prop β‡’ ordinal z ∧ PSNo z p ∈ y))
L1687
Hypothesis H5 : SNo (SNoCut x y)
L1688
Hypothesis H6 : SNoLev (SNoCut x y) = PNo_bd (Ξ»z : set β‡’ Ξ»p : set β†’ prop β‡’ ordinal z ∧ PSNo z p ∈ x) (Ξ»z : set β‡’ Ξ»p : set β†’ prop β‡’ ordinal z ∧ PSNo z p ∈ y)
L1689
Hypothesis H8 : PNoEq_ (PNo_bd (Ξ»z : set β‡’ Ξ»p : set β†’ prop β‡’ ordinal z ∧ PSNo z p ∈ x) (Ξ»z : set β‡’ Ξ»p : set β†’ prop β‡’ ordinal z ∧ PSNo z p ∈ y)) (Ξ»z : set β‡’ z ∈ SNoCut x y) (PNo_pred (Ξ»z : set β‡’ Ξ»p : set β†’ prop β‡’ ordinal z ∧ PSNo z p ∈ x) (Ξ»z : set β‡’ Ξ»p : set β†’ prop β‡’ ordinal z ∧ PSNo z p ∈ y))
L1690
Hypothesis H9 : (βˆ€z : set, z ∈ x β†’ z < SNoCut x y)
L1691
Theorem. (Conj_SNoCutP_SNoCut__21__7)
(βˆ€z : set, z ∈ y β†’ SNoCut x y < z) β†’ SNo (SNoCut x y) ∧ SNoLev (SNoCut x y) ∈ ordsucc (binunion (famunion x (Ξ»z : set β‡’ ordsucc (SNoLev z))) (famunion y (Ξ»z : set β‡’ ordsucc (SNoLev z)))) ∧ (βˆ€z : set, z ∈ x β†’ z < SNoCut x y) ∧ (βˆ€z : set, z ∈ y β†’ SNoCut x y < z) ∧ (βˆ€z : set, SNo z β†’ (βˆ€w : set, w ∈ x β†’ w < z) β†’ (βˆ€w : set, w ∈ y β†’ z < w) β†’ Subq (SNoLev (SNoCut x y)) (SNoLev z) ∧ PNoEq_ (SNoLev (SNoCut x y)) (Ξ»w : set β‡’ w ∈ SNoCut x y) (Ξ»w : set β‡’ w ∈ z))
In Proofgold the corresponding term root is c091db... and proposition id is 4be492...
Proof:
Proof not loaded.
End of Section Conj_SNoCutP_SNoCut__21__7
Beginning of Section Conj_SNoCutP_SNoCut__29__1
L1697
Variable x : set
(*** Conj_SNoCutP_SNoCut__29__1 TMLVpfJkzoimAXdGs8iGqtCYRLqYCmNZpyb bounty of about 25 bars ***)
L1698
Variable y : set
L1699
Hypothesis H0 : (βˆ€z : set, z ∈ x β†’ SNo z)
L1700
Hypothesis H2 : PNoLt_pwise (Ξ»z : set β‡’ Ξ»p : set β†’ prop β‡’ ordinal z ∧ PSNo z p ∈ x) (Ξ»z : set β‡’ Ξ»p : set β†’ prop β‡’ ordinal z ∧ PSNo z p ∈ y)
L1701
Hypothesis H3 : ordinal (binunion (famunion x (Ξ»z : set β‡’ ordsucc (SNoLev z))) (famunion y (Ξ»z : set β‡’ ordsucc (SNoLev z))))
L1702
Hypothesis H4 : PNo_lenbdd (binunion (famunion x (Ξ»z : set β‡’ ordsucc (SNoLev z))) (famunion y (Ξ»z : set β‡’ ordsucc (SNoLev z)))) (Ξ»z : set β‡’ Ξ»p : set β†’ prop β‡’ ordinal z ∧ PSNo z p ∈ x)
L1703
Theorem. (Conj_SNoCutP_SNoCut__29__1)
PNo_lenbdd (binunion (famunion x (Ξ»z : set β‡’ ordsucc (SNoLev z))) (famunion y (Ξ»z : set β‡’ ordsucc (SNoLev z)))) (Ξ»z : set β‡’ Ξ»p : set β†’ prop β‡’ ordinal z ∧ PSNo z p ∈ y) β†’ SNo (SNoCut x y) ∧ SNoLev (SNoCut x y) ∈ ordsucc (binunion (famunion x (Ξ»z : set β‡’ ordsucc (SNoLev z))) (famunion y (Ξ»z : set β‡’ ordsucc (SNoLev z)))) ∧ (βˆ€z : set, z ∈ x β†’ z < SNoCut x y) ∧ (βˆ€z : set, z ∈ y β†’ SNoCut x y < z) ∧ (βˆ€z : set, SNo z β†’ (βˆ€w : set, w ∈ x β†’ w < z) β†’ (βˆ€w : set, w ∈ y β†’ z < w) β†’ Subq (SNoLev (SNoCut x y)) (SNoLev z) ∧ PNoEq_ (SNoLev (SNoCut x y)) (Ξ»w : set β‡’ w ∈ SNoCut x y) (Ξ»w : set β‡’ w ∈ z))
In Proofgold the corresponding term root is fc51f0... and proposition id is 719171...
Proof:
Proof not loaded.
End of Section Conj_SNoCutP_SNoCut__29__1
Beginning of Section Conj_SNoCutP_SNoCut__34__2
L1709
Variable x : set
(*** Conj_SNoCutP_SNoCut__34__2 TMU8tJRdhMB8wNMgDDHkWXZvZxFQqnoVmNN bounty of about 25 bars ***)
L1710
Variable y : set
L1711
Hypothesis H0 : (βˆ€z : set, z ∈ x β†’ SNo z)
L1712
Hypothesis H1 : (βˆ€z : set, z ∈ y β†’ SNo z)
L1713
Theorem. (Conj_SNoCutP_SNoCut__34__2)
PNoLt_pwise (Ξ»z : set β‡’ Ξ»p : set β†’ prop β‡’ ordinal z ∧ PSNo z p ∈ x) (Ξ»z : set β‡’ Ξ»p : set β†’ prop β‡’ ordinal z ∧ PSNo z p ∈ y) β†’ SNo (SNoCut x y) ∧ SNoLev (SNoCut x y) ∈ ordsucc (binunion (famunion x (Ξ»z : set β‡’ ordsucc (SNoLev z))) (famunion y (Ξ»z : set β‡’ ordsucc (SNoLev z)))) ∧ (βˆ€z : set, z ∈ x β†’ z < SNoCut x y) ∧ (βˆ€z : set, z ∈ y β†’ SNoCut x y < z) ∧ (βˆ€z : set, SNo z β†’ (βˆ€w : set, w ∈ x β†’ w < z) β†’ (βˆ€w : set, w ∈ y β†’ z < w) β†’ Subq (SNoLev (SNoCut x y)) (SNoLev z) ∧ PNoEq_ (SNoLev (SNoCut x y)) (Ξ»w : set β‡’ w ∈ SNoCut x y) (Ξ»w : set β‡’ w ∈ z))
In Proofgold the corresponding term root is 98a148... and proposition id is 5978fc...
Proof:
Proof not loaded.
End of Section Conj_SNoCutP_SNoCut__34__2
Beginning of Section Conj_SNoCutP_SNoL_SNoR__5__1
L1719
Variable x : set
(*** Conj_SNoCutP_SNoL_SNoR__5__1 TMFbe6neX8VWV2t3CHXwkWXdRg3r7sUoJPX bounty of about 25 bars ***)
L1720
Hypothesis H0 : SNo x
L1721
Theorem. (Conj_SNoCutP_SNoL_SNoR__5__1)
(βˆ€y : set, y ∈ SNoL x β†’ SNo y) β†’ (βˆ€y : set, y ∈ SNoL x β†’ SNo y) ∧ (βˆ€y : set, y ∈ SNoR x β†’ SNo y) ∧ (βˆ€y : set, y ∈ SNoL x β†’ (βˆ€z : set, z ∈ SNoR x β†’ y < z))
In Proofgold the corresponding term root is 41fe53... and proposition id is f0704e...
Proof:
Proof not loaded.
End of Section Conj_SNoCutP_SNoL_SNoR__5__1
Beginning of Section Conj_SNo_eta__5__1
L1727
Variable x : set
(*** Conj_SNo_eta__5__1 TMRfQzg5NL8NU3amqk6E87dxtNkT7Hs1MHr bounty of about 25 bars ***)
L1728
Hypothesis H0 : SNo x
L1729
Theorem. (Conj_SNo_eta__5__1)
SNoCutP (SNoL x) (SNoR x) β†’ x = SNoCut (SNoL x) (SNoR x)
In Proofgold the corresponding term root is 3e42c5... and proposition id is 6cf6d8...
Proof:
Proof not loaded.
End of Section Conj_SNo_eta__5__1
Beginning of Section Conj_SNoCut_Le__3__5
L1735
Variable x : set
(*** Conj_SNoCut_Le__3__5 TMZfF8PwXiyBcFk2sujnRVmcZnxbez7xVBV bounty of about 25 bars ***)
L1736
Variable y : set
L1737
Variable z : set
L1738
Variable w : set
L1739
Variable u : set
L1740
Hypothesis H0 : (βˆ€v : set, v ∈ y β†’ SNo v)
L1741
Hypothesis H1 : SNo (SNoCut x y)
L1742
Hypothesis H2 : (βˆ€v : set, v ∈ y β†’ SNoCut x y < v)
L1743
Hypothesis H3 : (βˆ€v : set, SNo v β†’ (βˆ€x2 : set, x2 ∈ x β†’ x2 < v) β†’ (βˆ€x2 : set, x2 ∈ y β†’ v < x2) β†’ Subq (SNoLev (SNoCut x y)) (SNoLev v) ∧ SNoEq_ (SNoLev (SNoCut x y)) (SNoCut x y) v)
L1744
Hypothesis H4 : SNo u
L1745
Hypothesis H6 : u < SNoCut x y
L1746
Hypothesis H7 : (βˆ€v : set, v ∈ x β†’ v < u)
L1747
Theorem. (Conj_SNoCut_Le__3__5)
Β¬ (βˆ€v : set, v ∈ y β†’ u < v)
In Proofgold the corresponding term root is a1e40f... and proposition id is 4b294f...
Proof:
Proof not loaded.
End of Section Conj_SNoCut_Le__3__5
Beginning of Section Conj_SNoCut_ext__2__3
L1753
Variable x : set
(*** Conj_SNoCut_ext__2__3 TMVesvRm7y9ZgCK8rPkvkjttuyp38mhq1py bounty of about 25 bars ***)
L1754
Variable y : set
L1755
Variable z : set
L1756
Variable w : set
L1757
Hypothesis H0 : SNoCutP x y
L1758
Hypothesis H1 : SNoCutP z w
L1759
Hypothesis H2 : (βˆ€u : set, u ∈ x β†’ u < SNoCut z w)
L1760
Hypothesis H4 : (βˆ€u : set, u ∈ z β†’ u < SNoCut x y)
L1761
Hypothesis H5 : (βˆ€u : set, u ∈ w β†’ SNoCut x y < u)
L1762
Theorem. (Conj_SNoCut_ext__2__3)
SNo (SNoCut x y) β†’ SNoCut x y = SNoCut z w
In Proofgold the corresponding term root is cb3180... and proposition id is 205976...
Proof:
Proof not loaded.
End of Section Conj_SNoCut_ext__2__3
Beginning of Section Conj_SNoCut_ext__2__5
L1768
Variable x : set
(*** Conj_SNoCut_ext__2__5 TMNPnuT7rAi2w2HcfLcxg492v7UJby7KJ56 bounty of about 25 bars ***)
L1769
Variable y : set
L1770
Variable z : set
L1771
Variable w : set
L1772
Hypothesis H0 : SNoCutP x y
L1773
Hypothesis H1 : SNoCutP z w
L1774
Hypothesis H2 : (βˆ€u : set, u ∈ x β†’ u < SNoCut z w)
L1775
Hypothesis H3 : (βˆ€u : set, u ∈ y β†’ SNoCut z w < u)
L1776
Hypothesis H4 : (βˆ€u : set, u ∈ z β†’ u < SNoCut x y)
L1777
Theorem. (Conj_SNoCut_ext__2__5)
SNo (SNoCut x y) β†’ SNoCut x y = SNoCut z w
In Proofgold the corresponding term root is c5f161... and proposition id is c480b2...
Proof:
Proof not loaded.
End of Section Conj_SNoCut_ext__2__5
Beginning of Section Conj_ordinal_SNoR__1__0
L1783
Variable x : set
(*** Conj_ordinal_SNoR__1__0 TMVQwMnbn97bXvYgjVmrgVeK2dw2UrWSxMs bounty of about 25 bars ***)
L1784
Hypothesis H1 : SNo x
L1785
Theorem. (Conj_ordinal_SNoR__1__0)
SNoLev x = x β†’ SNoR x = Empty
In Proofgold the corresponding term root is a6a7f0... and proposition id is dd834e...
Proof:
Proof not loaded.
End of Section Conj_ordinal_SNoR__1__0
Beginning of Section Conj_ordinal_In_SNoLt__1__0
L1791
Variable x : set
(*** Conj_ordinal_In_SNoLt__1__0 TMUJE4wUTMt4gTiSmWQCcRMK1FsoL7qwcwi bounty of about 25 bars ***)
L1792
Variable y : set
L1793
Hypothesis H1 : y ∈ x
L1794
Hypothesis H2 : ordinal y
L1795
Hypothesis H3 : SNo y
L1796
Theorem. (Conj_ordinal_In_SNoLt__1__0)
SNoLev y = y β†’ y < x
In Proofgold the corresponding term root is 3b77e7... and proposition id is 7b81e0...
Proof:
Proof not loaded.
End of Section Conj_ordinal_In_SNoLt__1__0
Beginning of Section Conj_ordinal_SNoLev_max_2__5__0
L1802
Variable x : set
(*** Conj_ordinal_SNoLev_max_2__5__0 TMQX5WYsnF4PJHigkSixPiikryhh7HNusSo bounty of about 25 bars ***)
L1803
Variable y : set
L1804
Hypothesis H1 : TransSet x
L1805
Hypothesis H2 : SNo y
L1806
Hypothesis H3 : SNo x
L1807
Hypothesis H4 : SNoLev x = x
L1808
Hypothesis H5 : SNoLev y = x
L1809
Hypothesis H6 : Β¬ y ≀ x
L1810
Theorem. (Conj_ordinal_SNoLev_max_2__5__0)
Β¬ (βˆ€z : set, ordinal z β†’ z ∈ x β†’ z ∈ y)
In Proofgold the corresponding term root is f375f8... and proposition id is 63f7aa...
Proof:
Proof not loaded.
End of Section Conj_ordinal_SNoLev_max_2__5__0
Beginning of Section Conj_SNoL_1__1__0
L1816
Variable x : set
(*** Conj_SNoL_1__1__0 TMNKDisa8NheGiCvicXJ6wVH1obD3tZceHx bounty of about 25 bars ***)
L1817
Hypothesis H1 : SNoLev x ∈ ordsucc Empty
L1818
Theorem. (Conj_SNoL_1__1__0)
Empty = x β†’ x ∈ ordsucc Empty
In Proofgold the corresponding term root is a12e97... and proposition id is a558ae...
Proof:
Proof not loaded.
End of Section Conj_SNoL_1__1__0
Beginning of Section Conj_SNo__eps___3__3
L1824
Variable x : set
(*** Conj_SNo__eps___3__3 TMTRSHCcJvMLiFsE2YUk9Dx4EYihupZeRDj bounty of about 25 bars ***)
L1825
Variable y : set
L1826
Variable z : set
L1827
Hypothesis H0 : nat_p x
L1828
Hypothesis H1 : y ∈ ordsucc x
L1829
Hypothesis H2 : nat_p z
L1830
Theorem. (Conj_SNo__eps___3__3)
nat_p y β†’ exactly1of2 (SetAdjoin y (Sing (ordsucc Empty)) ∈ eps_ x) (y ∈ eps_ x)
In Proofgold the corresponding term root is 4bca0b... and proposition id is dcf98d...
Proof:
Proof not loaded.
End of Section Conj_SNo__eps___3__3
Beginning of Section Conj_SNo_pos_eps_Lt__1__3
L1836
Variable x : set
(*** Conj_SNo_pos_eps_Lt__1__3 TMS6aFHTyDgS14PKbJq5oEuZLpzjp1ZSKZE bounty of about 25 bars ***)
L1837
Variable y : set
L1838
Hypothesis H0 : Empty < y
L1839
Hypothesis H1 : ordinal (SNoLev y)
L1840
Hypothesis H2 : SNo y
L1841
Theorem. (Conj_SNo_pos_eps_Lt__1__3)
y β‰  Empty
In Proofgold the corresponding term root is 2632dc... and proposition id is d43ffe...
Proof:
Proof not loaded.
End of Section Conj_SNo_pos_eps_Lt__1__3
Beginning of Section Conj_SNo_pos_eps_Lt__2__3
L1847
Variable x : set
(*** Conj_SNo_pos_eps_Lt__2__3 TMQaNRh2ysPgPsg2g52o7ETFgCFiGcWZj6d bounty of about 25 bars ***)
L1848
Variable y : set
L1849
Variable z : set
L1850
Hypothesis H0 : Empty < y
L1851
Hypothesis H1 : SNo y
L1852
Hypothesis H2 : SNo z
L1853
Hypothesis H4 : SNoLev z ∈ eps_ x
L1854
Theorem. (Conj_SNo_pos_eps_Lt__2__3)
z β‰  Empty
In Proofgold the corresponding term root is 1d6443... and proposition id is 296c65...
Proof:
Proof not loaded.
End of Section Conj_SNo_pos_eps_Lt__2__3
Beginning of Section Conj_SNo_pos_eps_Le__1__3
L1860
Variable x : set
(*** Conj_SNo_pos_eps_Le__1__3 TMS6aFHTyDgS14PKbJq5oEuZLpzjp1ZSKZE bounty of about 25 bars ***)
L1861
Variable y : set
L1862
Hypothesis H0 : Empty < y
L1863
Hypothesis H1 : ordinal (SNoLev y)
L1864
Hypothesis H2 : SNo y
L1865
Theorem. (Conj_SNo_pos_eps_Le__1__3)
y β‰  Empty
In Proofgold the corresponding term root is 2632dc... and proposition id is d43ffe...
Proof:
Proof not loaded.
End of Section Conj_SNo_pos_eps_Le__1__3
Beginning of Section Conj_SNo_pos_eps_Le__2__3
L1871
Variable x : set
(*** Conj_SNo_pos_eps_Le__2__3 TMQaNRh2ysPgPsg2g52o7ETFgCFiGcWZj6d bounty of about 25 bars ***)
L1872
Variable y : set
L1873
Variable z : set
L1874
Hypothesis H0 : Empty < y
L1875
Hypothesis H1 : SNo y
L1876
Hypothesis H2 : SNo z
L1877
Hypothesis H4 : SNoLev z ∈ eps_ x
L1878
Theorem. (Conj_SNo_pos_eps_Le__2__3)
z β‰  Empty
In Proofgold the corresponding term root is 1d6443... and proposition id is 296c65...
Proof:
Proof not loaded.
End of Section Conj_SNo_pos_eps_Le__2__3
Beginning of Section Conj_eps_SNoCut__5__2
L1884
Variable x : set
(*** Conj_eps_SNoCut__5__2 TMRCi8wYQz3GsEqVAgiPR3oLt2viCevNkuo bounty of about 25 bars ***)
L1885
Variable y : set
L1886
Hypothesis H0 : (βˆ€z : set, z ∈ Repl x eps_ β†’ SNo z)
L1887
Hypothesis H1 : SNo (SNoCut (Sing Empty) (Repl x eps_))
L1888
Hypothesis H3 : (βˆ€z : set, SNo z β†’ (βˆ€w : set, w ∈ Sing Empty β†’ w < z) β†’ (βˆ€w : set, w ∈ Repl x eps_ β†’ z < w) β†’ Subq (SNoLev (SNoCut (Sing Empty) (Repl x eps_))) (SNoLev z) ∧ SNoEq_ (SNoLev (SNoCut (Sing Empty) (Repl x eps_))) (SNoCut (Sing Empty) (Repl x eps_)) z)
L1889
Hypothesis H4 : SNo y
L1890
Hypothesis H5 : SNoLev y ∈ binintersect (SNoLev (eps_ x)) (SNoLev (SNoCut (Sing Empty) (Repl x eps_)))
L1891
Hypothesis H6 : y < SNoCut (Sing Empty) (Repl x eps_)
L1892
Hypothesis H7 : (βˆ€z : set, z ∈ Sing Empty β†’ z < y)
L1893
Theorem. (Conj_eps_SNoCut__5__2)
Β¬ (βˆ€z : set, z ∈ Repl x eps_ β†’ y < z)
In Proofgold the corresponding term root is a34286... and proposition id is 0cfb50...
Proof:
Proof not loaded.
End of Section Conj_eps_SNoCut__5__2
Beginning of Section Conj_eps_SNoCut__6__5
L1899
Variable x : set
(*** Conj_eps_SNoCut__6__5 TMap7RCHu1waWo4TxmKcjmpbz8GKQY6tFqh bounty of about 25 bars ***)
L1900
Variable y : set
L1901
Hypothesis H0 : x ∈ Ο‰
L1902
Hypothesis H1 : (βˆ€z : set, z ∈ Repl x eps_ β†’ SNo z)
L1903
Hypothesis H2 : SNo (SNoCut (Sing Empty) (Repl x eps_))
L1904
Hypothesis H3 : (βˆ€z : set, z ∈ Repl x eps_ β†’ SNoCut (Sing Empty) (Repl x eps_) < z)
L1905
Hypothesis H4 : (βˆ€z : set, SNo z β†’ (βˆ€w : set, w ∈ Sing Empty β†’ w < z) β†’ (βˆ€w : set, w ∈ Repl x eps_ β†’ z < w) β†’ Subq (SNoLev (SNoCut (Sing Empty) (Repl x eps_))) (SNoLev z) ∧ SNoEq_ (SNoLev (SNoCut (Sing Empty) (Repl x eps_))) (SNoCut (Sing Empty) (Repl x eps_)) z)
L1906
Hypothesis H6 : SNoLev y ∈ binintersect (SNoLev (eps_ x)) (SNoLev (SNoCut (Sing Empty) (Repl x eps_)))
L1907
Hypothesis H7 : eps_ x < y
L1908
Hypothesis H8 : y < SNoCut (Sing Empty) (Repl x eps_)
L1909
Theorem. (Conj_eps_SNoCut__6__5)
Β¬ (βˆ€z : set, z ∈ Sing Empty β†’ z < y)
In Proofgold the corresponding term root is 25bf6f... and proposition id is 1b5b9c...
Proof:
Proof not loaded.
End of Section Conj_eps_SNoCut__6__5
Beginning of Section Conj_SNo_etaE__2__1
L1915
Variable x : set
(*** Conj_SNo_etaE__2__1 TMRNa2WLKRRLuwfLFughua4sbsJWH4q9QDk bounty of about 25 bars ***)
L1916
Variable y : set
L1917
Variable z : set
L1918
Hypothesis H0 : y < x
L1919
Hypothesis H2 : SNo_ z y
L1920
Hypothesis H3 : ordinal z
L1921
Hypothesis H4 : SNo y
L1922
Theorem. (Conj_SNo_etaE__2__1)
SNoLev y = z β†’ SNo y ∧ SNoLev y ∈ SNoLev x ∧ y < x
In Proofgold the corresponding term root is b6b811... and proposition id is 826135...
Proof:
Proof not loaded.
End of Section Conj_SNo_etaE__2__1
Beginning of Section Conj_SNo_etaE__3__2
L1928
Variable x : set
(*** Conj_SNo_etaE__3__2 TMNwQD2Q8KgikwWtQC3xRypmDaB9o2Qeoop bounty of about 25 bars ***)
L1929
Variable y : set
L1930
Variable z : set
L1931
Hypothesis H0 : y < x
L1932
Hypothesis H1 : z ∈ SNoLev x
L1933
Hypothesis H3 : ordinal z
L1934
Theorem. (Conj_SNo_etaE__3__2)
SNo y β†’ SNo y ∧ SNoLev y ∈ SNoLev x ∧ y < x
In Proofgold the corresponding term root is 9070ac... and proposition id is 46b34d...
Proof:
Proof not loaded.
End of Section Conj_SNo_etaE__3__2
Beginning of Section Conj_SNo_etaE__5__0
L1940
Variable x : set
(*** Conj_SNo_etaE__5__0 TMJitoTRoqNStjvXMfcHsD2CzNYzYn2sJsF bounty of about 25 bars ***)
L1941
Variable y : set
L1942
Variable z : set
L1943
Hypothesis H1 : z ∈ SNoLev x
L1944
Hypothesis H2 : SNo y
L1945
Hypothesis H3 : SNoLev y = z
L1946
Theorem. (Conj_SNo_etaE__5__0)
SNoLev y ∈ SNoLev x β†’ SNo y ∧ SNoLev y ∈ SNoLev x ∧ x < y
In Proofgold the corresponding term root is ce96a0... and proposition id is 65298d...
Proof:
Proof not loaded.
End of Section Conj_SNo_etaE__5__0
Beginning of Section Conj_SNo_etaE__5__1
L1952
Variable x : set
(*** Conj_SNo_etaE__5__1 TMMQBSNwvR4q8BZRJicSctkatQUj6pn9s4B bounty of about 25 bars ***)
L1953
Variable y : set
L1954
Variable z : set
L1955
Hypothesis H0 : x < y
L1956
Hypothesis H2 : SNo y
L1957
Hypothesis H3 : SNoLev y = z
L1958
Theorem. (Conj_SNo_etaE__5__1)
SNoLev y ∈ SNoLev x β†’ SNo y ∧ SNoLev y ∈ SNoLev x ∧ x < y
In Proofgold the corresponding term root is a9d1bd... and proposition id is 79465d...
Proof:
Proof not loaded.
End of Section Conj_SNo_etaE__5__1
Beginning of Section Conj_SNo_etaE__7__0
L1964
Variable x : set
(*** Conj_SNo_etaE__7__0 TMSVU1xcXQYeamkRBqXc9msW472NiSLYZMq bounty of about 25 bars ***)
L1965
Variable y : set
L1966
Variable z : set
L1967
Hypothesis H1 : z ∈ SNoLev x
L1968
Hypothesis H2 : SNo_ z y
L1969
Hypothesis H3 : ordinal z
L1970
Theorem. (Conj_SNo_etaE__7__0)
SNo y β†’ SNo y ∧ SNoLev y ∈ SNoLev x ∧ x < y
In Proofgold the corresponding term root is abe493... and proposition id is 8c3db9...
Proof:
Proof not loaded.
End of Section Conj_SNo_etaE__7__0
Beginning of Section Conj_SNo_etaE__12__1
L1976
Variable x : set
(*** Conj_SNo_etaE__12__1 TMZH7xP2gJz7gabR9dZUUqatMFhuQjeN12z bounty of about 25 bars ***)
L1977
Variable P : prop
L1978
Hypothesis H0 : SNo x
L1979
Theorem. (Conj_SNo_etaE__12__1)
ordinal (SNoLev x) β†’ P
In Proofgold the corresponding term root is 743d86... and proposition id is ad676b...
Proof:
Proof not loaded.
End of Section Conj_SNo_etaE__12__1
Beginning of Section Conj_SNo_rec2_eq_1__1__2
L1985
Variable P : (set β†’ (set β†’ ((set β†’ (set β†’ set)) β†’ set)))
(*** Conj_SNo_rec2_eq_1__1__2 TMMYsKaPdYQ7Txzch9GgTGJvPU1QBCfsTqg bounty of about 25 bars ***)
L1986
Variable x : set
L1987
Variable g : (set β†’ (set β†’ set))
L1988
Variable y : set
L1989
Variable f : (set β†’ set)
L1990
Variable f2 : (set β†’ set)
L1991
Hypothesis H0 : (βˆ€z : set, SNo z β†’ (βˆ€w : set, SNo w β†’ (βˆ€h : set β†’ set β†’ set, βˆ€g2 : set β†’ set β†’ set, (βˆ€u : set, u ∈ SNoS_ (SNoLev z) β†’ (βˆ€v : set, SNo v β†’ h u v = g2 u v)) β†’ (βˆ€u : set, u ∈ SNoS_ (SNoLev w) β†’ h z u = g2 z u) β†’ P z w h = P z w g2)))
L1992
Hypothesis H1 : SNo x
L1993
Hypothesis H3 : (βˆ€z : set, z ∈ SNoS_ (SNoLev y) β†’ f z = f2 z)
L1994
Theorem. (Conj_SNo_rec2_eq_1__1__2)
(βˆ€z : set, z ∈ SNoS_ (SNoLev x) β†’ g z = g z) β†’ P x y (Ξ»z : set β‡’ Ξ»w : set β‡’ If_i (z = x) (f w) (g z w)) = P x y (Ξ»z : set β‡’ Ξ»w : set β‡’ If_i (z = x) (f2 w) (g z w))
In Proofgold the corresponding term root is e9c444... and proposition id is 5fe1b1...
Proof:
Proof not loaded.
End of Section Conj_SNo_rec2_eq_1__1__2
Beginning of Section Conj_SNo_rec2_eq__1__1
L2000
Variable P : (set β†’ (set β†’ ((set β†’ (set β†’ set)) β†’ set)))
(*** Conj_SNo_rec2_eq__1__1 TML53N3CLmcpvTytFsYxprPnDpSX8nYwswb bounty of about 25 bars ***)
L2001
Variable x : set
L2002
Variable g : (set β†’ (set β†’ set))
L2003
Variable h : (set β†’ (set β†’ set))
L2004
Variable y : set
L2005
Hypothesis H0 : (βˆ€z : set, SNo z β†’ (βˆ€w : set, SNo w β†’ (βˆ€g2 : set β†’ set β†’ set, βˆ€h2 : set β†’ set β†’ set, (βˆ€u : set, u ∈ SNoS_ (SNoLev z) β†’ (βˆ€v : set, SNo v β†’ g2 u v = h2 u v)) β†’ (βˆ€u : set, u ∈ SNoS_ (SNoLev w) β†’ g2 z u = h2 z u) β†’ P z w g2 = P z w h2)))
L2006
Hypothesis H2 : (βˆ€z : set, z ∈ SNoS_ (SNoLev x) β†’ g z = h z)
L2007
Hypothesis H3 : SNo y
L2008
Theorem. (Conj_SNo_rec2_eq__1__1)
(βˆ€z : set, ordinal z β†’ (βˆ€w : set, w ∈ SNoS_ z β†’ SNo_rec_i (Ξ»u : set β‡’ Ξ»f : set β†’ set β‡’ P x u (Ξ»v : set β‡’ Ξ»x2 : set β‡’ If_i (v = x) (f x2) (g v x2))) w = SNo_rec_i (Ξ»u : set β‡’ Ξ»f : set β†’ set β‡’ P x u (Ξ»v : set β‡’ Ξ»x2 : set β‡’ If_i (v = x) (f x2) (h v x2))) w)) β†’ SNo_rec_i (Ξ»z : set β‡’ Ξ»f : set β†’ set β‡’ P x z (Ξ»w : set β‡’ Ξ»u : set β‡’ If_i (w = x) (f u) (g w u))) y = SNo_rec_i (Ξ»z : set β‡’ Ξ»f : set β†’ set β‡’ P x z (Ξ»w : set β‡’ Ξ»u : set β‡’ If_i (w = x) (f u) (h w u))) y
In Proofgold the corresponding term root is 1a8dec... and proposition id is 859184...
Proof:
Proof not loaded.
End of Section Conj_SNo_rec2_eq__1__1
Beginning of Section Conj_SNo_rec2_eq__4__1
L2014
Variable P : (set β†’ (set β†’ ((set β†’ (set β†’ set)) β†’ set)))
(*** Conj_SNo_rec2_eq__4__1 TMXh7nGgfUKMtiFGeTfpiavob3GMjgnCvYi bounty of about 25 bars ***)
L2015
Variable x : set
L2016
Variable y : set
L2017
Hypothesis H0 : (βˆ€z : set, SNo z β†’ (βˆ€w : set, SNo w β†’ (βˆ€g : set β†’ set β†’ set, βˆ€h : set β†’ set β†’ set, (βˆ€u : set, u ∈ SNoS_ (SNoLev z) β†’ (βˆ€v : set, SNo v β†’ g u v = h u v)) β†’ (βˆ€u : set, u ∈ SNoS_ (SNoLev w) β†’ g z u = h z u) β†’ P z w g = P z w h)))
L2018
Hypothesis H2 : SNo y
L2019
Hypothesis H3 : (βˆ€z : set, SNo z β†’ (βˆ€g : set β†’ set β†’ set, βˆ€h : set β†’ set β†’ set, (βˆ€w : set, w ∈ SNoS_ (SNoLev z) β†’ g w = h w) β†’ (Ξ»w : set β‡’ If_i (SNo w) (SNo_rec_i (Ξ»u : set β‡’ Ξ»f : set β†’ set β‡’ P z u (Ξ»v : set β‡’ Ξ»x2 : set β‡’ If_i (v = z) (f x2) (g v x2))) w) Empty) = (Ξ»w : set β‡’ If_i (SNo w) (SNo_rec_i (Ξ»u : set β‡’ Ξ»f : set β†’ set β‡’ P z u (Ξ»v : set β‡’ Ξ»x2 : set β‡’ If_i (v = z) (f x2) (h v x2))) w) Empty)))
L2020
Hypothesis H4 : (βˆ€z : set, z ∈ SNoS_ (SNoLev x) β†’ (βˆ€w : set, SNo w β†’ If_i (z = x) (SNo_rec_i (Ξ»u : set β‡’ Ξ»f : set β†’ set β‡’ P x u (Ξ»v : set β‡’ Ξ»x2 : set β‡’ If_i (v = x) (f x2) (SNo_rec_ii (Ξ»y2 : set β‡’ Ξ»g : set β†’ set β†’ set β‡’ Ξ»z2 : set β‡’ If_i (SNo z2) (SNo_rec_i (Ξ»w2 : set β‡’ Ξ»f2 : set β†’ set β‡’ P y2 w2 (Ξ»u2 : set β‡’ Ξ»v2 : set β‡’ If_i (u2 = y2) (f2 v2) (g u2 v2))) z2) Empty) v x2))) w) (SNo_rec_ii (Ξ»u : set β‡’ Ξ»g : set β†’ set β†’ set β‡’ Ξ»v : set β‡’ If_i (SNo v) (SNo_rec_i (Ξ»x2 : set β‡’ Ξ»f : set β†’ set β‡’ P u x2 (Ξ»y2 : set β‡’ Ξ»z2 : set β‡’ If_i (y2 = u) (f z2) (g y2 z2))) v) Empty) z w) = SNo_rec_ii (Ξ»u : set β‡’ Ξ»g : set β†’ set β†’ set β‡’ Ξ»v : set β‡’ If_i (SNo v) (SNo_rec_i (Ξ»x2 : set β‡’ Ξ»f : set β†’ set β‡’ P u x2 (Ξ»y2 : set β‡’ Ξ»z2 : set β‡’ If_i (y2 = u) (f z2) (g y2 z2))) v) Empty) z w))
L2021
Theorem. (Conj_SNo_rec2_eq__4__1)
(βˆ€z : set, z ∈ SNoS_ (SNoLev y) β†’ If_i (x = x) (SNo_rec_i (Ξ»w : set β‡’ Ξ»f : set β†’ set β‡’ P x w (Ξ»u : set β‡’ Ξ»v : set β‡’ If_i (u = x) (f v) (SNo_rec_ii (Ξ»x2 : set β‡’ Ξ»g : set β†’ set β†’ set β‡’ Ξ»y2 : set β‡’ If_i (SNo y2) (SNo_rec_i (Ξ»z2 : set β‡’ Ξ»f2 : set β†’ set β‡’ P x2 z2 (Ξ»w2 : set β‡’ Ξ»u2 : set β‡’ If_i (w2 = x2) (f2 u2) (g w2 u2))) y2) Empty) u v))) z) (SNo_rec_ii (Ξ»w : set β‡’ Ξ»g : set β†’ set β†’ set β‡’ Ξ»u : set β‡’ If_i (SNo u) (SNo_rec_i (Ξ»v : set β‡’ Ξ»f : set β†’ set β‡’ P w v (Ξ»x2 : set β‡’ Ξ»y2 : set β‡’ If_i (x2 = w) (f y2) (g x2 y2))) u) Empty) x z) = SNo_rec_ii (Ξ»w : set β‡’ Ξ»g : set β†’ set β†’ set β‡’ Ξ»u : set β‡’ If_i (SNo u) (SNo_rec_i (Ξ»v : set β‡’ Ξ»f : set β†’ set β‡’ P w v (Ξ»x2 : set β‡’ Ξ»y2 : set β‡’ If_i (x2 = w) (f y2) (g x2 y2))) u) Empty) x z) β†’ P x y (Ξ»z : set β‡’ Ξ»w : set β‡’ If_i (z = x) (SNo_rec_i (Ξ»u : set β‡’ Ξ»f : set β†’ set β‡’ P x u (Ξ»v : set β‡’ Ξ»x2 : set β‡’ If_i (v = x) (f x2) (SNo_rec_ii (Ξ»y2 : set β‡’ Ξ»g : set β†’ set β†’ set β‡’ Ξ»z2 : set β‡’ If_i (SNo z2) (SNo_rec_i (Ξ»w2 : set β‡’ Ξ»f2 : set β†’ set β‡’ P y2 w2 (Ξ»u2 : set β‡’ Ξ»v2 : set β‡’ If_i (u2 = y2) (f2 v2) (g u2 v2))) z2) Empty) v x2))) w) (SNo_rec_ii (Ξ»u : set β‡’ Ξ»g : set β†’ set β†’ set β‡’ Ξ»v : set β‡’ If_i (SNo v) (SNo_rec_i (Ξ»x2 : set β‡’ Ξ»f : set β†’ set β‡’ P u x2 (Ξ»y2 : set β‡’ Ξ»z2 : set β‡’ If_i (y2 = u) (f z2) (g y2 z2))) v) Empty) z w)) = P x y (SNo_rec_ii (Ξ»z : set β‡’ Ξ»g : set β†’ set β†’ set β‡’ Ξ»w : set β‡’ If_i (SNo w) (SNo_rec_i (Ξ»u : set β‡’ Ξ»f : set β†’ set β‡’ P z u (Ξ»v : set β‡’ Ξ»x2 : set β‡’ If_i (v = z) (f x2) (g v x2))) w) Empty))
In Proofgold the corresponding term root is 380df4... and proposition id is 211ef4...
Proof:
Proof not loaded.
End of Section Conj_SNo_rec2_eq__4__1
Beginning of Section Conj_SNo_ordinal_ind__2__1
L2027
Variable p : (set β†’ prop)
(*** Conj_SNo_ordinal_ind__2__1 TMaLcBMMLVMkDnt3hMFd2dG1B7cPd2cAwqi bounty of about 25 bars ***)
L2028
Variable x : set
L2029
Hypothesis H0 : (βˆ€y : set, ordinal y β†’ (βˆ€z : set, z ∈ SNoS_ y β†’ p z))
L2030
Hypothesis H2 : ordinal (SNoLev x)
L2031
Theorem. (Conj_SNo_ordinal_ind__2__1)
ordinal (ordsucc (SNoLev x)) β†’ p x
In Proofgold the corresponding term root is af5b2b... and proposition id is d73c1f...
Proof:
Proof not loaded.
End of Section Conj_SNo_ordinal_ind__2__1
Beginning of Section Conj_SNo_ordinal_ind2__5__1
L2037
Variable r : (set β†’ (set β†’ prop))
(*** Conj_SNo_ordinal_ind2__5__1 TMRV7jRCE2Hbm73kXk7UHgitPiadAhKSe5E bounty of about 25 bars ***)
L2038
Variable x : set
L2039
Variable y : set
L2040
Hypothesis H0 : (βˆ€z : set, ordinal z β†’ (βˆ€w : set, ordinal w β†’ (βˆ€u : set, u ∈ SNoS_ z β†’ (βˆ€v : set, v ∈ SNoS_ w β†’ r u v))))
L2041
Hypothesis H2 : SNo y
L2042
Hypothesis H3 : ordinal (SNoLev x)
L2043
Theorem. (Conj_SNo_ordinal_ind2__5__1)
ordinal (ordsucc (SNoLev x)) β†’ r x y
In Proofgold the corresponding term root is 81b3ce... and proposition id is 918cf1...
Proof:
Proof not loaded.
End of Section Conj_SNo_ordinal_ind2__5__1
Beginning of Section Conj_SNo_ordinal_ind3__6__1
L2049
Variable P : (set β†’ (set β†’ (set β†’ prop)))
(*** Conj_SNo_ordinal_ind3__6__1 TMdnNAhgcmZDNgNpP3FZXkPp4LVP7wZ4Zwy bounty of about 25 bars ***)
L2050
Variable x : set
L2051
Variable y : set
L2052
Variable z : set
L2053
Hypothesis H0 : (βˆ€w : set, ordinal w β†’ (βˆ€u : set, ordinal u β†’ (βˆ€v : set, ordinal v β†’ (βˆ€x2 : set, x2 ∈ SNoS_ w β†’ (βˆ€y2 : set, y2 ∈ SNoS_ u β†’ (βˆ€z2 : set, z2 ∈ SNoS_ v β†’ P x2 y2 z2))))))
L2054
Hypothesis H2 : SNo z
L2055
Hypothesis H3 : ordinal (ordsucc (SNoLev x))
L2056
Hypothesis H4 : x ∈ SNoS_ (ordsucc (SNoLev x))
L2057
Theorem. (Conj_SNo_ordinal_ind3__6__1)
ordinal (SNoLev y) β†’ P x y z
In Proofgold the corresponding term root is 86c6f8... and proposition id is cb0b1d...
Proof:
Proof not loaded.
End of Section Conj_SNo_ordinal_ind3__6__1
Beginning of Section Conj_restr_SNo__1__2
L2063
Variable x : set
(*** Conj_restr_SNo__1__2 TMGAwJLMQCYx8RJYcWuEpPGpVpvdFVZJEua bounty of about 25 bars ***)
L2064
Variable y : set
L2065
Hypothesis H0 : SNo x
L2066
Hypothesis H1 : y ∈ SNoLev x
L2067
Theorem. (Conj_restr_SNo__1__2)
SNo_ y (binintersect x (SNoElts_ y)) β†’ SNo (binintersect x (SNoElts_ y))
In Proofgold the corresponding term root is ad79ea... and proposition id is 614555...
Proof:
Proof not loaded.
End of Section Conj_restr_SNo__1__2
Beginning of Section Conj_minus_SNo_prop1__1__2
L2073
Variable x : set
(*** Conj_minus_SNo_prop1__1__2 TMYMaxzU2JUeroyyJpkPCqYSb9Pddj41aNu bounty of about 25 bars ***)
L2074
Variable y : set
L2075
Hypothesis H0 : SNo x
L2076
Hypothesis H1 : (βˆ€z : set, z ∈ SNoS_ (SNoLev x) β†’ SNo (- z) ∧ (βˆ€w : set, w ∈ SNoL z β†’ - z < - w) ∧ (βˆ€w : set, w ∈ SNoR z β†’ - w < - z) ∧ SNoCutP (Repl (SNoR z) minus_SNo) (Repl (SNoL z) minus_SNo))
L2077
Hypothesis H3 : SNoLev y ∈ SNoLev x
L2078
Theorem. (Conj_minus_SNo_prop1__1__2)
y ∈ SNoS_ (SNoLev x) β†’ SNo (- y) ∧ (βˆ€z : set, z ∈ SNoL y β†’ - y < - z) ∧ (βˆ€z : set, z ∈ SNoR y β†’ - z < - y)
In Proofgold the corresponding term root is 02f59e... and proposition id is e0a4af...
Proof:
Proof not loaded.
End of Section Conj_minus_SNo_prop1__1__2
Beginning of Section Conj_minus_SNo_prop1__2__2
L2084
Variable x : set
(*** Conj_minus_SNo_prop1__2__2 TMYMaxzU2JUeroyyJpkPCqYSb9Pddj41aNu bounty of about 25 bars ***)
L2085
Variable y : set
L2086
Hypothesis H0 : SNo x
L2087
Hypothesis H1 : (βˆ€z : set, z ∈ SNoS_ (SNoLev x) β†’ SNo (- z) ∧ (βˆ€w : set, w ∈ SNoL z β†’ - z < - w) ∧ (βˆ€w : set, w ∈ SNoR z β†’ - w < - z) ∧ SNoCutP (Repl (SNoR z) minus_SNo) (Repl (SNoL z) minus_SNo))
L2088
Hypothesis H3 : SNoLev y ∈ SNoLev x
L2089
Theorem. (Conj_minus_SNo_prop1__2__2)
y ∈ SNoS_ (SNoLev x) β†’ SNo (- y) ∧ (βˆ€z : set, z ∈ SNoL y β†’ - y < - z) ∧ (βˆ€z : set, z ∈ SNoR y β†’ - z < - y)
In Proofgold the corresponding term root is 02f59e... and proposition id is e0a4af...
Proof:
Proof not loaded.
End of Section Conj_minus_SNo_prop1__2__2
Beginning of Section Conj_minus_SNo_prop1__4__5
L2095
Variable x : set
(*** Conj_minus_SNo_prop1__4__5 TMc44dd7T2UPwDcwP5tCPP4eguntjsziPKv bounty of about 25 bars ***)
L2096
Variable y : set
L2097
Variable z : set
L2098
Variable w : set
L2099
Hypothesis H0 : SNo x
L2100
Hypothesis H1 : (βˆ€u : set, u ∈ SNoS_ (SNoLev x) β†’ SNo (- u) ∧ (βˆ€v : set, v ∈ SNoL u β†’ - u < - v) ∧ (βˆ€v : set, v ∈ SNoR u β†’ - v < - u) ∧ SNoCutP (Repl (SNoR u) minus_SNo) (Repl (SNoL u) minus_SNo))
L2101
Hypothesis H2 : SNo y
L2102
Hypothesis H3 : SNoLev y ∈ SNoLev x
L2103
Hypothesis H4 : SNo z
L2104
Hypothesis H6 : (βˆ€u : set, u ∈ SNoR z β†’ - u < - z)
L2105
Hypothesis H7 : SNo (- y)
L2106
Hypothesis H8 : (βˆ€u : set, u ∈ SNoL y β†’ - y < - u)
L2107
Hypothesis H9 : SNo w
L2108
Hypothesis H10 : z < w
L2109
Hypothesis H11 : w < y
L2110
Hypothesis H12 : SNoLev w ∈ SNoLev z
L2111
Hypothesis H13 : SNoLev w ∈ SNoLev y
L2112
Theorem. (Conj_minus_SNo_prop1__4__5)
w ∈ SNoS_ (SNoLev x) β†’ - y < - z
In Proofgold the corresponding term root is 0b4120... and proposition id is 4cc472...
Proof:
Proof not loaded.
End of Section Conj_minus_SNo_prop1__4__5
Beginning of Section Conj_minus_SNo_prop1__5__7
L2118
Variable x : set
(*** Conj_minus_SNo_prop1__5__7 TMRCM2vqC6pwsN5o8v87EbyJTHhZscDb9Qx bounty of about 25 bars ***)
L2119
Variable y : set
L2120
Variable z : set
L2121
Hypothesis H0 : SNo x
L2122
Hypothesis H1 : (βˆ€w : set, w ∈ SNoS_ (SNoLev x) β†’ SNo (- w) ∧ (βˆ€u : set, u ∈ SNoL w β†’ - w < - u) ∧ (βˆ€u : set, u ∈ SNoR w β†’ - u < - w) ∧ SNoCutP (Repl (SNoR w) minus_SNo) (Repl (SNoL w) minus_SNo))
L2123
Hypothesis H2 : SNo y
L2124
Hypothesis H3 : SNoLev y ∈ SNoLev x
L2125
Hypothesis H4 : x < y
L2126
Hypothesis H5 : SNo z
L2127
Hypothesis H6 : z < x
L2128
Hypothesis H8 : (βˆ€w : set, w ∈ SNoR z β†’ - w < - z)
L2129
Hypothesis H9 : SNo (- y)
L2130
Hypothesis H10 : (βˆ€w : set, w ∈ SNoL y β†’ - y < - w)
L2131
Theorem. (Conj_minus_SNo_prop1__5__7)
z < y β†’ - y < - z
In Proofgold the corresponding term root is 62be8a... and proposition id is ec8fd6...
Proof:
Proof not loaded.
End of Section Conj_minus_SNo_prop1__5__7
Beginning of Section Conj_minus_SNo_prop1__5__9
L2137
Variable x : set
(*** Conj_minus_SNo_prop1__5__9 TMGsHhRAEY3AHtueVYYnzTnFQtJyjgBz993 bounty of about 25 bars ***)
L2138
Variable y : set
L2139
Variable z : set
L2140
Hypothesis H0 : SNo x
L2141
Hypothesis H1 : (βˆ€w : set, w ∈ SNoS_ (SNoLev x) β†’ SNo (- w) ∧ (βˆ€u : set, u ∈ SNoL w β†’ - w < - u) ∧ (βˆ€u : set, u ∈ SNoR w β†’ - u < - w) ∧ SNoCutP (Repl (SNoR w) minus_SNo) (Repl (SNoL w) minus_SNo))
L2142
Hypothesis H2 : SNo y
L2143
Hypothesis H3 : SNoLev y ∈ SNoLev x
L2144
Hypothesis H4 : x < y
L2145
Hypothesis H5 : SNo z
L2146
Hypothesis H6 : z < x
L2147
Hypothesis H7 : SNo (- z)
L2148
Hypothesis H8 : (βˆ€w : set, w ∈ SNoR z β†’ - w < - z)
L2149
Hypothesis H10 : (βˆ€w : set, w ∈ SNoL y β†’ - y < - w)
L2150
Theorem. (Conj_minus_SNo_prop1__5__9)
z < y β†’ - y < - z
In Proofgold the corresponding term root is c6de85... and proposition id is d0d3de...
Proof:
Proof not loaded.
End of Section Conj_minus_SNo_prop1__5__9
Beginning of Section Conj_minus_SNo_prop1__9__3
L2156
Variable x : set
(*** Conj_minus_SNo_prop1__9__3 TMNfkJud6RAy5eEa4GpHHy4Qyj1bK18ndcM bounty of about 25 bars ***)
L2157
Hypothesis H0 : SNo x
L2158
Hypothesis H1 : (βˆ€y : set, y ∈ SNoS_ (SNoLev x) β†’ SNo (- y) ∧ (βˆ€z : set, z ∈ SNoL y β†’ - y < - z) ∧ (βˆ€z : set, z ∈ SNoR y β†’ - z < - y) ∧ SNoCutP (Repl (SNoR y) minus_SNo) (Repl (SNoL y) minus_SNo))
L2159
Hypothesis H2 : (βˆ€y : set, y ∈ SNoL x β†’ SNo (- y) ∧ (βˆ€z : set, z ∈ SNoL y β†’ - y < - z) ∧ (βˆ€z : set, z ∈ SNoR y β†’ - z < - y))
L2160
Theorem. (Conj_minus_SNo_prop1__9__3)
SNoCutP (Repl (SNoR x) minus_SNo) (Repl (SNoL x) minus_SNo) β†’ SNo (- x) ∧ (βˆ€y : set, y ∈ SNoL x β†’ - x < - y) ∧ (βˆ€y : set, y ∈ SNoR x β†’ - y < - x) ∧ SNoCutP (Repl (SNoR x) minus_SNo) (Repl (SNoL x) minus_SNo)
In Proofgold the corresponding term root is 1db218... and proposition id is cf6dd2...
Proof:
Proof not loaded.
End of Section Conj_minus_SNo_prop1__9__3
Beginning of Section Conj_minus_SNo_prop1__11__0
L2166
Variable x : set
(*** Conj_minus_SNo_prop1__11__0 TMGmmVHFG98AkaZHQiK8reWvC1kS6umcyRh bounty of about 25 bars ***)
L2167
Hypothesis H1 : (βˆ€y : set, y ∈ SNoS_ (SNoLev x) β†’ SNo (- y) ∧ (βˆ€z : set, z ∈ SNoL y β†’ - y < - z) ∧ (βˆ€z : set, z ∈ SNoR y β†’ - z < - y) ∧ SNoCutP (Repl (SNoR y) minus_SNo) (Repl (SNoL y) minus_SNo))
L2168
Theorem. (Conj_minus_SNo_prop1__11__0)
(βˆ€y : set, y ∈ SNoL x β†’ SNo (- y) ∧ (βˆ€z : set, z ∈ SNoL y β†’ - y < - z) ∧ (βˆ€z : set, z ∈ SNoR y β†’ - z < - y)) β†’ SNo (- x) ∧ (βˆ€y : set, y ∈ SNoL x β†’ - x < - y) ∧ (βˆ€y : set, y ∈ SNoR x β†’ - y < - x) ∧ SNoCutP (Repl (SNoR x) minus_SNo) (Repl (SNoL x) minus_SNo)
In Proofgold the corresponding term root is cf94fc... and proposition id is aad4a2...
Proof:
Proof not loaded.
End of Section Conj_minus_SNo_prop1__11__0
Beginning of Section Conj_minus_SNo_Lev_lem1__1__2
L2174
Variable x : set
(*** Conj_minus_SNo_Lev_lem1__1__2 TMbz4EZRNA5qWZytgvu5NQSdFz3cs5xQT4X bounty of about 25 bars ***)
L2175
Variable y : set
L2176
Variable z : set
L2177
Variable w : set
L2178
Hypothesis H0 : y ∈ ordsucc (SNoLev z)
L2179
Hypothesis H1 : z = - w
L2180
Hypothesis H3 : Subq (SNoLev (- w)) (SNoLev w)
L2181
Hypothesis H4 : Subq (SNoLev x) y
L2182
Theorem. (Conj_minus_SNo_Lev_lem1__1__2)
¬ SNoLev w ∈ y
In Proofgold the corresponding term root is 9113ae... and proposition id is 658123...
Proof:
Proof not loaded.
End of Section Conj_minus_SNo_Lev_lem1__1__2
Beginning of Section Conj_minus_SNo_Lev_lem1__3__1
L2188
Variable x : set
(*** Conj_minus_SNo_Lev_lem1__3__1 TMTcdQMXjPauR7ukwDWvXgM1CQD5RfWRvMK bounty of about 25 bars ***)
L2189
Variable y : set
L2190
Variable z : set
L2191
Variable w : set
L2192
Hypothesis H0 : ordinal (SNoLev x)
L2193
Hypothesis H2 : z = - w
L2194
Hypothesis H3 : SNoLev w ∈ SNoLev x
L2195
Hypothesis H4 : Subq (SNoLev (- w)) (SNoLev w)
L2196
Hypothesis H5 : ordinal (SNoLev (- w))
L2197
Theorem. (Conj_minus_SNo_Lev_lem1__3__1)
ordinal (ordsucc (SNoLev z)) β†’ y ∈ SNoLev x
In Proofgold the corresponding term root is d42ebf... and proposition id is 3489a1...
Proof:
Proof not loaded.
End of Section Conj_minus_SNo_Lev_lem1__3__1
Beginning of Section Conj_minus_SNo_Lev_lem1__3__3
L2203
Variable x : set
(*** Conj_minus_SNo_Lev_lem1__3__3 TMGKC3265aZ5x4ucFBVkQ6VSH6Lf3yW87sQ bounty of about 25 bars ***)
L2204
Variable y : set
L2205
Variable z : set
L2206
Variable w : set
L2207
Hypothesis H0 : ordinal (SNoLev x)
L2208
Hypothesis H1 : y ∈ ordsucc (SNoLev z)
L2209
Hypothesis H2 : z = - w
L2210
Hypothesis H4 : Subq (SNoLev (- w)) (SNoLev w)
L2211
Hypothesis H5 : ordinal (SNoLev (- w))
L2212
Theorem. (Conj_minus_SNo_Lev_lem1__3__3)
ordinal (ordsucc (SNoLev z)) β†’ y ∈ SNoLev x
In Proofgold the corresponding term root is 07ccff... and proposition id is f77619...
Proof:
Proof not loaded.
End of Section Conj_minus_SNo_Lev_lem1__3__3
Beginning of Section Conj_minus_SNo_Lev_lem1__3__5
L2218
Variable x : set
(*** Conj_minus_SNo_Lev_lem1__3__5 TMMiWkkRySxd5enup94kHp8QmguThngroFh bounty of about 25 bars ***)
L2219
Variable y : set
L2220
Variable z : set
L2221
Variable w : set
L2222
Hypothesis H0 : ordinal (SNoLev x)
L2223
Hypothesis H1 : y ∈ ordsucc (SNoLev z)
L2224
Hypothesis H2 : z = - w
L2225
Hypothesis H3 : SNoLev w ∈ SNoLev x
L2226
Hypothesis H4 : Subq (SNoLev (- w)) (SNoLev w)
L2227
Theorem. (Conj_minus_SNo_Lev_lem1__3__5)
ordinal (ordsucc (SNoLev z)) β†’ y ∈ SNoLev x
In Proofgold the corresponding term root is 390b65... and proposition id is 39e564...
Proof:
Proof not loaded.
End of Section Conj_minus_SNo_Lev_lem1__3__5
Beginning of Section Conj_minus_SNo_Lev_lem1__4__5
L2233
Variable x : set
(*** Conj_minus_SNo_Lev_lem1__4__5 TMRUHMoq6t53Y4jjAmfAhBWst6aAzq3dPTJ bounty of about 25 bars ***)
L2234
Variable y : set
L2235
Variable z : set
L2236
Variable w : set
L2237
Hypothesis H0 : ordinal (SNoLev x)
L2238
Hypothesis H1 : y ∈ ordsucc (SNoLev z)
L2239
Hypothesis H2 : z = - w
L2240
Hypothesis H3 : SNoLev w ∈ SNoLev x
L2241
Hypothesis H4 : Subq (SNoLev (- w)) (SNoLev w)
L2242
Theorem. (Conj_minus_SNo_Lev_lem1__4__5)
ordinal (SNoLev (- w)) β†’ y ∈ SNoLev x
In Proofgold the corresponding term root is 6a63ca... and proposition id is 348a3b...
Proof:
Proof not loaded.
End of Section Conj_minus_SNo_Lev_lem1__4__5
Beginning of Section Conj_minus_SNo_Lev_lem1__6__0
L2248
Variable x : set
(*** Conj_minus_SNo_Lev_lem1__6__0 TMZeWUTfWrms4y6Mkb9RwkpPMAGKzAPahBB bounty of about 25 bars ***)
L2249
Variable y : set
L2250
Variable z : set
L2251
Variable w : set
L2252
Hypothesis H1 : y ∈ ordsucc (SNoLev z)
L2253
Hypothesis H2 : z = - w
L2254
Hypothesis H3 : SNo w
L2255
Hypothesis H4 : SNoLev w ∈ SNoLev x
L2256
Hypothesis H5 : Subq (SNoLev (- w)) (SNoLev w)
L2257
Theorem. (Conj_minus_SNo_Lev_lem1__6__0)
ordinal (SNoLev w) β†’ y ∈ SNoLev x
In Proofgold the corresponding term root is b7ded1... and proposition id is 066f94...
Proof:
Proof not loaded.
End of Section Conj_minus_SNo_Lev_lem1__6__0
Beginning of Section Conj_minus_SNo_Lev_lem1__6__4
L2263
Variable x : set
(*** Conj_minus_SNo_Lev_lem1__6__4 TMTi4TJu6UYp4Rdsc3HT5DJUXUGHZgEX1ad bounty of about 25 bars ***)
L2264
Variable y : set
L2265
Variable z : set
L2266
Variable w : set
L2267
Hypothesis H0 : ordinal (SNoLev x)
L2268
Hypothesis H1 : y ∈ ordsucc (SNoLev z)
L2269
Hypothesis H2 : z = - w
L2270
Hypothesis H3 : SNo w
L2271
Hypothesis H5 : Subq (SNoLev (- w)) (SNoLev w)
L2272
Theorem. (Conj_minus_SNo_Lev_lem1__6__4)
ordinal (SNoLev w) β†’ y ∈ SNoLev x
In Proofgold the corresponding term root is 90ad34... and proposition id is 85401b...
Proof:
Proof not loaded.
End of Section Conj_minus_SNo_Lev_lem1__6__4
Beginning of Section Conj_minus_SNo_Lev_lem1__7__1
L2278
Variable x : set
(*** Conj_minus_SNo_Lev_lem1__7__1 TMNmDQRRPPFEkN5jEiHbXYwEFqRQ9pJKxNT bounty of about 25 bars ***)
L2279
Variable y : set
L2280
Variable z : set
L2281
Variable w : set
L2282
Variable u : set
L2283
Hypothesis H0 : (βˆ€v : set, v ∈ x β†’ (βˆ€x2 : set, x2 ∈ SNoS_ v β†’ Subq (SNoLev (- x2)) (SNoLev x2)))
L2284
Hypothesis H2 : z ∈ ordsucc (SNoLev w)
L2285
Hypothesis H3 : w = - u
L2286
Hypothesis H4 : SNo u
L2287
Hypothesis H5 : SNoLev u ∈ SNoLev y
L2288
Hypothesis H6 : u ∈ SNoS_ (ordsucc (SNoLev u))
L2289
Hypothesis H7 : ordsucc (SNoLev u) ∈ x
L2290
Theorem. (Conj_minus_SNo_Lev_lem1__7__1)
Subq (SNoLev (- u)) (SNoLev u) β†’ z ∈ SNoLev y
In Proofgold the corresponding term root is b7a42e... and proposition id is 43b48c...
Proof:
Proof not loaded.
End of Section Conj_minus_SNo_Lev_lem1__7__1
Beginning of Section Conj_minus_SNo_Lev_lem1__10__2
L2296
Variable x : set
(*** Conj_minus_SNo_Lev_lem1__10__2 TMbz4EZRNA5qWZytgvu5NQSdFz3cs5xQT4X bounty of about 25 bars ***)
L2297
Variable y : set
L2298
Variable z : set
L2299
Variable w : set
L2300
Hypothesis H0 : y ∈ ordsucc (SNoLev z)
L2301
Hypothesis H1 : z = - w
L2302
Hypothesis H3 : Subq (SNoLev (- w)) (SNoLev w)
L2303
Hypothesis H4 : Subq (SNoLev x) y
L2304
Theorem. (Conj_minus_SNo_Lev_lem1__10__2)
¬ SNoLev w ∈ y
In Proofgold the corresponding term root is 9113ae... and proposition id is 658123...
Proof:
Proof not loaded.
End of Section Conj_minus_SNo_Lev_lem1__10__2
Beginning of Section Conj_minus_SNo_Lev_lem1__12__1
L2310
Variable x : set
(*** Conj_minus_SNo_Lev_lem1__12__1 TMTcdQMXjPauR7ukwDWvXgM1CQD5RfWRvMK bounty of about 25 bars ***)
L2311
Variable y : set
L2312
Variable z : set
L2313
Variable w : set
L2314
Hypothesis H0 : ordinal (SNoLev x)
L2315
Hypothesis H2 : z = - w
L2316
Hypothesis H3 : SNoLev w ∈ SNoLev x
L2317
Hypothesis H4 : Subq (SNoLev (- w)) (SNoLev w)
L2318
Hypothesis H5 : ordinal (SNoLev (- w))
L2319
Theorem. (Conj_minus_SNo_Lev_lem1__12__1)
ordinal (ordsucc (SNoLev z)) β†’ y ∈ SNoLev x
In Proofgold the corresponding term root is d42ebf... and proposition id is 3489a1...
Proof:
Proof not loaded.
End of Section Conj_minus_SNo_Lev_lem1__12__1
Beginning of Section Conj_minus_SNo_Lev_lem1__12__3
L2325
Variable x : set
(*** Conj_minus_SNo_Lev_lem1__12__3 TMGKC3265aZ5x4ucFBVkQ6VSH6Lf3yW87sQ bounty of about 25 bars ***)
L2326
Variable y : set
L2327
Variable z : set
L2328
Variable w : set
L2329
Hypothesis H0 : ordinal (SNoLev x)
L2330
Hypothesis H1 : y ∈ ordsucc (SNoLev z)
L2331
Hypothesis H2 : z = - w
L2332
Hypothesis H4 : Subq (SNoLev (- w)) (SNoLev w)
L2333
Hypothesis H5 : ordinal (SNoLev (- w))
L2334
Theorem. (Conj_minus_SNo_Lev_lem1__12__3)
ordinal (ordsucc (SNoLev z)) β†’ y ∈ SNoLev x
In Proofgold the corresponding term root is 07ccff... and proposition id is f77619...
Proof:
Proof not loaded.
End of Section Conj_minus_SNo_Lev_lem1__12__3
Beginning of Section Conj_minus_SNo_Lev_lem1__12__5
L2340
Variable x : set
(*** Conj_minus_SNo_Lev_lem1__12__5 TMMiWkkRySxd5enup94kHp8QmguThngroFh bounty of about 25 bars ***)
L2341
Variable y : set
L2342
Variable z : set
L2343
Variable w : set
L2344
Hypothesis H0 : ordinal (SNoLev x)
L2345
Hypothesis H1 : y ∈ ordsucc (SNoLev z)
L2346
Hypothesis H2 : z = - w
L2347
Hypothesis H3 : SNoLev w ∈ SNoLev x
L2348
Hypothesis H4 : Subq (SNoLev (- w)) (SNoLev w)
L2349
Theorem. (Conj_minus_SNo_Lev_lem1__12__5)
ordinal (ordsucc (SNoLev z)) β†’ y ∈ SNoLev x
In Proofgold the corresponding term root is 390b65... and proposition id is 39e564...
Proof:
Proof not loaded.
End of Section Conj_minus_SNo_Lev_lem1__12__5
Beginning of Section Conj_minus_SNo_Lev_lem1__13__5
L2355
Variable x : set
(*** Conj_minus_SNo_Lev_lem1__13__5 TMRUHMoq6t53Y4jjAmfAhBWst6aAzq3dPTJ bounty of about 25 bars ***)
L2356
Variable y : set
L2357
Variable z : set
L2358
Variable w : set
L2359
Hypothesis H0 : ordinal (SNoLev x)
L2360
Hypothesis H1 : y ∈ ordsucc (SNoLev z)
L2361
Hypothesis H2 : z = - w
L2362
Hypothesis H3 : SNoLev w ∈ SNoLev x
L2363
Hypothesis H4 : Subq (SNoLev (- w)) (SNoLev w)
L2364
Theorem. (Conj_minus_SNo_Lev_lem1__13__5)
ordinal (SNoLev (- w)) β†’ y ∈ SNoLev x
In Proofgold the corresponding term root is 6a63ca... and proposition id is 348a3b...
Proof:
Proof not loaded.
End of Section Conj_minus_SNo_Lev_lem1__13__5
Beginning of Section Conj_minus_SNo_Lev_lem1__15__0
L2370
Variable x : set
(*** Conj_minus_SNo_Lev_lem1__15__0 TMZeWUTfWrms4y6Mkb9RwkpPMAGKzAPahBB bounty of about 25 bars ***)
L2371
Variable y : set
L2372
Variable z : set
L2373
Variable w : set
L2374
Hypothesis H1 : y ∈ ordsucc (SNoLev z)
L2375
Hypothesis H2 : z = - w
L2376
Hypothesis H3 : SNo w
L2377
Hypothesis H4 : SNoLev w ∈ SNoLev x
L2378
Hypothesis H5 : Subq (SNoLev (- w)) (SNoLev w)
L2379
Theorem. (Conj_minus_SNo_Lev_lem1__15__0)
ordinal (SNoLev w) β†’ y ∈ SNoLev x
In Proofgold the corresponding term root is b7ded1... and proposition id is 066f94...
Proof:
Proof not loaded.
End of Section Conj_minus_SNo_Lev_lem1__15__0
Beginning of Section Conj_minus_SNo_Lev_lem1__15__4
L2385
Variable x : set
(*** Conj_minus_SNo_Lev_lem1__15__4 TMTi4TJu6UYp4Rdsc3HT5DJUXUGHZgEX1ad bounty of about 25 bars ***)
L2386
Variable y : set
L2387
Variable z : set
L2388
Variable w : set
L2389
Hypothesis H0 : ordinal (SNoLev x)
L2390
Hypothesis H1 : y ∈ ordsucc (SNoLev z)
L2391
Hypothesis H2 : z = - w
L2392
Hypothesis H3 : SNo w
L2393
Hypothesis H5 : Subq (SNoLev (- w)) (SNoLev w)
L2394
Theorem. (Conj_minus_SNo_Lev_lem1__15__4)
ordinal (SNoLev w) β†’ y ∈ SNoLev x
In Proofgold the corresponding term root is 90ad34... and proposition id is 85401b...
Proof:
Proof not loaded.
End of Section Conj_minus_SNo_Lev_lem1__15__4
Beginning of Section Conj_minus_SNo_Lev_lem1__16__1
L2400
Variable x : set
(*** Conj_minus_SNo_Lev_lem1__16__1 TMNmDQRRPPFEkN5jEiHbXYwEFqRQ9pJKxNT bounty of about 25 bars ***)
L2401
Variable y : set
L2402
Variable z : set
L2403
Variable w : set
L2404
Variable u : set
L2405
Hypothesis H0 : (βˆ€v : set, v ∈ x β†’ (βˆ€x2 : set, x2 ∈ SNoS_ v β†’ Subq (SNoLev (- x2)) (SNoLev x2)))
L2406
Hypothesis H2 : z ∈ ordsucc (SNoLev w)
L2407
Hypothesis H3 : w = - u
L2408
Hypothesis H4 : SNo u
L2409
Hypothesis H5 : SNoLev u ∈ SNoLev y
L2410
Hypothesis H6 : u ∈ SNoS_ (ordsucc (SNoLev u))
L2411
Hypothesis H7 : ordsucc (SNoLev u) ∈ x
L2412
Theorem. (Conj_minus_SNo_Lev_lem1__16__1)
Subq (SNoLev (- u)) (SNoLev u) β†’ z ∈ SNoLev y
In Proofgold the corresponding term root is b7a42e... and proposition id is 43b48c...
Proof:
Proof not loaded.
End of Section Conj_minus_SNo_Lev_lem1__16__1
Beginning of Section Conj_minus_SNo_Lev_lem1__22__2
L2418
Variable x : set
(*** Conj_minus_SNo_Lev_lem1__22__2 TMU48qLBFhsvPGcawgwznT1u9bcVtEjztaz bounty of about 25 bars ***)
L2419
Variable y : set
L2420
Hypothesis H0 : TransSet x
L2421
Hypothesis H1 : (βˆ€z : set, z ∈ x β†’ (βˆ€w : set, w ∈ SNoS_ z β†’ Subq (SNoLev (- w)) (SNoLev w)))
L2422
Hypothesis H3 : ordinal (SNoLev y)
L2423
Hypothesis H4 : SNo y
L2424
Theorem. (Conj_minus_SNo_Lev_lem1__22__2)
SNoCutP (Repl (SNoR y) minus_SNo) (Repl (SNoL y) minus_SNo) β†’ Subq (SNoLev (- y)) (SNoLev y)
In Proofgold the corresponding term root is 7239d9... and proposition id is 49f1d3...
Proof:
Proof not loaded.
End of Section Conj_minus_SNo_Lev_lem1__22__2
Beginning of Section Conj_minus_SNo_invol__5__6
L2430
Variable x : set
(*** Conj_minus_SNo_invol__5__6 TMJwhNkd4azSgkPG1XNNhfDTvh9J7Bk89Kk bounty of about 25 bars ***)
L2431
Variable y : set
L2432
Hypothesis H0 : SNoCutP x y
L2433
Hypothesis H1 : (βˆ€z : set, z ∈ x β†’ - (- z) = z)
L2434
Hypothesis H2 : (βˆ€z : set, z ∈ y β†’ - (- z) = z)
L2435
Hypothesis H3 : (βˆ€z : set, z ∈ x β†’ SNo z)
L2436
Hypothesis H4 : (βˆ€z : set, z ∈ y β†’ SNo z)
L2437
Hypothesis H5 : SNo (SNoCut x y)
L2438
Hypothesis H7 : SNo (- (- (SNoCut x y)))
L2439
Theorem. (Conj_minus_SNo_invol__5__6)
Subq (SNoLev (SNoCut x y)) (SNoLev (- (- (SNoCut x y)))) ∧ SNoEq_ (SNoLev (SNoCut x y)) (SNoCut x y) (- (- (SNoCut x y))) β†’ - (- (SNoCut x y)) = SNoCut x y
In Proofgold the corresponding term root is 6ebb6d... and proposition id is 59be3a...
Proof:
Proof not loaded.
End of Section Conj_minus_SNo_invol__5__6
Beginning of Section Conj_minus_SNo_invol__8__0
L2445
Variable x : set
(*** Conj_minus_SNo_invol__8__0 TMHW2CxWY88b3KHixxku1tPi4ij2bKsnqZg bounty of about 25 bars ***)
L2446
Variable y : set
L2447
Hypothesis H1 : (βˆ€z : set, z ∈ x β†’ - (- z) = z)
L2448
Hypothesis H2 : (βˆ€z : set, z ∈ y β†’ - (- z) = z)
L2449
Hypothesis H3 : (βˆ€z : set, z ∈ x β†’ SNo z)
L2450
Hypothesis H4 : (βˆ€z : set, z ∈ y β†’ SNo z)
L2451
Theorem. (Conj_minus_SNo_invol__8__0)
SNo (SNoCut x y) β†’ - (- (SNoCut x y)) = SNoCut x y
In Proofgold the corresponding term root is 138f41... and proposition id is 687fa7...
Proof:
Proof not loaded.
End of Section Conj_minus_SNo_invol__8__0
Beginning of Section Conj_minus_SNo_invol__8__2
L2457
Variable x : set
(*** Conj_minus_SNo_invol__8__2 TMXsedNVbmFjSwfHzrhgczVZTRQVZfS1BLt bounty of about 25 bars ***)
L2458
Variable y : set
L2459
Hypothesis H0 : SNoCutP x y
L2460
Hypothesis H1 : (βˆ€z : set, z ∈ x β†’ - (- z) = z)
L2461
Hypothesis H3 : (βˆ€z : set, z ∈ x β†’ SNo z)
L2462
Hypothesis H4 : (βˆ€z : set, z ∈ y β†’ SNo z)
L2463
Theorem. (Conj_minus_SNo_invol__8__2)
SNo (SNoCut x y) β†’ - (- (SNoCut x y)) = SNoCut x y
In Proofgold the corresponding term root is 1195c2... and proposition id is a064b7...
Proof:
Proof not loaded.
End of Section Conj_minus_SNo_invol__8__2
Beginning of Section Conj_minus_SNoCut_eq_lem__5__2
L2469
Variable x : set
(*** Conj_minus_SNoCut_eq_lem__5__2 TMJY5ReEfidqQ4c388Zwk6gKtPa5ataYPgZ bounty of about 25 bars ***)
L2470
Variable y : set
L2471
Variable z : set
L2472
Hypothesis H0 : SNo x
L2473
Hypothesis H1 : (βˆ€w : set, w ∈ SNoS_ (SNoLev x) β†’ (βˆ€u : set, βˆ€v : set, SNoCutP u v β†’ w = SNoCut u v β†’ - w = SNoCut (Repl v minus_SNo) (Repl u minus_SNo)))
L2474
Hypothesis H3 : (βˆ€w : set, w ∈ z β†’ SNo w)
L2475
Hypothesis H4 : x = SNoCut y z
L2476
Hypothesis H5 : SNoCutP (Repl z minus_SNo) (Repl y minus_SNo)
L2477
Hypothesis H6 : SNo (SNoCut (Repl z minus_SNo) (Repl y minus_SNo))
L2478
Hypothesis H7 : SNoLev (SNoCut (Repl z (Ξ»w : set β‡’ - w)) (Repl y (Ξ»w : set β‡’ - w))) ∈ SNoLev (- x)
L2479
Theorem. (Conj_minus_SNoCut_eq_lem__5__2)
¬ SNoCut (Repl z minus_SNo) (Repl y minus_SNo) ∈ SNoS_ (SNoLev x)
In Proofgold the corresponding term root is 068a0e... and proposition id is b7b604...
Proof:
Proof not loaded.
End of Section Conj_minus_SNoCut_eq_lem__5__2
Beginning of Section Conj_minus_SNoCut_eq_lem__6__2
L2485
Variable x : set
(*** Conj_minus_SNoCut_eq_lem__6__2 TMQs6FJxebzQkiULnjLJv2kNcZzc1tm3Gpp bounty of about 25 bars ***)
L2486
Variable y : set
L2487
Variable z : set
L2488
Hypothesis H0 : SNo x
L2489
Hypothesis H1 : (βˆ€w : set, w ∈ SNoS_ (SNoLev x) β†’ (βˆ€u : set, βˆ€v : set, SNoCutP u v β†’ w = SNoCut u v β†’ - w = SNoCut (Repl v minus_SNo) (Repl u minus_SNo)))
L2490
Hypothesis H3 : (βˆ€w : set, w ∈ z β†’ SNo w)
L2491
Hypothesis H4 : x = SNoCut y z
L2492
Hypothesis H5 : SNoCutP (Repl z minus_SNo) (Repl y minus_SNo)
L2493
Hypothesis H6 : SNo (SNoCut (Repl z minus_SNo) (Repl y minus_SNo))
L2494
Hypothesis H7 : Subq (SNoLev (SNoCut (Repl z (Ξ»w : set β‡’ - w)) (Repl y (Ξ»w : set β‡’ - w)))) (SNoLev (- x))
L2495
Hypothesis H8 : SNoEq_ (SNoLev (SNoCut (Repl z (Ξ»w : set β‡’ - w)) (Repl y (Ξ»w : set β‡’ - w)))) (SNoCut (Repl z (Ξ»w : set β‡’ - w)) (Repl y (Ξ»w : set β‡’ - w))) (- x)
L2496
Hypothesis H9 : ordinal (SNoLev (SNoCut (Repl z minus_SNo) (Repl y minus_SNo)))
L2497
Theorem. (Conj_minus_SNoCut_eq_lem__6__2)
ordinal (SNoLev (- x)) β†’ - x = SNoCut (Repl z minus_SNo) (Repl y minus_SNo)
In Proofgold the corresponding term root is baab8a... and proposition id is 9b6ecd...
Proof:
Proof not loaded.
End of Section Conj_minus_SNoCut_eq_lem__6__2
Beginning of Section Conj_minus_SNoCut_eq_lem__6__9
L2503
Variable x : set
(*** Conj_minus_SNoCut_eq_lem__6__9 TMNSN1otPMhzUcCre292qvZS7oWjpSniFJR bounty of about 25 bars ***)
L2504
Variable y : set
L2505
Variable z : set
L2506
Hypothesis H0 : SNo x
L2507
Hypothesis H1 : (βˆ€w : set, w ∈ SNoS_ (SNoLev x) β†’ (βˆ€u : set, βˆ€v : set, SNoCutP u v β†’ w = SNoCut u v β†’ - w = SNoCut (Repl v minus_SNo) (Repl u minus_SNo)))
L2508
Hypothesis H2 : (βˆ€w : set, w ∈ y β†’ SNo w)
L2509
Hypothesis H3 : (βˆ€w : set, w ∈ z β†’ SNo w)
L2510
Hypothesis H4 : x = SNoCut y z
L2511
Hypothesis H5 : SNoCutP (Repl z minus_SNo) (Repl y minus_SNo)
L2512
Hypothesis H6 : SNo (SNoCut (Repl z minus_SNo) (Repl y minus_SNo))
L2513
Hypothesis H7 : Subq (SNoLev (SNoCut (Repl z (Ξ»w : set β‡’ - w)) (Repl y (Ξ»w : set β‡’ - w)))) (SNoLev (- x))
L2514
Hypothesis H8 : SNoEq_ (SNoLev (SNoCut (Repl z (Ξ»w : set β‡’ - w)) (Repl y (Ξ»w : set β‡’ - w)))) (SNoCut (Repl z (Ξ»w : set β‡’ - w)) (Repl y (Ξ»w : set β‡’ - w))) (- x)
L2515
Theorem. (Conj_minus_SNoCut_eq_lem__6__9)
ordinal (SNoLev (- x)) β†’ - x = SNoCut (Repl z minus_SNo) (Repl y minus_SNo)
In Proofgold the corresponding term root is 9c762b... and proposition id is c30e63...
Proof:
Proof not loaded.
End of Section Conj_minus_SNoCut_eq_lem__6__9
Beginning of Section Conj_minus_SNoCut_eq_lem__7__5
L2521
Variable x : set
(*** Conj_minus_SNoCut_eq_lem__7__5 TMUYD4A4JwUUvGwfXE2qtXXpxcVNTrsFbcD bounty of about 25 bars ***)
L2522
Variable y : set
L2523
Variable z : set
L2524
Hypothesis H0 : SNo x
L2525
Hypothesis H1 : (βˆ€w : set, w ∈ SNoS_ (SNoLev x) β†’ (βˆ€u : set, βˆ€v : set, SNoCutP u v β†’ w = SNoCut u v β†’ - w = SNoCut (Repl v minus_SNo) (Repl u minus_SNo)))
L2526
Hypothesis H2 : (βˆ€w : set, w ∈ y β†’ SNo w)
L2527
Hypothesis H3 : (βˆ€w : set, w ∈ z β†’ SNo w)
L2528
Hypothesis H4 : x = SNoCut y z
L2529
Hypothesis H6 : SNo (SNoCut (Repl z minus_SNo) (Repl y minus_SNo))
L2530
Hypothesis H7 : Subq (SNoLev (SNoCut (Repl z (Ξ»w : set β‡’ - w)) (Repl y (Ξ»w : set β‡’ - w)))) (SNoLev (- x))
L2531
Hypothesis H8 : SNoEq_ (SNoLev (SNoCut (Repl z (Ξ»w : set β‡’ - w)) (Repl y (Ξ»w : set β‡’ - w)))) (SNoCut (Repl z (Ξ»w : set β‡’ - w)) (Repl y (Ξ»w : set β‡’ - w))) (- x)
L2532
Theorem. (Conj_minus_SNoCut_eq_lem__7__5)
ordinal (SNoLev (SNoCut (Repl z minus_SNo) (Repl y minus_SNo))) β†’ - x = SNoCut (Repl z minus_SNo) (Repl y minus_SNo)
In Proofgold the corresponding term root is 5304c2... and proposition id is 1bc03b...
Proof:
Proof not loaded.
End of Section Conj_minus_SNoCut_eq_lem__7__5
Beginning of Section Conj_minus_SNoCut_eq_lem__8__3
L2538
Variable x : set
(*** Conj_minus_SNoCut_eq_lem__8__3 TMX2KpJVpuZxNFRvX767d4WM8rsqHBM25nQ bounty of about 25 bars ***)
L2539
Variable y : set
L2540
Variable z : set
L2541
Hypothesis H0 : SNo x
L2542
Hypothesis H1 : (βˆ€w : set, w ∈ SNoS_ (SNoLev x) β†’ (βˆ€u : set, βˆ€v : set, SNoCutP u v β†’ w = SNoCut u v β†’ - w = SNoCut (Repl v minus_SNo) (Repl u minus_SNo)))
L2543
Hypothesis H2 : SNoCutP y z
L2544
Hypothesis H4 : (βˆ€w : set, w ∈ z β†’ SNo w)
L2545
Hypothesis H5 : x = SNoCut y z
L2546
Hypothesis H6 : SNoCutP (Repl z minus_SNo) (Repl y minus_SNo)
L2547
Hypothesis H7 : SNo (SNoCut (Repl z minus_SNo) (Repl y minus_SNo))
L2548
Hypothesis H8 : (βˆ€w : set, SNo w β†’ (βˆ€u : set, u ∈ Repl z minus_SNo β†’ u < w) β†’ (βˆ€u : set, u ∈ Repl y minus_SNo β†’ w < u) β†’ Subq (SNoLev (SNoCut (Repl z minus_SNo) (Repl y minus_SNo))) (SNoLev w) ∧ SNoEq_ (SNoLev (SNoCut (Repl z minus_SNo) (Repl y minus_SNo))) (SNoCut (Repl z minus_SNo) (Repl y minus_SNo)) w)
L2549
Hypothesis H9 : (βˆ€w : set, w ∈ Repl z minus_SNo β†’ w < - x)
L2550
Theorem. (Conj_minus_SNoCut_eq_lem__8__3)
(βˆ€w : set, w ∈ Repl y minus_SNo β†’ - x < w) β†’ - x = SNoCut (Repl z minus_SNo) (Repl y minus_SNo)
In Proofgold the corresponding term root is 34872f... and proposition id is da648d...
Proof:
Proof not loaded.
End of Section Conj_minus_SNoCut_eq_lem__8__3
Beginning of Section Conj_minus_SNoCut_eq_lem__11__3
L2556
Variable x : set
(*** Conj_minus_SNoCut_eq_lem__11__3 TMKGQeB1XUrhM9NfGuRKuUiMi2HJpfwBiic bounty of about 25 bars ***)
L2557
Variable y : set
L2558
Variable z : set
L2559
Hypothesis H0 : SNo x
L2560
Hypothesis H1 : (βˆ€w : set, w ∈ SNoS_ (SNoLev x) β†’ (βˆ€u : set, βˆ€v : set, SNoCutP u v β†’ w = SNoCut u v β†’ - w = SNoCut (Repl v minus_SNo) (Repl u minus_SNo)))
L2561
Hypothesis H2 : SNoCutP y z
L2562
Hypothesis H4 : (βˆ€w : set, w ∈ z β†’ SNo w)
L2563
Hypothesis H5 : x = SNoCut y z
L2564
Theorem. (Conj_minus_SNoCut_eq_lem__11__3)
SNo (SNoCut y z) β†’ - x = SNoCut (Repl z minus_SNo) (Repl y minus_SNo)
In Proofgold the corresponding term root is 30015a... and proposition id is 9d1d72...
Proof:
Proof not loaded.
End of Section Conj_minus_SNoCut_eq_lem__11__3
Beginning of Section Conj_minus_SNoCut_eq_lem__11__5
L2570
Variable x : set
(*** Conj_minus_SNoCut_eq_lem__11__5 TMbRofhjHsY9yRSNkXz96bUsSSPJxZGpbt4 bounty of about 25 bars ***)
L2571
Variable y : set
L2572
Variable z : set
L2573
Hypothesis H0 : SNo x
L2574
Hypothesis H1 : (βˆ€w : set, w ∈ SNoS_ (SNoLev x) β†’ (βˆ€u : set, βˆ€v : set, SNoCutP u v β†’ w = SNoCut u v β†’ - w = SNoCut (Repl v minus_SNo) (Repl u minus_SNo)))
L2575
Hypothesis H2 : SNoCutP y z
L2576
Hypothesis H3 : (βˆ€w : set, w ∈ y β†’ SNo w)
L2577
Hypothesis H4 : (βˆ€w : set, w ∈ z β†’ SNo w)
L2578
Theorem. (Conj_minus_SNoCut_eq_lem__11__5)
SNo (SNoCut y z) β†’ - x = SNoCut (Repl z minus_SNo) (Repl y minus_SNo)
In Proofgold the corresponding term root is cab5bc... and proposition id is 2e22cf...
Proof:
Proof not loaded.
End of Section Conj_minus_SNoCut_eq_lem__11__5
Beginning of Section Conj_add_SNo_prop1__1__1
L2584
Variable x : set
(*** Conj_add_SNo_prop1__1__1 TMQbpF2vyGvRi3eTTudZXQqybuv9mEMX1Nf bounty of about 25 bars ***)
L2585
Variable y : set
L2586
Variable z : set
L2587
Hypothesis H0 : SNo x
L2588
Hypothesis H2 : SNo z
L2589
Hypothesis H3 : SNoLev z ∈ SNoLev x
L2590
Theorem. (Conj_add_SNo_prop1__1__1)
z ∈ SNoS_ (SNoLev x) β†’ SNo (z + y) ∧ (βˆ€w : set, w ∈ SNoL z β†’ (w + y) < z + y) ∧ (βˆ€w : set, w ∈ SNoR z β†’ (z + y) < w + y) ∧ (βˆ€w : set, w ∈ SNoL y β†’ (z + w) < z + y) ∧ (βˆ€w : set, w ∈ SNoR y β†’ (z + y) < z + w) ∧ SNoCutP (binunion (Repl (SNoL z) (Ξ»w : set β‡’ w + y)) (Repl (SNoL y) (add_SNo z))) (binunion (Repl (SNoR z) (Ξ»w : set β‡’ w + y)) (Repl (SNoR y) (add_SNo z)))
In Proofgold the corresponding term root is 6e0d02... and proposition id is 7b0caf...
Proof:
Proof not loaded.
End of Section Conj_add_SNo_prop1__1__1
Beginning of Section Conj_add_SNo_prop1__2__1
L2596
Variable x : set
(*** Conj_add_SNo_prop1__2__1 TMQbpF2vyGvRi3eTTudZXQqybuv9mEMX1Nf bounty of about 25 bars ***)
L2597
Variable y : set
L2598
Variable z : set
L2599
Hypothesis H0 : SNo x
L2600
Hypothesis H2 : SNo z
L2601
Hypothesis H3 : SNoLev z ∈ SNoLev x
L2602
Theorem. (Conj_add_SNo_prop1__2__1)
z ∈ SNoS_ (SNoLev x) β†’ SNo (z + y) ∧ (βˆ€w : set, w ∈ SNoL z β†’ (w + y) < z + y) ∧ (βˆ€w : set, w ∈ SNoR z β†’ (z + y) < w + y) ∧ (βˆ€w : set, w ∈ SNoL y β†’ (z + w) < z + y) ∧ (βˆ€w : set, w ∈ SNoR y β†’ (z + y) < z + w) ∧ SNoCutP (binunion (Repl (SNoL z) (Ξ»w : set β‡’ w + y)) (Repl (SNoL y) (add_SNo z))) (binunion (Repl (SNoR z) (Ξ»w : set β‡’ w + y)) (Repl (SNoR y) (add_SNo z)))
In Proofgold the corresponding term root is 6e0d02... and proposition id is 7b0caf...
Proof:
Proof not loaded.
End of Section Conj_add_SNo_prop1__2__1
Beginning of Section Conj_add_SNo_prop1__3__2
L2608
Variable x : set
(*** Conj_add_SNo_prop1__3__2 TMHM1qpBJnAKvuTu4rk82JYirdUm9tPbxMS bounty of about 25 bars ***)
L2609
Variable y : set
L2610
Variable z : set
L2611
Hypothesis H0 : SNo y
L2612
Hypothesis H1 : (βˆ€w : set, w ∈ SNoS_ (SNoLev y) β†’ SNo (x + w) ∧ (βˆ€u : set, u ∈ SNoL x β†’ (u + w) < x + w) ∧ (βˆ€u : set, u ∈ SNoR x β†’ (x + w) < u + w) ∧ (βˆ€u : set, u ∈ SNoL w β†’ (x + u) < x + w) ∧ (βˆ€u : set, u ∈ SNoR w β†’ (x + w) < x + u) ∧ SNoCutP (binunion (Repl (SNoL x) (Ξ»u : set β‡’ u + w)) (Repl (SNoL w) (add_SNo x))) (binunion (Repl (SNoR x) (Ξ»u : set β‡’ u + w)) (Repl (SNoR w) (add_SNo x))))
L2613
Hypothesis H3 : SNoLev z ∈ SNoLev y
L2614
Theorem. (Conj_add_SNo_prop1__3__2)
z ∈ SNoS_ (SNoLev y) β†’ SNo (x + z) ∧ (βˆ€w : set, w ∈ SNoL x β†’ (w + z) < x + z) ∧ (βˆ€w : set, w ∈ SNoR x β†’ (x + z) < w + z) ∧ (βˆ€w : set, w ∈ SNoL z β†’ (x + w) < x + z) ∧ (βˆ€w : set, w ∈ SNoR z β†’ (x + z) < x + w) ∧ SNoCutP (binunion (Repl (SNoL x) (Ξ»w : set β‡’ w + z)) (Repl (SNoL z) (add_SNo x))) (binunion (Repl (SNoR x) (Ξ»w : set β‡’ w + z)) (Repl (SNoR z) (add_SNo x)))
In Proofgold the corresponding term root is 9bef0a... and proposition id is 5ece17...
Proof:
Proof not loaded.
End of Section Conj_add_SNo_prop1__3__2
Beginning of Section Conj_add_SNo_prop1__4__2
L2620
Variable x : set
(*** Conj_add_SNo_prop1__4__2 TMHM1qpBJnAKvuTu4rk82JYirdUm9tPbxMS bounty of about 25 bars ***)
L2621
Variable y : set
L2622
Variable z : set
L2623
Hypothesis H0 : SNo y
L2624
Hypothesis H1 : (βˆ€w : set, w ∈ SNoS_ (SNoLev y) β†’ SNo (x + w) ∧ (βˆ€u : set, u ∈ SNoL x β†’ (u + w) < x + w) ∧ (βˆ€u : set, u ∈ SNoR x β†’ (x + w) < u + w) ∧ (βˆ€u : set, u ∈ SNoL w β†’ (x + u) < x + w) ∧ (βˆ€u : set, u ∈ SNoR w β†’ (x + w) < x + u) ∧ SNoCutP (binunion (Repl (SNoL x) (Ξ»u : set β‡’ u + w)) (Repl (SNoL w) (add_SNo x))) (binunion (Repl (SNoR x) (Ξ»u : set β‡’ u + w)) (Repl (SNoR w) (add_SNo x))))
L2625
Hypothesis H3 : SNoLev z ∈ SNoLev y
L2626
Theorem. (Conj_add_SNo_prop1__4__2)
z ∈ SNoS_ (SNoLev y) β†’ SNo (x + z) ∧ (βˆ€w : set, w ∈ SNoL x β†’ (w + z) < x + z) ∧ (βˆ€w : set, w ∈ SNoR x β†’ (x + z) < w + z) ∧ (βˆ€w : set, w ∈ SNoL z β†’ (x + w) < x + z) ∧ (βˆ€w : set, w ∈ SNoR z β†’ (x + z) < x + w) ∧ SNoCutP (binunion (Repl (SNoL x) (Ξ»w : set β‡’ w + z)) (Repl (SNoL z) (add_SNo x))) (binunion (Repl (SNoR x) (Ξ»w : set β‡’ w + z)) (Repl (SNoR z) (add_SNo x)))
In Proofgold the corresponding term root is 9bef0a... and proposition id is 5ece17...
Proof:
Proof not loaded.
End of Section Conj_add_SNo_prop1__4__2
Beginning of Section Conj_add_SNo_prop1__5__5
L2632
Variable x : set
(*** Conj_add_SNo_prop1__5__5 TMLYedyBf7ofvYkyoZuiAKYb5HbYK1aogzd bounty of about 25 bars ***)
L2633
Variable y : set
L2634
Variable z : set
L2635
Variable w : set
L2636
Variable u : set
L2637
Hypothesis H0 : (βˆ€v : set, βˆ€x2 : set, SNo (v + x2) ∧ (βˆ€y2 : set, y2 ∈ SNoL v β†’ (y2 + x2) < v + x2) ∧ (βˆ€y2 : set, y2 ∈ SNoR v β†’ (v + x2) < y2 + x2) ∧ (βˆ€y2 : set, y2 ∈ SNoL x2 β†’ (v + y2) < v + x2) ∧ (βˆ€y2 : set, y2 ∈ SNoR x2 β†’ (v + x2) < v + y2) ∧ SNoCutP (binunion (Repl (SNoL v) (Ξ»y2 : set β‡’ y2 + x2)) (Repl (SNoL x2) (add_SNo v))) (binunion (Repl (SNoR v) (Ξ»y2 : set β‡’ y2 + x2)) (Repl (SNoR x2) (add_SNo v))) β†’ (βˆ€P : prop, (SNo (v + x2) β†’ (βˆ€y2 : set, y2 ∈ SNoL v β†’ (y2 + x2) < v + x2) β†’ (βˆ€y2 : set, y2 ∈ SNoR v β†’ (v + x2) < y2 + x2) β†’ (βˆ€y2 : set, y2 ∈ SNoL x2 β†’ (v + y2) < v + x2) β†’ (βˆ€y2 : set, y2 ∈ SNoR x2 β†’ (v + x2) < v + y2) β†’ P) β†’ P))
L2638
Hypothesis H1 : (βˆ€v : set, v ∈ SNoS_ (SNoLev y) β†’ SNo (x + v) ∧ (βˆ€x2 : set, x2 ∈ SNoL x β†’ (x2 + v) < x + v) ∧ (βˆ€x2 : set, x2 ∈ SNoR x β†’ (x + v) < x2 + v) ∧ (βˆ€x2 : set, x2 ∈ SNoL v β†’ (x + x2) < x + v) ∧ (βˆ€x2 : set, x2 ∈ SNoR v β†’ (x + v) < x + x2) ∧ SNoCutP (binunion (Repl (SNoL x) (Ξ»x2 : set β‡’ x2 + v)) (Repl (SNoL v) (add_SNo x))) (binunion (Repl (SNoR x) (Ξ»x2 : set β‡’ x2 + v)) (Repl (SNoR v) (add_SNo x))))
L2639
Hypothesis H2 : SNo z
L2640
Hypothesis H3 : SNo (x + z)
L2641
Hypothesis H4 : (βˆ€v : set, v ∈ SNoR z β†’ (x + z) < x + v)
L2642
Hypothesis H6 : SNo (x + w)
L2643
Hypothesis H7 : (βˆ€v : set, v ∈ SNoL w β†’ (x + v) < x + w)
L2644
Hypothesis H8 : SNo u
L2645
Hypothesis H9 : z < u
L2646
Hypothesis H10 : u < w
L2647
Hypothesis H11 : SNoLev u ∈ SNoLev z
L2648
Hypothesis H12 : SNoLev u ∈ SNoLev w
L2649
Hypothesis H13 : u ∈ SNoS_ (SNoLev y)
L2650
Theorem. (Conj_add_SNo_prop1__5__5)
SNo (x + u) β†’ (x + z) < x + w
In Proofgold the corresponding term root is 4866ff... and proposition id is def21a...
Proof:
Proof not loaded.
End of Section Conj_add_SNo_prop1__5__5
Beginning of Section Conj_add_SNo_prop1__5__13
L2656
Variable x : set
(*** Conj_add_SNo_prop1__5__13 TMGiWmj3rRpgA2Fo8vBHq3ob2rkhDY6VsTe bounty of about 25 bars ***)
L2657
Variable y : set
L2658
Variable z : set
L2659
Variable w : set
L2660
Variable u : set
L2661
Hypothesis H0 : (βˆ€v : set, βˆ€x2 : set, SNo (v + x2) ∧ (βˆ€y2 : set, y2 ∈ SNoL v β†’ (y2 + x2) < v + x2) ∧ (βˆ€y2 : set, y2 ∈ SNoR v β†’ (v + x2) < y2 + x2) ∧ (βˆ€y2 : set, y2 ∈ SNoL x2 β†’ (v + y2) < v + x2) ∧ (βˆ€y2 : set, y2 ∈ SNoR x2 β†’ (v + x2) < v + y2) ∧ SNoCutP (binunion (Repl (SNoL v) (Ξ»y2 : set β‡’ y2 + x2)) (Repl (SNoL x2) (add_SNo v))) (binunion (Repl (SNoR v) (Ξ»y2 : set β‡’ y2 + x2)) (Repl (SNoR x2) (add_SNo v))) β†’ (βˆ€P : prop, (SNo (v + x2) β†’ (βˆ€y2 : set, y2 ∈ SNoL v β†’ (y2 + x2) < v + x2) β†’ (βˆ€y2 : set, y2 ∈ SNoR v β†’ (v + x2) < y2 + x2) β†’ (βˆ€y2 : set, y2 ∈ SNoL x2 β†’ (v + y2) < v + x2) β†’ (βˆ€y2 : set, y2 ∈ SNoR x2 β†’ (v + x2) < v + y2) β†’ P) β†’ P))
L2662
Hypothesis H1 : (βˆ€v : set, v ∈ SNoS_ (SNoLev y) β†’ SNo (x + v) ∧ (βˆ€x2 : set, x2 ∈ SNoL x β†’ (x2 + v) < x + v) ∧ (βˆ€x2 : set, x2 ∈ SNoR x β†’ (x + v) < x2 + v) ∧ (βˆ€x2 : set, x2 ∈ SNoL v β†’ (x + x2) < x + v) ∧ (βˆ€x2 : set, x2 ∈ SNoR v β†’ (x + v) < x + x2) ∧ SNoCutP (binunion (Repl (SNoL x) (Ξ»x2 : set β‡’ x2 + v)) (Repl (SNoL v) (add_SNo x))) (binunion (Repl (SNoR x) (Ξ»x2 : set β‡’ x2 + v)) (Repl (SNoR v) (add_SNo x))))
L2663
Hypothesis H2 : SNo z
L2664
Hypothesis H3 : SNo (x + z)
L2665
Hypothesis H4 : (βˆ€v : set, v ∈ SNoR z β†’ (x + z) < x + v)
L2666
Hypothesis H5 : SNo w
L2667
Hypothesis H6 : SNo (x + w)
L2668
Hypothesis H7 : (βˆ€v : set, v ∈ SNoL w β†’ (x + v) < x + w)
L2669
Hypothesis H8 : SNo u
L2670
Hypothesis H9 : z < u
L2671
Hypothesis H10 : u < w
L2672
Hypothesis H11 : SNoLev u ∈ SNoLev z
L2673
Hypothesis H12 : SNoLev u ∈ SNoLev w
L2674
Theorem. (Conj_add_SNo_prop1__5__13)
SNo (x + u) β†’ (x + z) < x + w
In Proofgold the corresponding term root is ebf76b... and proposition id is d2c42e...
Proof:
Proof not loaded.
End of Section Conj_add_SNo_prop1__5__13
Beginning of Section Conj_add_SNo_prop1__6__4
L2680
Variable x : set
(*** Conj_add_SNo_prop1__6__4 TMX1z1mBc5iGivDCfsZzNaVQQjP2eDTttnk bounty of about 25 bars ***)
L2681
Variable y : set
L2682
Variable z : set
L2683
Variable w : set
L2684
Variable u : set
L2685
Hypothesis H0 : (βˆ€v : set, βˆ€x2 : set, SNo (v + x2) ∧ (βˆ€y2 : set, y2 ∈ SNoL v β†’ (y2 + x2) < v + x2) ∧ (βˆ€y2 : set, y2 ∈ SNoR v β†’ (v + x2) < y2 + x2) ∧ (βˆ€y2 : set, y2 ∈ SNoL x2 β†’ (v + y2) < v + x2) ∧ (βˆ€y2 : set, y2 ∈ SNoR x2 β†’ (v + x2) < v + y2) ∧ SNoCutP (binunion (Repl (SNoL v) (Ξ»y2 : set β‡’ y2 + x2)) (Repl (SNoL x2) (add_SNo v))) (binunion (Repl (SNoR v) (Ξ»y2 : set β‡’ y2 + x2)) (Repl (SNoR x2) (add_SNo v))) β†’ (βˆ€P : prop, (SNo (v + x2) β†’ (βˆ€y2 : set, y2 ∈ SNoL v β†’ (y2 + x2) < v + x2) β†’ (βˆ€y2 : set, y2 ∈ SNoR v β†’ (v + x2) < y2 + x2) β†’ (βˆ€y2 : set, y2 ∈ SNoL x2 β†’ (v + y2) < v + x2) β†’ (βˆ€y2 : set, y2 ∈ SNoR x2 β†’ (v + x2) < v + y2) β†’ P) β†’ P))
L2686
Hypothesis H1 : SNo y
L2687
Hypothesis H2 : (βˆ€v : set, v ∈ SNoS_ (SNoLev y) β†’ SNo (x + v) ∧ (βˆ€x2 : set, x2 ∈ SNoL x β†’ (x2 + v) < x + v) ∧ (βˆ€x2 : set, x2 ∈ SNoR x β†’ (x + v) < x2 + v) ∧ (βˆ€x2 : set, x2 ∈ SNoL v β†’ (x + x2) < x + v) ∧ (βˆ€x2 : set, x2 ∈ SNoR v β†’ (x + v) < x + x2) ∧ SNoCutP (binunion (Repl (SNoL x) (Ξ»x2 : set β‡’ x2 + v)) (Repl (SNoL v) (add_SNo x))) (binunion (Repl (SNoR x) (Ξ»x2 : set β‡’ x2 + v)) (Repl (SNoR v) (add_SNo x))))
L2688
Hypothesis H3 : SNo z
L2689
Hypothesis H5 : (βˆ€v : set, v ∈ SNoR z β†’ (x + z) < x + v)
L2690
Hypothesis H6 : SNo w
L2691
Hypothesis H7 : SNo (x + w)
L2692
Hypothesis H8 : (βˆ€v : set, v ∈ SNoL w β†’ (x + v) < x + w)
L2693
Hypothesis H9 : SNo u
L2694
Hypothesis H10 : z < u
L2695
Hypothesis H11 : u < w
L2696
Hypothesis H12 : SNoLev u ∈ SNoLev z
L2697
Hypothesis H13 : SNoLev u ∈ SNoLev w
L2698
Hypothesis H14 : SNoLev u ∈ SNoLev y
L2699
Theorem. (Conj_add_SNo_prop1__6__4)
u ∈ SNoS_ (SNoLev y) β†’ (x + z) < x + w
In Proofgold the corresponding term root is cec830... and proposition id is ebe2dd...
Proof:
Proof not loaded.
End of Section Conj_add_SNo_prop1__6__4
Beginning of Section Conj_add_SNo_prop1__8__1
L2705
Variable x : set
(*** Conj_add_SNo_prop1__8__1 TMZJETyhtdueWQKMyKqiHb2zZA88t4QosBt bounty of about 25 bars ***)
L2706
Variable y : set
L2707
Variable z : set
L2708
Variable w : set
L2709
Hypothesis H0 : (βˆ€u : set, βˆ€v : set, SNo (u + v) ∧ (βˆ€x2 : set, x2 ∈ SNoL u β†’ (x2 + v) < u + v) ∧ (βˆ€x2 : set, x2 ∈ SNoR u β†’ (u + v) < x2 + v) ∧ (βˆ€x2 : set, x2 ∈ SNoL v β†’ (u + x2) < u + v) ∧ (βˆ€x2 : set, x2 ∈ SNoR v β†’ (u + v) < u + x2) ∧ SNoCutP (binunion (Repl (SNoL u) (Ξ»x2 : set β‡’ x2 + v)) (Repl (SNoL v) (add_SNo u))) (binunion (Repl (SNoR u) (Ξ»x2 : set β‡’ x2 + v)) (Repl (SNoR v) (add_SNo u))) β†’ (βˆ€P : prop, (SNo (u + v) β†’ (βˆ€x2 : set, x2 ∈ SNoL u β†’ (x2 + v) < u + v) β†’ (βˆ€x2 : set, x2 ∈ SNoR u β†’ (u + v) < x2 + v) β†’ (βˆ€x2 : set, x2 ∈ SNoL v β†’ (u + x2) < u + v) β†’ (βˆ€x2 : set, x2 ∈ SNoR v β†’ (u + v) < u + x2) β†’ P) β†’ P))
L2710
Hypothesis H2 : (βˆ€u : set, u ∈ SNoS_ (SNoLev y) β†’ SNo (x + u) ∧ (βˆ€v : set, v ∈ SNoL x β†’ (v + u) < x + u) ∧ (βˆ€v : set, v ∈ SNoR x β†’ (x + u) < v + u) ∧ (βˆ€v : set, v ∈ SNoL u β†’ (x + v) < x + u) ∧ (βˆ€v : set, v ∈ SNoR u β†’ (x + u) < x + v) ∧ SNoCutP (binunion (Repl (SNoL x) (Ξ»v : set β‡’ v + u)) (Repl (SNoL u) (add_SNo x))) (binunion (Repl (SNoR x) (Ξ»v : set β‡’ v + u)) (Repl (SNoR u) (add_SNo x))))
L2711
Hypothesis H3 : TransSet (SNoLev y)
L2712
Hypothesis H4 : SNo z
L2713
Hypothesis H5 : z < y
L2714
Hypothesis H6 : SNo (x + z)
L2715
Hypothesis H7 : (βˆ€u : set, u ∈ SNoR z β†’ (x + z) < x + u)
L2716
Hypothesis H8 : SNo w
L2717
Hypothesis H9 : SNoLev w ∈ SNoLev y
L2718
Hypothesis H10 : y < w
L2719
Hypothesis H11 : SNo (x + w)
L2720
Hypothesis H12 : (βˆ€u : set, u ∈ SNoL w β†’ (x + u) < x + w)
L2721
Theorem. (Conj_add_SNo_prop1__8__1)
z < w β†’ (x + z) < x + w
In Proofgold the corresponding term root is 9c23c1... and proposition id is 89392f...
Proof:
Proof not loaded.
End of Section Conj_add_SNo_prop1__8__1
Beginning of Section Conj_add_SNo_prop1__8__10
L2727
Variable x : set
(*** Conj_add_SNo_prop1__8__10 TMQ2FVWnir38Dfj24h3U6hRgZoHCxjMHyvD bounty of about 25 bars ***)
L2728
Variable y : set
L2729
Variable z : set
L2730
Variable w : set
L2731
Hypothesis H0 : (βˆ€u : set, βˆ€v : set, SNo (u + v) ∧ (βˆ€x2 : set, x2 ∈ SNoL u β†’ (x2 + v) < u + v) ∧ (βˆ€x2 : set, x2 ∈ SNoR u β†’ (u + v) < x2 + v) ∧ (βˆ€x2 : set, x2 ∈ SNoL v β†’ (u + x2) < u + v) ∧ (βˆ€x2 : set, x2 ∈ SNoR v β†’ (u + v) < u + x2) ∧ SNoCutP (binunion (Repl (SNoL u) (Ξ»x2 : set β‡’ x2 + v)) (Repl (SNoL v) (add_SNo u))) (binunion (Repl (SNoR u) (Ξ»x2 : set β‡’ x2 + v)) (Repl (SNoR v) (add_SNo u))) β†’ (βˆ€P : prop, (SNo (u + v) β†’ (βˆ€x2 : set, x2 ∈ SNoL u β†’ (x2 + v) < u + v) β†’ (βˆ€x2 : set, x2 ∈ SNoR u β†’ (u + v) < x2 + v) β†’ (βˆ€x2 : set, x2 ∈ SNoL v β†’ (u + x2) < u + v) β†’ (βˆ€x2 : set, x2 ∈ SNoR v β†’ (u + v) < u + x2) β†’ P) β†’ P))
L2732
Hypothesis H1 : SNo y
L2733
Hypothesis H2 : (βˆ€u : set, u ∈ SNoS_ (SNoLev y) β†’ SNo (x + u) ∧ (βˆ€v : set, v ∈ SNoL x β†’ (v + u) < x + u) ∧ (βˆ€v : set, v ∈ SNoR x β†’ (x + u) < v + u) ∧ (βˆ€v : set, v ∈ SNoL u β†’ (x + v) < x + u) ∧ (βˆ€v : set, v ∈ SNoR u β†’ (x + u) < x + v) ∧ SNoCutP (binunion (Repl (SNoL x) (Ξ»v : set β‡’ v + u)) (Repl (SNoL u) (add_SNo x))) (binunion (Repl (SNoR x) (Ξ»v : set β‡’ v + u)) (Repl (SNoR u) (add_SNo x))))
L2734
Hypothesis H3 : TransSet (SNoLev y)
L2735
Hypothesis H4 : SNo z
L2736
Hypothesis H5 : z < y
L2737
Hypothesis H6 : SNo (x + z)
L2738
Hypothesis H7 : (βˆ€u : set, u ∈ SNoR z β†’ (x + z) < x + u)
L2739
Hypothesis H8 : SNo w
L2740
Hypothesis H9 : SNoLev w ∈ SNoLev y
L2741
Hypothesis H11 : SNo (x + w)
L2742
Hypothesis H12 : (βˆ€u : set, u ∈ SNoL w β†’ (x + u) < x + w)
L2743
Theorem. (Conj_add_SNo_prop1__8__10)
z < w β†’ (x + z) < x + w
In Proofgold the corresponding term root is 0308e2... and proposition id is 959578...
Proof:
Proof not loaded.
End of Section Conj_add_SNo_prop1__8__10
Beginning of Section Conj_add_SNo_prop1__10__9
L2749
Variable x : set
(*** Conj_add_SNo_prop1__10__9 TMHm7uoYzB4dWZTHHTnrEDXfuube3z1xSUY bounty of about 25 bars ***)
L2750
Variable y : set
L2751
Variable z : set
L2752
Variable w : set
L2753
Variable u : set
L2754
Hypothesis H0 : (βˆ€v : set, βˆ€x2 : set, SNo (v + x2) ∧ (βˆ€y2 : set, y2 ∈ SNoL v β†’ (y2 + x2) < v + x2) ∧ (βˆ€y2 : set, y2 ∈ SNoR v β†’ (v + x2) < y2 + x2) ∧ (βˆ€y2 : set, y2 ∈ SNoL x2 β†’ (v + y2) < v + x2) ∧ (βˆ€y2 : set, y2 ∈ SNoR x2 β†’ (v + x2) < v + y2) ∧ SNoCutP (binunion (Repl (SNoL v) (Ξ»y2 : set β‡’ y2 + x2)) (Repl (SNoL x2) (add_SNo v))) (binunion (Repl (SNoR v) (Ξ»y2 : set β‡’ y2 + x2)) (Repl (SNoR x2) (add_SNo v))) β†’ (βˆ€P : prop, (SNo (v + x2) β†’ (βˆ€y2 : set, y2 ∈ SNoL v β†’ (y2 + x2) < v + x2) β†’ (βˆ€y2 : set, y2 ∈ SNoR v β†’ (v + x2) < y2 + x2) β†’ (βˆ€y2 : set, y2 ∈ SNoL x2 β†’ (v + y2) < v + x2) β†’ (βˆ€y2 : set, y2 ∈ SNoR x2 β†’ (v + x2) < v + y2) β†’ P) β†’ P))
L2755
Hypothesis H1 : SNo x
L2756
Hypothesis H2 : SNo y
L2757
Hypothesis H3 : (βˆ€v : set, v ∈ SNoS_ (SNoLev y) β†’ SNo (x + v) ∧ (βˆ€x2 : set, x2 ∈ SNoL x β†’ (x2 + v) < x + v) ∧ (βˆ€x2 : set, x2 ∈ SNoR x β†’ (x + v) < x2 + v) ∧ (βˆ€x2 : set, x2 ∈ SNoL v β†’ (x + x2) < x + v) ∧ (βˆ€x2 : set, x2 ∈ SNoR v β†’ (x + v) < x + x2) ∧ SNoCutP (binunion (Repl (SNoL x) (Ξ»x2 : set β‡’ x2 + v)) (Repl (SNoL v) (add_SNo x))) (binunion (Repl (SNoR x) (Ξ»x2 : set β‡’ x2 + v)) (Repl (SNoR v) (add_SNo x))))
L2758
Hypothesis H4 : (βˆ€v : set, v ∈ SNoS_ (SNoLev x) β†’ (βˆ€x2 : set, x2 ∈ SNoS_ (SNoLev y) β†’ SNo (v + x2) ∧ (βˆ€y2 : set, y2 ∈ SNoL v β†’ (y2 + x2) < v + x2) ∧ (βˆ€y2 : set, y2 ∈ SNoR v β†’ (v + x2) < y2 + x2) ∧ (βˆ€y2 : set, y2 ∈ SNoL x2 β†’ (v + y2) < v + x2) ∧ (βˆ€y2 : set, y2 ∈ SNoR x2 β†’ (v + x2) < v + y2) ∧ SNoCutP (binunion (Repl (SNoL v) (Ξ»y2 : set β‡’ y2 + x2)) (Repl (SNoL x2) (add_SNo v))) (binunion (Repl (SNoR v) (Ξ»y2 : set β‡’ y2 + x2)) (Repl (SNoR x2) (add_SNo v)))))
L2759
Hypothesis H5 : TransSet (SNoLev y)
L2760
Hypothesis H6 : (βˆ€v : set, v ∈ SNoR x β†’ SNo (v + y) ∧ (βˆ€x2 : set, x2 ∈ SNoL v β†’ (x2 + y) < v + y) ∧ (βˆ€x2 : set, x2 ∈ SNoR v β†’ (v + y) < x2 + y) ∧ (βˆ€x2 : set, x2 ∈ SNoL y β†’ (v + x2) < v + y) ∧ (βˆ€x2 : set, x2 ∈ SNoR y β†’ (v + y) < v + x2) ∧ SNoCutP (binunion (Repl (SNoL v) (Ξ»x2 : set β‡’ x2 + y)) (Repl (SNoL y) (add_SNo v))) (binunion (Repl (SNoR v) (Ξ»x2 : set β‡’ x2 + y)) (Repl (SNoR y) (add_SNo v))))
L2761
Hypothesis H7 : (βˆ€v : set, v ∈ SNoL y β†’ SNo (x + v) ∧ (βˆ€x2 : set, x2 ∈ SNoL x β†’ (x2 + v) < x + v) ∧ (βˆ€x2 : set, x2 ∈ SNoR x β†’ (x + v) < x2 + v) ∧ (βˆ€x2 : set, x2 ∈ SNoL v β†’ (x + x2) < x + v) ∧ (βˆ€x2 : set, x2 ∈ SNoR v β†’ (x + v) < x + x2) ∧ SNoCutP (binunion (Repl (SNoL x) (Ξ»x2 : set β‡’ x2 + v)) (Repl (SNoL v) (add_SNo x))) (binunion (Repl (SNoR x) (Ξ»x2 : set β‡’ x2 + v)) (Repl (SNoR v) (add_SNo x))))
L2762
Hypothesis H8 : (βˆ€v : set, v ∈ SNoR y β†’ SNo (x + v) ∧ (βˆ€x2 : set, x2 ∈ SNoL x β†’ (x2 + v) < x + v) ∧ (βˆ€x2 : set, x2 ∈ SNoR x β†’ (x + v) < x2 + v) ∧ (βˆ€x2 : set, x2 ∈ SNoL v β†’ (x + x2) < x + v) ∧ (βˆ€x2 : set, x2 ∈ SNoR v β†’ (x + v) < x + x2) ∧ SNoCutP (binunion (Repl (SNoL x) (Ξ»x2 : set β‡’ x2 + v)) (Repl (SNoL v) (add_SNo x))) (binunion (Repl (SNoR x) (Ξ»x2 : set β‡’ x2 + v)) (Repl (SNoR v) (add_SNo x))))
L2763
Hypothesis H10 : u ∈ SNoL y
L2764
Hypothesis H11 : z = x + u
L2765
Hypothesis H12 : SNo u
L2766
Hypothesis H13 : SNoLev u ∈ SNoLev y
L2767
Hypothesis H14 : u < y
L2768
Theorem. (Conj_add_SNo_prop1__10__9)
u ∈ SNoS_ (SNoLev y) β†’ z < w
In Proofgold the corresponding term root is 3e613f... and proposition id is 821f21...
Proof:
Proof not loaded.
End of Section Conj_add_SNo_prop1__10__9
Beginning of Section Conj_add_SNo_prop1__11__8
L2774
Variable x : set
(*** Conj_add_SNo_prop1__11__8 TMJjD8VWjPjQsD8BY5Cf4iNQ2SyAZD3eguq bounty of about 25 bars ***)
L2775
Variable y : set
L2776
Variable z : set
L2777
Variable w : set
L2778
Variable u : set
L2779
Hypothesis H0 : (βˆ€v : set, βˆ€x2 : set, SNo (v + x2) ∧ (βˆ€y2 : set, y2 ∈ SNoL v β†’ (y2 + x2) < v + x2) ∧ (βˆ€y2 : set, y2 ∈ SNoR v β†’ (v + x2) < y2 + x2) ∧ (βˆ€y2 : set, y2 ∈ SNoL x2 β†’ (v + y2) < v + x2) ∧ (βˆ€y2 : set, y2 ∈ SNoR x2 β†’ (v + x2) < v + y2) ∧ SNoCutP (binunion (Repl (SNoL v) (Ξ»y2 : set β‡’ y2 + x2)) (Repl (SNoL x2) (add_SNo v))) (binunion (Repl (SNoR v) (Ξ»y2 : set β‡’ y2 + x2)) (Repl (SNoR x2) (add_SNo v))) β†’ (βˆ€P : prop, (SNo (v + x2) β†’ (βˆ€y2 : set, y2 ∈ SNoL v β†’ (y2 + x2) < v + x2) β†’ (βˆ€y2 : set, y2 ∈ SNoR v β†’ (v + x2) < y2 + x2) β†’ (βˆ€y2 : set, y2 ∈ SNoL x2 β†’ (v + y2) < v + x2) β†’ (βˆ€y2 : set, y2 ∈ SNoR x2 β†’ (v + x2) < v + y2) β†’ P) β†’ P))
L2780
Hypothesis H1 : SNo x
L2781
Hypothesis H2 : SNo y
L2782
Hypothesis H3 : (βˆ€v : set, v ∈ SNoS_ (SNoLev x) β†’ (βˆ€x2 : set, x2 ∈ SNoS_ (SNoLev y) β†’ SNo (v + x2) ∧ (βˆ€y2 : set, y2 ∈ SNoL v β†’ (y2 + x2) < v + x2) ∧ (βˆ€y2 : set, y2 ∈ SNoR v β†’ (v + x2) < y2 + x2) ∧ (βˆ€y2 : set, y2 ∈ SNoL x2 β†’ (v + y2) < v + x2) ∧ (βˆ€y2 : set, y2 ∈ SNoR x2 β†’ (v + x2) < v + y2) ∧ SNoCutP (binunion (Repl (SNoL v) (Ξ»y2 : set β‡’ y2 + x2)) (Repl (SNoL x2) (add_SNo v))) (binunion (Repl (SNoR v) (Ξ»y2 : set β‡’ y2 + x2)) (Repl (SNoR x2) (add_SNo v)))))
L2783
Hypothesis H4 : (βˆ€v : set, v ∈ SNoR y β†’ SNo (x + v) ∧ (βˆ€x2 : set, x2 ∈ SNoL x β†’ (x2 + v) < x + v) ∧ (βˆ€x2 : set, x2 ∈ SNoR x β†’ (x + v) < x2 + v) ∧ (βˆ€x2 : set, x2 ∈ SNoL v β†’ (x + x2) < x + v) ∧ (βˆ€x2 : set, x2 ∈ SNoR v β†’ (x + v) < x + x2) ∧ SNoCutP (binunion (Repl (SNoL x) (Ξ»x2 : set β‡’ x2 + v)) (Repl (SNoL v) (add_SNo x))) (binunion (Repl (SNoR x) (Ξ»x2 : set β‡’ x2 + v)) (Repl (SNoR v) (add_SNo x))))
L2784
Hypothesis H5 : SNo w
L2785
Hypothesis H6 : SNoLev w ∈ SNoLev x
L2786
Hypothesis H7 : w < x
L2787
Hypothesis H9 : SNo (w + y)
L2788
Hypothesis H10 : (βˆ€v : set, v ∈ SNoR y β†’ (w + y) < w + v)
L2789
Hypothesis H11 : u ∈ SNoR y
L2790
Hypothesis H12 : z = x + u
L2791
Hypothesis H13 : SNo u
L2792
Hypothesis H14 : SNoLev u ∈ SNoLev y
L2793
Hypothesis H15 : y < u
L2794
Theorem. (Conj_add_SNo_prop1__11__8)
u ∈ SNoS_ (SNoLev y) β†’ (w + y) < z
In Proofgold the corresponding term root is 42e809... and proposition id is a6fba0...
Proof:
Proof not loaded.
End of Section Conj_add_SNo_prop1__11__8
Beginning of Section Conj_add_SNo_prop1__13__13
L2800
Variable x : set
(*** Conj_add_SNo_prop1__13__13 TMXhoDqXufNnYUqK65nAXv3zKC1pzTtgBvN bounty of about 25 bars ***)
L2801
Variable y : set
L2802
Variable z : set
L2803
Variable w : set
L2804
Variable u : set
L2805
Hypothesis H0 : (βˆ€v : set, βˆ€x2 : set, SNo (v + x2) ∧ (βˆ€y2 : set, y2 ∈ SNoL v β†’ (y2 + x2) < v + x2) ∧ (βˆ€y2 : set, y2 ∈ SNoR v β†’ (v + x2) < y2 + x2) ∧ (βˆ€y2 : set, y2 ∈ SNoL x2 β†’ (v + y2) < v + x2) ∧ (βˆ€y2 : set, y2 ∈ SNoR x2 β†’ (v + x2) < v + y2) ∧ SNoCutP (binunion (Repl (SNoL v) (Ξ»y2 : set β‡’ y2 + x2)) (Repl (SNoL x2) (add_SNo v))) (binunion (Repl (SNoR v) (Ξ»y2 : set β‡’ y2 + x2)) (Repl (SNoR x2) (add_SNo v))) β†’ (βˆ€P : prop, (SNo (v + x2) β†’ (βˆ€y2 : set, y2 ∈ SNoL v β†’ (y2 + x2) < v + x2) β†’ (βˆ€y2 : set, y2 ∈ SNoR v β†’ (v + x2) < y2 + x2) β†’ (βˆ€y2 : set, y2 ∈ SNoL x2 β†’ (v + y2) < v + x2) β†’ (βˆ€y2 : set, y2 ∈ SNoR x2 β†’ (v + x2) < v + y2) β†’ P) β†’ P))
L2806
Hypothesis H1 : SNo x
L2807
Hypothesis H2 : (βˆ€v : set, v ∈ SNoS_ (SNoLev x) β†’ SNo (v + y) ∧ (βˆ€x2 : set, x2 ∈ SNoL v β†’ (x2 + y) < v + y) ∧ (βˆ€x2 : set, x2 ∈ SNoR v β†’ (v + y) < x2 + y) ∧ (βˆ€x2 : set, x2 ∈ SNoL y β†’ (v + x2) < v + y) ∧ (βˆ€x2 : set, x2 ∈ SNoR y β†’ (v + y) < v + x2) ∧ SNoCutP (binunion (Repl (SNoL v) (Ξ»x2 : set β‡’ x2 + y)) (Repl (SNoL y) (add_SNo v))) (binunion (Repl (SNoR v) (Ξ»x2 : set β‡’ x2 + y)) (Repl (SNoR y) (add_SNo v))))
L2808
Hypothesis H3 : SNo z
L2809
Hypothesis H4 : SNo (z + y)
L2810
Hypothesis H5 : (βˆ€v : set, v ∈ SNoR z β†’ (z + y) < v + y)
L2811
Hypothesis H6 : SNo w
L2812
Hypothesis H7 : SNo (w + y)
L2813
Hypothesis H8 : (βˆ€v : set, v ∈ SNoL w β†’ (v + y) < w + y)
L2814
Hypothesis H9 : SNo u
L2815
Hypothesis H10 : z < u
L2816
Hypothesis H11 : u < w
L2817
Hypothesis H12 : SNoLev u ∈ SNoLev z
L2818
Hypothesis H14 : SNoLev u ∈ SNoLev x
L2819
Theorem. (Conj_add_SNo_prop1__13__13)
u ∈ SNoS_ (SNoLev x) β†’ (z + y) < w + y
In Proofgold the corresponding term root is 5c0b9b... and proposition id is 354f83...
Proof:
Proof not loaded.
End of Section Conj_add_SNo_prop1__13__13
Beginning of Section Conj_add_SNo_prop1__14__3
L2825
Variable x : set
(*** Conj_add_SNo_prop1__14__3 TMbHrxQUgmMPHhFFRw5GuT9N9eLNYCzrWbx bounty of about 25 bars ***)
L2826
Variable y : set
L2827
Variable z : set
L2828
Variable w : set
L2829
Variable u : set
L2830
Hypothesis H0 : (βˆ€v : set, βˆ€x2 : set, SNo (v + x2) ∧ (βˆ€y2 : set, y2 ∈ SNoL v β†’ (y2 + x2) < v + x2) ∧ (βˆ€y2 : set, y2 ∈ SNoR v β†’ (v + x2) < y2 + x2) ∧ (βˆ€y2 : set, y2 ∈ SNoL x2 β†’ (v + y2) < v + x2) ∧ (βˆ€y2 : set, y2 ∈ SNoR x2 β†’ (v + x2) < v + y2) ∧ SNoCutP (binunion (Repl (SNoL v) (Ξ»y2 : set β‡’ y2 + x2)) (Repl (SNoL x2) (add_SNo v))) (binunion (Repl (SNoR v) (Ξ»y2 : set β‡’ y2 + x2)) (Repl (SNoR x2) (add_SNo v))) β†’ (βˆ€P : prop, (SNo (v + x2) β†’ (βˆ€y2 : set, y2 ∈ SNoL v β†’ (y2 + x2) < v + x2) β†’ (βˆ€y2 : set, y2 ∈ SNoR v β†’ (v + x2) < y2 + x2) β†’ (βˆ€y2 : set, y2 ∈ SNoL x2 β†’ (v + y2) < v + x2) β†’ (βˆ€y2 : set, y2 ∈ SNoR x2 β†’ (v + x2) < v + y2) β†’ P) β†’ P))
L2831
Hypothesis H1 : SNo x
L2832
Hypothesis H2 : (βˆ€v : set, v ∈ SNoS_ (SNoLev x) β†’ SNo (v + y) ∧ (βˆ€x2 : set, x2 ∈ SNoL v β†’ (x2 + y) < v + y) ∧ (βˆ€x2 : set, x2 ∈ SNoR v β†’ (v + y) < x2 + y) ∧ (βˆ€x2 : set, x2 ∈ SNoL y β†’ (v + x2) < v + y) ∧ (βˆ€x2 : set, x2 ∈ SNoR y β†’ (v + y) < v + x2) ∧ SNoCutP (binunion (Repl (SNoL v) (Ξ»x2 : set β‡’ x2 + y)) (Repl (SNoL y) (add_SNo v))) (binunion (Repl (SNoR v) (Ξ»x2 : set β‡’ x2 + y)) (Repl (SNoR y) (add_SNo v))))
L2833
Hypothesis H4 : SNo z
L2834
Hypothesis H5 : SNoLev z ∈ SNoLev x
L2835
Hypothesis H6 : SNo (z + y)
L2836
Hypothesis H7 : (βˆ€v : set, v ∈ SNoR z β†’ (z + y) < v + y)
L2837
Hypothesis H8 : SNo w
L2838
Hypothesis H9 : SNo (w + y)
L2839
Hypothesis H10 : (βˆ€v : set, v ∈ SNoL w β†’ (v + y) < w + y)
L2840
Hypothesis H11 : SNo u
L2841
Hypothesis H12 : z < u
L2842
Hypothesis H13 : u < w
L2843
Hypothesis H14 : SNoLev u ∈ SNoLev z
L2844
Hypothesis H15 : SNoLev u ∈ SNoLev w
L2845
Theorem. (Conj_add_SNo_prop1__14__3)
SNoLev u ∈ SNoLev x β†’ (z + y) < w + y
In Proofgold the corresponding term root is 4e3d09... and proposition id is 1239a1...
Proof:
Proof not loaded.
End of Section Conj_add_SNo_prop1__14__3
Beginning of Section Conj_add_SNo_prop1__16__5
L2851
Variable x : set
(*** Conj_add_SNo_prop1__16__5 TMNFMUEv3yFcwmtyjwitM6wH1sFvdWwjuxG bounty of about 25 bars ***)
L2852
Variable y : set
L2853
Variable z : set
L2854
Variable w : set
L2855
Variable u : set
L2856
Hypothesis H0 : (βˆ€v : set, βˆ€x2 : set, SNo (v + x2) ∧ (βˆ€y2 : set, y2 ∈ SNoL v β†’ (y2 + x2) < v + x2) ∧ (βˆ€y2 : set, y2 ∈ SNoR v β†’ (v + x2) < y2 + x2) ∧ (βˆ€y2 : set, y2 ∈ SNoL x2 β†’ (v + y2) < v + x2) ∧ (βˆ€y2 : set, y2 ∈ SNoR x2 β†’ (v + x2) < v + y2) ∧ SNoCutP (binunion (Repl (SNoL v) (Ξ»y2 : set β‡’ y2 + x2)) (Repl (SNoL x2) (add_SNo v))) (binunion (Repl (SNoR v) (Ξ»y2 : set β‡’ y2 + x2)) (Repl (SNoR x2) (add_SNo v))) β†’ (βˆ€P : prop, (SNo (v + x2) β†’ (βˆ€y2 : set, y2 ∈ SNoL v β†’ (y2 + x2) < v + x2) β†’ (βˆ€y2 : set, y2 ∈ SNoR v β†’ (v + x2) < y2 + x2) β†’ (βˆ€y2 : set, y2 ∈ SNoL x2 β†’ (v + y2) < v + x2) β†’ (βˆ€y2 : set, y2 ∈ SNoR x2 β†’ (v + x2) < v + y2) β†’ P) β†’ P))
L2857
Hypothesis H1 : SNo x
L2858
Hypothesis H2 : SNo y
L2859
Hypothesis H3 : (βˆ€v : set, v ∈ SNoS_ (SNoLev x) β†’ SNo (v + y) ∧ (βˆ€x2 : set, x2 ∈ SNoL v β†’ (x2 + y) < v + y) ∧ (βˆ€x2 : set, x2 ∈ SNoR v β†’ (v + y) < x2 + y) ∧ (βˆ€x2 : set, x2 ∈ SNoL y β†’ (v + x2) < v + y) ∧ (βˆ€x2 : set, x2 ∈ SNoR y β†’ (v + y) < v + x2) ∧ SNoCutP (binunion (Repl (SNoL v) (Ξ»x2 : set β‡’ x2 + y)) (Repl (SNoL y) (add_SNo v))) (binunion (Repl (SNoR v) (Ξ»x2 : set β‡’ x2 + y)) (Repl (SNoR y) (add_SNo v))))
L2860
Hypothesis H4 : (βˆ€v : set, v ∈ SNoS_ (SNoLev x) β†’ (βˆ€x2 : set, x2 ∈ SNoS_ (SNoLev y) β†’ SNo (v + x2) ∧ (βˆ€y2 : set, y2 ∈ SNoL v β†’ (y2 + x2) < v + x2) ∧ (βˆ€y2 : set, y2 ∈ SNoR v β†’ (v + x2) < y2 + x2) ∧ (βˆ€y2 : set, y2 ∈ SNoL x2 β†’ (v + y2) < v + x2) ∧ (βˆ€y2 : set, y2 ∈ SNoR x2 β†’ (v + x2) < v + y2) ∧ SNoCutP (binunion (Repl (SNoL v) (Ξ»y2 : set β‡’ y2 + x2)) (Repl (SNoL x2) (add_SNo v))) (binunion (Repl (SNoR v) (Ξ»y2 : set β‡’ y2 + x2)) (Repl (SNoR x2) (add_SNo v)))))
L2861
Hypothesis H6 : (βˆ€v : set, v ∈ SNoL x β†’ SNo (v + y) ∧ (βˆ€x2 : set, x2 ∈ SNoL v β†’ (x2 + y) < v + y) ∧ (βˆ€x2 : set, x2 ∈ SNoR v β†’ (v + y) < x2 + y) ∧ (βˆ€x2 : set, x2 ∈ SNoL y β†’ (v + x2) < v + y) ∧ (βˆ€x2 : set, x2 ∈ SNoR y β†’ (v + y) < v + x2) ∧ SNoCutP (binunion (Repl (SNoL v) (Ξ»x2 : set β‡’ x2 + y)) (Repl (SNoL y) (add_SNo v))) (binunion (Repl (SNoR v) (Ξ»x2 : set β‡’ x2 + y)) (Repl (SNoR y) (add_SNo v))))
L2862
Hypothesis H7 : (βˆ€v : set, v ∈ SNoR x β†’ SNo (v + y) ∧ (βˆ€x2 : set, x2 ∈ SNoL v β†’ (x2 + y) < v + y) ∧ (βˆ€x2 : set, x2 ∈ SNoR v β†’ (v + y) < x2 + y) ∧ (βˆ€x2 : set, x2 ∈ SNoL y β†’ (v + x2) < v + y) ∧ (βˆ€x2 : set, x2 ∈ SNoR y β†’ (v + y) < v + x2) ∧ SNoCutP (binunion (Repl (SNoL v) (Ξ»x2 : set β‡’ x2 + y)) (Repl (SNoL y) (add_SNo v))) (binunion (Repl (SNoR v) (Ξ»x2 : set β‡’ x2 + y)) (Repl (SNoR y) (add_SNo v))))
L2863
Hypothesis H8 : (βˆ€v : set, v ∈ SNoR y β†’ SNo (x + v) ∧ (βˆ€x2 : set, x2 ∈ SNoL x β†’ (x2 + v) < x + v) ∧ (βˆ€x2 : set, x2 ∈ SNoR x β†’ (x + v) < x2 + v) ∧ (βˆ€x2 : set, x2 ∈ SNoL v β†’ (x + x2) < x + v) ∧ (βˆ€x2 : set, x2 ∈ SNoR v β†’ (x + v) < x + x2) ∧ SNoCutP (binunion (Repl (SNoL x) (Ξ»x2 : set β‡’ x2 + v)) (Repl (SNoL v) (add_SNo x))) (binunion (Repl (SNoR x) (Ξ»x2 : set β‡’ x2 + v)) (Repl (SNoR v) (add_SNo x))))
L2864
Hypothesis H9 : w ∈ binunion (Repl (SNoR x) (Ξ»v : set β‡’ v + y)) (Repl (SNoR y) (add_SNo x))
L2865
Hypothesis H10 : u ∈ SNoL x
L2866
Hypothesis H11 : z = u + y
L2867
Hypothesis H12 : SNo u
L2868
Hypothesis H13 : SNoLev u ∈ SNoLev x
L2869
Hypothesis H14 : u < x
L2870
Theorem. (Conj_add_SNo_prop1__16__5)
u ∈ SNoS_ (SNoLev x) β†’ z < w
In Proofgold the corresponding term root is 4a144b... and proposition id is 9f3e3d...
Proof:
Proof not loaded.
End of Section Conj_add_SNo_prop1__16__5
Beginning of Section Conj_add_SNo_prop1__21__2
L2876
Variable x : set
(*** Conj_add_SNo_prop1__21__2 TMQC3GwuMGCnP4M83sEPpvZeFb6PB2LFTra bounty of about 25 bars ***)
L2877
Variable y : set
L2878
Hypothesis H0 : SNo x
L2879
Hypothesis H1 : SNo y
L2880
Theorem. (Conj_add_SNo_prop1__21__2)
SNo (SNoCut (binunion (Repl (SNoL x) (Ξ»z : set β‡’ z + y)) (Repl (SNoL y) (add_SNo x))) (binunion (Repl (SNoR x) (Ξ»z : set β‡’ z + y)) (Repl (SNoR y) (add_SNo x)))) β†’ SNo (x + y) ∧ (βˆ€z : set, z ∈ SNoL x β†’ (z + y) < x + y) ∧ (βˆ€z : set, z ∈ SNoR x β†’ (x + y) < z + y) ∧ (βˆ€z : set, z ∈ SNoL y β†’ (x + z) < x + y) ∧ (βˆ€z : set, z ∈ SNoR y β†’ (x + y) < x + z) ∧ SNoCutP (binunion (Repl (SNoL x) (Ξ»z : set β‡’ z + y)) (Repl (SNoL y) (add_SNo x))) (binunion (Repl (SNoR x) (Ξ»z : set β‡’ z + y)) (Repl (SNoR y) (add_SNo x)))
In Proofgold the corresponding term root is 4858bc... and proposition id is 5467b7...
Proof:
Proof not loaded.
End of Section Conj_add_SNo_prop1__21__2
Beginning of Section Conj_add_SNo_prop1__28__1
L2886
Variable x : set
(*** Conj_add_SNo_prop1__28__1 TMUWb5q21SbYbkqj22czGW7myn98AaLuYsZ bounty of about 25 bars ***)
L2887
Variable y : set
L2888
Hypothesis H0 : (βˆ€z : set, βˆ€w : set, SNo (z + w) ∧ (βˆ€u : set, u ∈ SNoL z β†’ (u + w) < z + w) ∧ (βˆ€u : set, u ∈ SNoR z β†’ (z + w) < u + w) ∧ (βˆ€u : set, u ∈ SNoL w β†’ (z + u) < z + w) ∧ (βˆ€u : set, u ∈ SNoR w β†’ (z + w) < z + u) ∧ SNoCutP (binunion (Repl (SNoL z) (Ξ»u : set β‡’ u + w)) (Repl (SNoL w) (add_SNo z))) (binunion (Repl (SNoR z) (Ξ»u : set β‡’ u + w)) (Repl (SNoR w) (add_SNo z))) β†’ (βˆ€P : prop, (SNo (z + w) β†’ (βˆ€u : set, u ∈ SNoL z β†’ (u + w) < z + w) β†’ (βˆ€u : set, u ∈ SNoR z β†’ (z + w) < u + w) β†’ (βˆ€u : set, u ∈ SNoL w β†’ (z + u) < z + w) β†’ (βˆ€u : set, u ∈ SNoR w β†’ (z + w) < z + u) β†’ P) β†’ P))
L2889
Hypothesis H2 : SNo y
L2890
Hypothesis H3 : (βˆ€z : set, z ∈ SNoS_ (SNoLev x) β†’ SNo (z + y) ∧ (βˆ€w : set, w ∈ SNoL z β†’ (w + y) < z + y) ∧ (βˆ€w : set, w ∈ SNoR z β†’ (z + y) < w + y) ∧ (βˆ€w : set, w ∈ SNoL y β†’ (z + w) < z + y) ∧ (βˆ€w : set, w ∈ SNoR y β†’ (z + y) < z + w) ∧ SNoCutP (binunion (Repl (SNoL z) (Ξ»w : set β‡’ w + y)) (Repl (SNoL y) (add_SNo z))) (binunion (Repl (SNoR z) (Ξ»w : set β‡’ w + y)) (Repl (SNoR y) (add_SNo z))))
L2891
Hypothesis H4 : (βˆ€z : set, z ∈ SNoS_ (SNoLev y) β†’ SNo (x + z) ∧ (βˆ€w : set, w ∈ SNoL x β†’ (w + z) < x + z) ∧ (βˆ€w : set, w ∈ SNoR x β†’ (x + z) < w + z) ∧ (βˆ€w : set, w ∈ SNoL z β†’ (x + w) < x + z) ∧ (βˆ€w : set, w ∈ SNoR z β†’ (x + z) < x + w) ∧ SNoCutP (binunion (Repl (SNoL x) (Ξ»w : set β‡’ w + z)) (Repl (SNoL z) (add_SNo x))) (binunion (Repl (SNoR x) (Ξ»w : set β‡’ w + z)) (Repl (SNoR z) (add_SNo x))))
L2892
Hypothesis H5 : (βˆ€z : set, z ∈ SNoS_ (SNoLev x) β†’ (βˆ€w : set, w ∈ SNoS_ (SNoLev y) β†’ SNo (z + w) ∧ (βˆ€u : set, u ∈ SNoL z β†’ (u + w) < z + w) ∧ (βˆ€u : set, u ∈ SNoR z β†’ (z + w) < u + w) ∧ (βˆ€u : set, u ∈ SNoL w β†’ (z + u) < z + w) ∧ (βˆ€u : set, u ∈ SNoR w β†’ (z + w) < z + u) ∧ SNoCutP (binunion (Repl (SNoL z) (Ξ»u : set β‡’ u + w)) (Repl (SNoL w) (add_SNo z))) (binunion (Repl (SNoR z) (Ξ»u : set β‡’ u + w)) (Repl (SNoR w) (add_SNo z)))))
L2893
Hypothesis H6 : TransSet (SNoLev x)
L2894
Theorem. (Conj_add_SNo_prop1__28__1)
ordinal (SNoLev y) β†’ SNo (x + y) ∧ (βˆ€z : set, z ∈ SNoL x β†’ (z + y) < x + y) ∧ (βˆ€z : set, z ∈ SNoR x β†’ (x + y) < z + y) ∧ (βˆ€z : set, z ∈ SNoL y β†’ (x + z) < x + y) ∧ (βˆ€z : set, z ∈ SNoR y β†’ (x + y) < x + z) ∧ SNoCutP (binunion (Repl (SNoL x) (Ξ»z : set β‡’ z + y)) (Repl (SNoL y) (add_SNo x))) (binunion (Repl (SNoR x) (Ξ»z : set β‡’ z + y)) (Repl (SNoR y) (add_SNo x)))
In Proofgold the corresponding term root is e7d4a6... and proposition id is 6386e5...
Proof:
Proof not loaded.
End of Section Conj_add_SNo_prop1__28__1
Beginning of Section Conj_add_SNo_com__1__1
L2900
Variable x : set
(*** Conj_add_SNo_com__1__1 TMMr9BebDgWDvTzt31sDU87QgwsZdMcb1By bounty of about 25 bars ***)
L2901
Variable y : set
L2902
Variable z : set
L2903
Hypothesis H0 : SNo x
L2904
Hypothesis H2 : SNo z
L2905
Hypothesis H3 : SNoLev z ∈ SNoLev x
L2906
Theorem. (Conj_add_SNo_com__1__1)
z ∈ SNoS_ (SNoLev x) β†’ z + y = y + z
In Proofgold the corresponding term root is 93fc44... and proposition id is a0faea...
Proof:
Proof not loaded.
End of Section Conj_add_SNo_com__1__1
Beginning of Section Conj_add_SNo_com__1__3
L2912
Variable x : set
(*** Conj_add_SNo_com__1__3 TMQ3dCmi7VjLptv3kYXBZ9NZ9a7XTTVzHa5 bounty of about 25 bars ***)
L2913
Variable y : set
L2914
Variable z : set
L2915
Hypothesis H0 : SNo x
L2916
Hypothesis H1 : (βˆ€w : set, w ∈ SNoS_ (SNoLev x) β†’ w + y = y + w)
L2917
Hypothesis H2 : SNo z
L2918
Theorem. (Conj_add_SNo_com__1__3)
z ∈ SNoS_ (SNoLev x) β†’ z + y = y + z
In Proofgold the corresponding term root is 26f467... and proposition id is 70a693...
Proof:
Proof not loaded.
End of Section Conj_add_SNo_com__1__3
Beginning of Section Conj_add_SNo_com__2__1
L2924
Variable x : set
(*** Conj_add_SNo_com__2__1 TMMr9BebDgWDvTzt31sDU87QgwsZdMcb1By bounty of about 25 bars ***)
L2925
Variable y : set
L2926
Variable z : set
L2927
Hypothesis H0 : SNo x
L2928
Hypothesis H2 : SNo z
L2929
Hypothesis H3 : SNoLev z ∈ SNoLev x
L2930
Theorem. (Conj_add_SNo_com__2__1)
z ∈ SNoS_ (SNoLev x) β†’ z + y = y + z
In Proofgold the corresponding term root is 93fc44... and proposition id is a0faea...
Proof:
Proof not loaded.
End of Section Conj_add_SNo_com__2__1
Beginning of Section Conj_add_SNo_com__2__3
L2936
Variable x : set
(*** Conj_add_SNo_com__2__3 TMQ3dCmi7VjLptv3kYXBZ9NZ9a7XTTVzHa5 bounty of about 25 bars ***)
L2937
Variable y : set
L2938
Variable z : set
L2939
Hypothesis H0 : SNo x
L2940
Hypothesis H1 : (βˆ€w : set, w ∈ SNoS_ (SNoLev x) β†’ w + y = y + w)
L2941
Hypothesis H2 : SNo z
L2942
Theorem. (Conj_add_SNo_com__2__3)
z ∈ SNoS_ (SNoLev x) β†’ z + y = y + z
In Proofgold the corresponding term root is 26f467... and proposition id is 70a693...
Proof:
Proof not loaded.
End of Section Conj_add_SNo_com__2__3
Beginning of Section Conj_add_SNo_com__6__2
L2948
Variable x : set
(*** Conj_add_SNo_com__6__2 TMSdbAk47ZEakrp4zJaRfdbpr9C6kXQmx7r bounty of about 25 bars ***)
L2949
Variable y : set
L2950
Hypothesis H0 : (βˆ€z : set, z ∈ SNoR x β†’ z + y = y + z)
L2951
Hypothesis H1 : (βˆ€z : set, z ∈ SNoR y β†’ x + z = z + x)
L2952
Hypothesis H3 : Repl (SNoL y) (add_SNo x) = Repl (SNoL y) (Ξ»z : set β‡’ z + x)
L2953
Theorem. (Conj_add_SNo_com__6__2)
Repl (SNoR x) (Ξ»z : set β‡’ z + y) = Repl (SNoR x) (add_SNo y) β†’ SNoCut (binunion (Repl (SNoL x) (Ξ»z : set β‡’ z + y)) (Repl (SNoL y) (add_SNo x))) (binunion (Repl (SNoR x) (Ξ»z : set β‡’ z + y)) (Repl (SNoR y) (add_SNo x))) = SNoCut (binunion (Repl (SNoL y) (Ξ»z : set β‡’ z + x)) (Repl (SNoL x) (add_SNo y))) (binunion (Repl (SNoR y) (Ξ»z : set β‡’ z + x)) (Repl (SNoR x) (add_SNo y)))
In Proofgold the corresponding term root is 8a9177... and proposition id is 7d56eb...
Proof:
Proof not loaded.
End of Section Conj_add_SNo_com__6__2
Beginning of Section Conj_add_SNo_com__9__0
L2959
Variable x : set
(*** Conj_add_SNo_com__9__0 TMX6ruLrKqAb7bzfgTV3uomLxXeLCg5jjrU bounty of about 25 bars ***)
L2960
Variable y : set
L2961
Hypothesis H1 : SNo y
L2962
Hypothesis H2 : (βˆ€z : set, z ∈ SNoS_ (SNoLev y) β†’ x + z = z + x)
L2963
Hypothesis H3 : (βˆ€z : set, z ∈ SNoL x β†’ z + y = y + z)
L2964
Hypothesis H4 : (βˆ€z : set, z ∈ SNoR x β†’ z + y = y + z)
L2965
Hypothesis H5 : (βˆ€z : set, z ∈ SNoL y β†’ x + z = z + x)
L2966
Theorem. (Conj_add_SNo_com__9__0)
(βˆ€z : set, z ∈ SNoR y β†’ x + z = z + x) β†’ x + y = y + x
In Proofgold the corresponding term root is 865e2d... and proposition id is 4c69e2...
Proof:
Proof not loaded.
End of Section Conj_add_SNo_com__9__0
Beginning of Section Conj_add_SNo_com__10__1
L2972
Variable x : set
(*** Conj_add_SNo_com__10__1 TMTGNuBh48GHgeyLGF4bb4Gdq3hWYJcMXea bounty of about 25 bars ***)
L2973
Variable y : set
L2974
Hypothesis H0 : SNo x
L2975
Hypothesis H2 : (βˆ€z : set, z ∈ SNoS_ (SNoLev y) β†’ x + z = z + x)
L2976
Hypothesis H3 : (βˆ€z : set, z ∈ SNoL x β†’ z + y = y + z)
L2977
Hypothesis H4 : (βˆ€z : set, z ∈ SNoR x β†’ z + y = y + z)
L2978
Theorem. (Conj_add_SNo_com__10__1)
(βˆ€z : set, z ∈ SNoL y β†’ x + z = z + x) β†’ x + y = y + x
In Proofgold the corresponding term root is 13b2b7... and proposition id is 77f7f7...
Proof:
Proof not loaded.
End of Section Conj_add_SNo_com__10__1
Beginning of Section Conj_add_SNo_minus_SNo_linv__4__0
L2984
Variable x : set
(*** Conj_add_SNo_minus_SNo_linv__4__0 TMGF59iGxxsUQDcsZi8viHS8DHm43emcanL bounty of about 25 bars ***)
L2985
Variable y : set
L2986
Variable z : set
L2987
Hypothesis H1 : SNo (- x)
L2988
Hypothesis H2 : y = z + x
L2989
Hypothesis H3 : SNo z
L2990
Hypothesis H4 : - x < z
L2991
Hypothesis H5 : SNo (- z)
L2992
Hypothesis H6 : - z + z = Empty
L2993
Theorem. (Conj_add_SNo_minus_SNo_linv__4__0)
- z < x β†’ Empty < y
In Proofgold the corresponding term root is 645386... and proposition id is f04f61...
Proof:
Proof not loaded.
End of Section Conj_add_SNo_minus_SNo_linv__4__0
Beginning of Section Conj_add_SNo_minus_SNo_linv__8__6
L2999
Variable x : set
(*** Conj_add_SNo_minus_SNo_linv__8__6 TMTCBnUbcVnTAyupe8YC4zvAzBURvfcHWyD bounty of about 25 bars ***)
L3000
Variable y : set
L3001
Variable z : set
L3002
Hypothesis H0 : SNo x
L3003
Hypothesis H1 : (βˆ€w : set, w ∈ SNoS_ (SNoLev x) β†’ - w + w = Empty)
L3004
Hypothesis H2 : SNo (- x)
L3005
Hypothesis H3 : y = - x + z
L3006
Hypothesis H4 : SNo z
L3007
Hypothesis H5 : SNoLev z ∈ SNoLev x
L3008
Hypothesis H7 : SNo (- z)
L3009
Theorem. (Conj_add_SNo_minus_SNo_linv__8__6)
- z + z = Empty β†’ y < Empty
In Proofgold the corresponding term root is e3ef4f... and proposition id is cb65a8...
Proof:
Proof not loaded.
End of Section Conj_add_SNo_minus_SNo_linv__8__6
Beginning of Section Conj_add_SNo_minus_SNo_linv__9__5
L3015
Variable x : set
(*** Conj_add_SNo_minus_SNo_linv__9__5 TMdGzUydD3SDVNxX8zyvzhrTjpwWMiWYUnd bounty of about 25 bars ***)
L3016
Variable y : set
L3017
Variable z : set
L3018
Hypothesis H0 : SNo x
L3019
Hypothesis H1 : (βˆ€w : set, w ∈ SNoS_ (SNoLev x) β†’ - w + w = Empty)
L3020
Hypothesis H2 : SNo (- x)
L3021
Hypothesis H3 : y = - x + z
L3022
Hypothesis H4 : SNo z
L3023
Hypothesis H6 : z < x
L3024
Theorem. (Conj_add_SNo_minus_SNo_linv__9__5)
SNo (- z) β†’ y < Empty
In Proofgold the corresponding term root is 9f61aa... and proposition id is 6cfb53...
Proof:
Proof not loaded.
End of Section Conj_add_SNo_minus_SNo_linv__9__5
Beginning of Section Conj_add_SNo_ordinal_ordinal__3__3
L3030
Variable x : set
(*** Conj_add_SNo_ordinal_ordinal__3__3 TMaFBae8DHzqkLw9HWhhDtDy7yU2BLJEJV3 bounty of about 25 bars ***)
L3031
Variable y : set
L3032
Hypothesis H0 : ordinal x
L3033
Hypothesis H1 : ordinal y
L3034
Hypothesis H2 : SNo x
L3035
Theorem. (Conj_add_SNo_ordinal_ordinal__3__3)
SNo (x + y) β†’ ordinal (x + y)
In Proofgold the corresponding term root is b4fe2a... and proposition id is df8dd9...
Proof:
Proof not loaded.
End of Section Conj_add_SNo_ordinal_ordinal__3__3
Beginning of Section Conj_add_SNo_ordinal_ordinal__4__1
L3041
Variable x : set
(*** Conj_add_SNo_ordinal_ordinal__4__1 TMHuBSyrVwnSJLJFTifwFTvXkDUk3LqmWFr bounty of about 25 bars ***)
L3042
Variable y : set
L3043
Hypothesis H0 : ordinal x
L3044
Hypothesis H2 : SNo x
L3045
Theorem. (Conj_add_SNo_ordinal_ordinal__4__1)
SNo y β†’ ordinal (x + y)
In Proofgold the corresponding term root is b2ae97... and proposition id is d0ad0e...
Proof:
Proof not loaded.
End of Section Conj_add_SNo_ordinal_ordinal__4__1
Beginning of Section Conj_add_SNo_ordinal_ordinal__4__2
L3051
Variable x : set
(*** Conj_add_SNo_ordinal_ordinal__4__2 TMXMHL28V7j1WUH1azsPwPtybhuppowfTjv bounty of about 25 bars ***)
L3052
Variable y : set
L3053
Hypothesis H0 : ordinal x
L3054
Hypothesis H1 : ordinal y
L3055
Theorem. (Conj_add_SNo_ordinal_ordinal__4__2)
SNo y β†’ ordinal (x + y)
In Proofgold the corresponding term root is ad88d2... and proposition id is d2f6bb...
Proof:
Proof not loaded.
End of Section Conj_add_SNo_ordinal_ordinal__4__2
Beginning of Section Conj_add_SNo_ordinal_SL__1__0
L3061
Variable x : set
(*** Conj_add_SNo_ordinal_SL__1__0 TMaTFttmxYnPgmEbgWdLQb9Ke5tzkbq2atf bounty of about 25 bars ***)
L3062
Variable y : set
L3063
Variable z : set
L3064
Hypothesis H1 : SNo x
L3065
Hypothesis H2 : SNo y
L3066
Hypothesis H3 : ordinal (x + y)
L3067
Hypothesis H4 : SNo (ordsucc x)
L3068
Hypothesis H5 : ordinal (ordsucc (x + y))
L3069
Hypothesis H6 : SNo (ordsucc (x + y))
L3070
Hypothesis H7 : SNoLev z ∈ y
L3071
Hypothesis H8 : ordinal (SNoLev z)
L3072
Hypothesis H9 : SNo z
L3073
Hypothesis H10 : ordsucc x + SNoLev z = ordsucc (x + SNoLev z)
L3074
Hypothesis H11 : SNo (ordsucc x + z)
L3075
Hypothesis H12 : ordinal (x + SNoLev z)
L3076
Hypothesis H13 : ordinal (ordsucc x + SNoLev z)
L3077
Theorem. (Conj_add_SNo_ordinal_SL__1__0)
SNo (ordsucc x + SNoLev z) β†’ (ordsucc x + z) < ordsucc (x + y)
In Proofgold the corresponding term root is eebf2b... and proposition id is 959830...
Proof:
Proof not loaded.
End of Section Conj_add_SNo_ordinal_SL__1__0
Beginning of Section Conj_add_SNo_ordinal_SL__1__4
L3083
Variable x : set
(*** Conj_add_SNo_ordinal_SL__1__4 TMagMnTJRWaacTqz4M68DbjRns82xUdAymA bounty of about 25 bars ***)
L3084
Variable y : set
L3085
Variable z : set
L3086
Hypothesis H0 : ordinal y
L3087
Hypothesis H1 : SNo x
L3088
Hypothesis H2 : SNo y
L3089
Hypothesis H3 : ordinal (x + y)
L3090
Hypothesis H5 : ordinal (ordsucc (x + y))
L3091
Hypothesis H6 : SNo (ordsucc (x + y))
L3092
Hypothesis H7 : SNoLev z ∈ y
L3093
Hypothesis H8 : ordinal (SNoLev z)
L3094
Hypothesis H9 : SNo z
L3095
Hypothesis H10 : ordsucc x + SNoLev z = ordsucc (x + SNoLev z)
L3096
Hypothesis H11 : SNo (ordsucc x + z)
L3097
Hypothesis H12 : ordinal (x + SNoLev z)
L3098
Hypothesis H13 : ordinal (ordsucc x + SNoLev z)
L3099
Theorem. (Conj_add_SNo_ordinal_SL__1__4)
SNo (ordsucc x + SNoLev z) β†’ (ordsucc x + z) < ordsucc (x + y)
In Proofgold the corresponding term root is eb3930... and proposition id is 8914fd...
Proof:
Proof not loaded.
End of Section Conj_add_SNo_ordinal_SL__1__4
Beginning of Section Conj_add_SNo_ordinal_SL__6__8
L3105
Variable x : set
(*** Conj_add_SNo_ordinal_SL__6__8 TMJE6wA9R179pjakH1EFJLShz58GZK1AykM bounty of about 25 bars ***)
L3106
Variable y : set
L3107
Variable z : set
L3108
Hypothesis H0 : ordinal x
L3109
Hypothesis H1 : SNo x
L3110
Hypothesis H2 : SNo y
L3111
Hypothesis H3 : ordinal (x + y)
L3112
Hypothesis H4 : ordinal (ordsucc (x + y))
L3113
Hypothesis H5 : SNo (ordsucc (x + y))
L3114
Hypothesis H6 : SNoLev z ∈ ordsucc x
L3115
Hypothesis H7 : ordinal (SNoLev z)
L3116
Hypothesis H9 : ordinal (SNoLev z + y)
L3117
Theorem. (Conj_add_SNo_ordinal_SL__6__8)
SNo (SNoLev z + y) β†’ (z + y) < ordsucc (x + y)
In Proofgold the corresponding term root is b45f72... and proposition id is 3db4ce...
Proof:
Proof not loaded.
End of Section Conj_add_SNo_ordinal_SL__6__8
Beginning of Section Conj_add_SNo_ordinal_SL__7__1
L3123
Variable x : set
(*** Conj_add_SNo_ordinal_SL__7__1 TMT6Q1Ne4tKhtSHYyhJb8Q5KXRQw1TfrEdJ bounty of about 25 bars ***)
L3124
Variable y : set
L3125
Variable z : set
L3126
Hypothesis H0 : ordinal x
L3127
Hypothesis H2 : SNo x
L3128
Hypothesis H3 : SNo y
L3129
Hypothesis H4 : ordinal (x + y)
L3130
Hypothesis H5 : ordinal (ordsucc (x + y))
L3131
Hypothesis H6 : SNo (ordsucc (x + y))
L3132
Hypothesis H7 : SNoLev z ∈ ordsucc x
L3133
Hypothesis H8 : ordinal (SNoLev z)
L3134
Hypothesis H9 : SNo z
L3135
Theorem. (Conj_add_SNo_ordinal_SL__7__1)
ordinal (SNoLev z + y) β†’ (z + y) < ordsucc (x + y)
In Proofgold the corresponding term root is 3cb115... and proposition id is 2c4c4a...
Proof:
Proof not loaded.
End of Section Conj_add_SNo_ordinal_SL__7__1
Beginning of Section Conj_add_SNo_ordinal_SL__7__3
L3141
Variable x : set
(*** Conj_add_SNo_ordinal_SL__7__3 TMYFj8hNJwk8usjNAB1DzkWx51ZNtFzpYqJ bounty of about 25 bars ***)
L3142
Variable y : set
L3143
Variable z : set
L3144
Hypothesis H0 : ordinal x
L3145
Hypothesis H1 : ordinal y
L3146
Hypothesis H2 : SNo x
L3147
Hypothesis H4 : ordinal (x + y)
L3148
Hypothesis H5 : ordinal (ordsucc (x + y))
L3149
Hypothesis H6 : SNo (ordsucc (x + y))
L3150
Hypothesis H7 : SNoLev z ∈ ordsucc x
L3151
Hypothesis H8 : ordinal (SNoLev z)
L3152
Hypothesis H9 : SNo z
L3153
Theorem. (Conj_add_SNo_ordinal_SL__7__3)
ordinal (SNoLev z + y) β†’ (z + y) < ordsucc (x + y)
In Proofgold the corresponding term root is 40084b... and proposition id is d6040b...
Proof:
Proof not loaded.
End of Section Conj_add_SNo_ordinal_SL__7__3
Beginning of Section Conj_add_SNo_ordinal_SL__11__9
L3159
Variable x : set
(*** Conj_add_SNo_ordinal_SL__11__9 TMdkjd9fhMh4ZvHPtn1sLrBJEBDBJSZkrLz bounty of about 25 bars ***)
L3160
Variable y : set
L3161
Hypothesis H0 : ordinal x
L3162
Hypothesis H1 : ordinal y
L3163
Hypothesis H2 : (βˆ€z : set, z ∈ y β†’ ordsucc x + z = ordsucc (x + z))
L3164
Hypothesis H3 : SNo x
L3165
Hypothesis H4 : SNo y
L3166
Hypothesis H5 : ordinal (x + y)
L3167
Hypothesis H6 : ordinal (ordsucc x)
L3168
Hypothesis H7 : SNo (ordsucc x)
L3169
Hypothesis H8 : ordinal (ordsucc x + y)
L3170
Hypothesis H10 : ordsucc (x + y) ∈ ordsucc x + y
L3171
Theorem. (Conj_add_SNo_ordinal_SL__11__9)
Β¬ ordinal (ordsucc (x + y))
In Proofgold the corresponding term root is 4e932a... and proposition id is 8c7322...
Proof:
Proof not loaded.
End of Section Conj_add_SNo_ordinal_SL__11__9
Beginning of Section Conj_add_SNo_ordinal_SL__14__0
L3177
Variable x : set
(*** Conj_add_SNo_ordinal_SL__14__0 TMPuzCEbNAmuev4kGXCya7U7chkJW3MfJyN bounty of about 25 bars ***)
L3178
Variable y : set
L3179
Hypothesis H1 : ordinal y
L3180
Hypothesis H2 : (βˆ€z : set, z ∈ y β†’ ordsucc x + z = ordsucc (x + z))
L3181
Hypothesis H3 : SNo x
L3182
Hypothesis H4 : SNo y
L3183
Hypothesis H5 : ordinal (x + y)
L3184
Hypothesis H6 : ordinal (ordsucc x)
L3185
Theorem. (Conj_add_SNo_ordinal_SL__14__0)
SNo (ordsucc x) β†’ ordsucc x + y = ordsucc (x + y)
In Proofgold the corresponding term root is 99b3d4... and proposition id is 4d514b...
Proof:
Proof not loaded.
End of Section Conj_add_SNo_ordinal_SL__14__0
Beginning of Section Conj_add_SNo_ordinal_SR__4__0
L3191
Variable x : set
(*** Conj_add_SNo_ordinal_SR__4__0 TMKzrR1a9aqcMhLSQeiyFYgwCQHvProN8Bg bounty of about 25 bars ***)
L3192
Variable y : set
L3193
Hypothesis H1 : ordinal y
L3194
Theorem. (Conj_add_SNo_ordinal_SR__4__0)
SNo y β†’ x + ordsucc y = ordsucc (x + y)
In Proofgold the corresponding term root is d784d4... and proposition id is b9ae95...
Proof:
Proof not loaded.
End of Section Conj_add_SNo_ordinal_SR__4__0
Beginning of Section Conj_add_SNo_ordinal_SR__5__1
L3200
Variable x : set
(*** Conj_add_SNo_ordinal_SR__5__1 TMQ3ME9j8CRf3C1j5v6Xf3dGj6SwY5FjpeC bounty of about 25 bars ***)
L3201
Variable y : set
L3202
Hypothesis H0 : ordinal x
L3203
Theorem. (Conj_add_SNo_ordinal_SR__5__1)
SNo x β†’ x + ordsucc y = ordsucc (x + y)
In Proofgold the corresponding term root is a0db9a... and proposition id is 6ffc39...
Proof:
Proof not loaded.
End of Section Conj_add_SNo_ordinal_SR__5__1
Beginning of Section Conj_add_SNo_ordinal_InR__1__1
L3209
Variable x : set
(*** Conj_add_SNo_ordinal_InR__1__1 TMVjjW9pc9n7Ayt6bsCFaGJNhJ2xjvMPjUj bounty of about 25 bars ***)
L3210
Variable y : set
L3211
Variable z : set
L3212
Hypothesis H0 : ordinal x
L3213
Hypothesis H2 : z ∈ y
L3214
Hypothesis H3 : SNo x
L3215
Hypothesis H4 : SNo y
L3216
Hypothesis H5 : ordinal z
L3217
Theorem. (Conj_add_SNo_ordinal_InR__1__1)
SNo z β†’ x + z ∈ x + y
In Proofgold the corresponding term root is aa9906... and proposition id is e51a9a...
Proof:
Proof not loaded.
End of Section Conj_add_SNo_ordinal_InR__1__1
Beginning of Section Conj_add_nat_add_SNo__1__1
L3223
Variable x : set
(*** Conj_add_nat_add_SNo__1__1 TMUWXmW5XTH3eedUHHdLT1fgohTdeoTbsLV bounty of about 25 bars ***)
L3224
Hypothesis H0 : ordinal x
L3225
Theorem. (Conj_add_nat_add_SNo__1__1)
(βˆ€y : set, nat_p y β†’ add_nat x y = x + y) β†’ (βˆ€y : set, y ∈ Ο‰ β†’ add_nat x y = x + y)
In Proofgold the corresponding term root is d6ce08... and proposition id is b53aa2...
Proof:
Proof not loaded.
End of Section Conj_add_nat_add_SNo__1__1
Beginning of Section Conj_add_SNo_SNoL_interpolate__2__11
L3231
Variable x : set
(*** Conj_add_SNo_SNoL_interpolate__2__11 TMPDccnwSGkXWNQfdRA9SDRCoa7d9E87qp8 bounty of about 25 bars ***)
L3232
Variable y : set
L3233
Variable z : set
L3234
Variable w : set
L3235
Hypothesis H0 : SNo x
L3236
Hypothesis H1 : SNo y
L3237
Hypothesis H2 : SNo (x + y)
L3238
Hypothesis H3 : SNo z
L3239
Hypothesis H4 : (βˆ€u : set, u ∈ SNoS_ (SNoLev z) β†’ SNoLev u ∈ SNoLev (x + y) β†’ u < x + y β†’ (βˆƒv : set, v ∈ SNoL x ∧ u ≀ v + y) ∨ (βˆƒv : set, v ∈ SNoL y ∧ u ≀ x + v))
L3240
Hypothesis H5 : SNoLev z ∈ SNoLev (x + y)
L3241
Hypothesis H6 : Β¬ ((βˆƒu : set, u ∈ SNoL x ∧ z ≀ u + y) ∨ (βˆƒu : set, u ∈ SNoL y ∧ z ≀ x + u))
L3242
Hypothesis H7 : w ∈ SNoR z
L3243
Hypothesis H8 : SNo w
L3244
Hypothesis H9 : SNoLev w ∈ SNoLev z
L3245
Hypothesis H10 : z < w
L3246
Theorem. (Conj_add_SNo_SNoL_interpolate__2__11)
¬ w ∈ SNoS_ (SNoLev z)
In Proofgold the corresponding term root is 46345c... and proposition id is b1fc76...
Proof:
Proof not loaded.
End of Section Conj_add_SNo_SNoL_interpolate__2__11
Beginning of Section Conj_add_SNo_SNoR_interpolate__1__1
L3252
Variable x : set
(*** Conj_add_SNo_SNoR_interpolate__1__1 TMVJUyQgEeBpr2MSP1ZC2ZHKS62rPQcU8Dp bounty of about 25 bars ***)
L3253
Variable y : set
L3254
Variable z : set
L3255
Variable w : set
L3256
Hypothesis H0 : SNo x
L3257
Hypothesis H2 : Β¬ ((βˆƒu : set, u ∈ SNoR x ∧ (u + y) ≀ z) ∨ (βˆƒu : set, u ∈ SNoR y ∧ (x + u) ≀ z))
L3258
Hypothesis H3 : w ∈ SNoR y
L3259
Hypothesis H4 : SNo w
L3260
Theorem. (Conj_add_SNo_SNoR_interpolate__1__1)
SNo (x + w) β†’ z < x + w
In Proofgold the corresponding term root is 5e3928... and proposition id is 19eea3...
Proof:
Proof not loaded.
End of Section Conj_add_SNo_SNoR_interpolate__1__1
Beginning of Section Conj_add_SNo_assoc__3__0
L3266
Variable x : set
(*** Conj_add_SNo_assoc__3__0 TMXsit1Vzi1NNsJuNcvocHtXeNgLzJvooae bounty of about 25 bars ***)
L3267
Variable y : set
L3268
Variable z : set
L3269
Variable w : set
L3270
Variable u : set
L3271
Hypothesis H1 : SNo y
L3272
Hypothesis H2 : SNo z
L3273
Hypothesis H3 : (βˆ€v : set, v ∈ SNoS_ (SNoLev x) β†’ v + y + z = (v + y) + z)
L3274
Hypothesis H4 : SNo (y + z)
L3275
Hypothesis H5 : SNo w
L3276
Hypothesis H6 : u ∈ SNoR x
L3277
Hypothesis H7 : (u + y) ≀ w
L3278
Hypothesis H8 : SNo u
L3279
Hypothesis H9 : x < u
L3280
Theorem. (Conj_add_SNo_assoc__3__0)
u + y + z = (u + y) + z β†’ (x + y + z) < w + z
In Proofgold the corresponding term root is d647d4... and proposition id is 52df83...
Proof:
Proof not loaded.
End of Section Conj_add_SNo_assoc__3__0
Beginning of Section Conj_add_SNo_assoc__6__7
L3286
Variable x : set
(*** Conj_add_SNo_assoc__6__7 TMW1Jjp6Nvat9sxCvmBheLCQTXhsEDkviwo bounty of about 25 bars ***)
L3287
Variable y : set
L3288
Variable z : set
L3289
Variable w : set
L3290
Variable u : set
L3291
Hypothesis H0 : SNo x
L3292
Hypothesis H1 : SNo y
L3293
Hypothesis H2 : SNo z
L3294
Hypothesis H3 : (βˆ€v : set, v ∈ SNoS_ (SNoLev x) β†’ v + y + z = (v + y) + z)
L3295
Hypothesis H4 : SNo (y + z)
L3296
Hypothesis H5 : SNo w
L3297
Hypothesis H6 : u ∈ SNoL x
L3298
Hypothesis H8 : SNo u
L3299
Hypothesis H9 : u < x
L3300
Theorem. (Conj_add_SNo_assoc__6__7)
u + y + z = (u + y) + z β†’ (w + z) < x + y + z
In Proofgold the corresponding term root is e820fd... and proposition id is 214a95...
Proof:
Proof not loaded.
End of Section Conj_add_SNo_assoc__6__7
Beginning of Section Conj_add_SNo_assoc__7__3
L3306
Variable x : set
(*** Conj_add_SNo_assoc__7__3 TMXsWRSQ997vqfdp7f9S4kbfZfvgqKkxoAZ bounty of about 25 bars ***)
L3307
Variable y : set
L3308
Variable z : set
L3309
Variable w : set
L3310
Variable u : set
L3311
Hypothesis H0 : SNo x
L3312
Hypothesis H1 : SNo y
L3313
Hypothesis H2 : SNo z
L3314
Hypothesis H4 : SNo (x + y)
L3315
Hypothesis H5 : SNo w
L3316
Hypothesis H6 : u ∈ SNoR z
L3317
Hypothesis H7 : (y + u) ≀ w
L3318
Hypothesis H8 : SNo u
L3319
Hypothesis H9 : z < u
L3320
Theorem. (Conj_add_SNo_assoc__7__3)
x + y + u = (x + y) + u β†’ ((x + y) + z) < x + w
In Proofgold the corresponding term root is 86edac... and proposition id is e5d8e6...
Proof:
Proof not loaded.
End of Section Conj_add_SNo_assoc__7__3
Beginning of Section Conj_add_SNo_assoc__10__0
L3326
Variable x : set
(*** Conj_add_SNo_assoc__10__0 TMa6BZWrSrLGV7YSyMBPpeMvY3rLHXxFPSC bounty of about 25 bars ***)
L3327
Variable y : set
L3328
Variable z : set
L3329
Variable w : set
L3330
Variable u : set
L3331
Hypothesis H1 : SNo y
L3332
Hypothesis H2 : SNo z
L3333
Hypothesis H3 : (βˆ€v : set, v ∈ SNoS_ (SNoLev z) β†’ x + y + v = (x + y) + v)
L3334
Hypothesis H4 : SNo (x + y)
L3335
Hypothesis H5 : SNo w
L3336
Hypothesis H6 : u ∈ SNoL z
L3337
Hypothesis H7 : w ≀ y + u
L3338
Hypothesis H8 : SNo u
L3339
Hypothesis H9 : u < z
L3340
Theorem. (Conj_add_SNo_assoc__10__0)
x + y + u = (x + y) + u β†’ (x + w) < (x + y) + z
In Proofgold the corresponding term root is 2b0565... and proposition id is 019879...
Proof:
Proof not loaded.
End of Section Conj_add_SNo_assoc__10__0
Beginning of Section Conj_add_SNo_assoc__14__4
L3346
Variable x : set
(*** Conj_add_SNo_assoc__14__4 TMTuKP6ArXdEmRoimjWtdgMJZ4HRoGW4P4f bounty of about 25 bars ***)
L3347
Variable y : set
L3348
Variable z : set
L3349
Hypothesis H0 : SNo x
L3350
Hypothesis H1 : SNo y
L3351
Hypothesis H2 : SNo z
L3352
Hypothesis H3 : (βˆ€w : set, w ∈ SNoS_ (SNoLev x) β†’ w + y + z = (w + y) + z)
L3353
Hypothesis H5 : (βˆ€w : set, w ∈ SNoS_ (SNoLev z) β†’ x + y + w = (x + y) + w)
L3354
Hypothesis H6 : SNo (x + y)
L3355
Hypothesis H7 : SNo (y + z)
L3356
Theorem. (Conj_add_SNo_assoc__14__4)
SNoCutP (binunion (Repl (SNoL x) (Ξ»w : set β‡’ w + y + z)) (Repl (SNoL (y + z)) (add_SNo x))) (binunion (Repl (SNoR x) (Ξ»w : set β‡’ w + y + z)) (Repl (SNoR (y + z)) (add_SNo x))) β†’ SNoCut (binunion (Repl (SNoL x) (Ξ»w : set β‡’ w + y + z)) (Repl (SNoL (y + z)) (add_SNo x))) (binunion (Repl (SNoR x) (Ξ»w : set β‡’ w + y + z)) (Repl (SNoR (y + z)) (add_SNo x))) = SNoCut (binunion (Repl (SNoL (x + y)) (Ξ»w : set β‡’ w + z)) (Repl (SNoL z) (add_SNo (x + y)))) (binunion (Repl (SNoR (x + y)) (Ξ»w : set β‡’ w + z)) (Repl (SNoR z) (add_SNo (x + y))))
In Proofgold the corresponding term root is 11b8db... and proposition id is 5aa7b6...
Proof:
Proof not loaded.
End of Section Conj_add_SNo_assoc__14__4
Beginning of Section Conj_add_SNo_cancel_L__2__3
L3362
Variable x : set
(*** Conj_add_SNo_cancel_L__2__3 TMQ3vxQuuyn1Q4vk43S1xSA8Zz5J8ZaofiG bounty of about 25 bars ***)
L3363
Variable y : set
L3364
Variable z : set
L3365
Hypothesis H0 : SNo x
L3366
Hypothesis H1 : SNo y
L3367
Hypothesis H2 : SNo z
L3368
Hypothesis H4 : SNo (- x)
L3369
Theorem. (Conj_add_SNo_cancel_L__2__3)
- x + x + y = y β†’ y = z
In Proofgold the corresponding term root is c17b67... and proposition id is c2b6b1...
Proof:
Proof not loaded.
End of Section Conj_add_SNo_cancel_L__2__3
Beginning of Section Conj_minus_add_SNo_distr__1__0
L3375
Variable x : set
(*** Conj_minus_add_SNo_distr__1__0 TMJEeETSrTdiebrNZo2udReSYGb37mJUNY6 bounty of about 25 bars ***)
L3376
Variable y : set
L3377
Hypothesis H1 : SNo y
L3378
Hypothesis H2 : SNo (- x)
L3379
Hypothesis H3 : SNo (- y)
L3380
Hypothesis H4 : SNo (x + y)
L3381
Theorem. (Conj_minus_add_SNo_distr__1__0)
(x + y) + - (x + y) = (x + y) + - x + - y β†’ - (x + y) = - x + - y
In Proofgold the corresponding term root is 77189e... and proposition id is 66ed72...
Proof:
Proof not loaded.
End of Section Conj_minus_add_SNo_distr__1__0
Beginning of Section Conj_minus_add_SNo_distr__3__2
L3387
Variable x : set
(*** Conj_minus_add_SNo_distr__3__2 TMQeaR1rvGbFrxrXdp8YaE98CMMTSoiTegz bounty of about 25 bars ***)
L3388
Variable y : set
L3389
Hypothesis H0 : SNo x
L3390
Hypothesis H1 : SNo y
L3391
Theorem. (Conj_minus_add_SNo_distr__3__2)
SNo (- y) β†’ - (x + y) = - x + - y
In Proofgold the corresponding term root is e06335... and proposition id is 75d78f...
Proof:
Proof not loaded.
End of Section Conj_minus_add_SNo_distr__3__2
Beginning of Section Conj_add_SNo_Lev_bd__3__0
L3397
Variable x : set
(*** Conj_add_SNo_Lev_bd__3__0 TMbuRaCZBt6Gr3LVPVb1rwEULiauyDEENss bounty of about 25 bars ***)
L3398
Variable y : set
L3399
Variable z : set
L3400
Variable p : (set β†’ prop)
L3401
Variable w : set
L3402
Hypothesis H1 : w ∈ SNoR x
L3403
Hypothesis H2 : z = w + y
L3404
Theorem. (Conj_add_SNo_Lev_bd__3__0)
w ∈ SNoS_ (SNoLev x) β†’ SNo w β†’ SNoLev w ∈ SNoLev x β†’ x < w β†’ p (w + y)
In Proofgold the corresponding term root is 41ac1a... and proposition id is 10aafd...
Proof:
Proof not loaded.
End of Section Conj_add_SNo_Lev_bd__3__0
Beginning of Section Conj_add_SNo_Lev_bd__6__2
L3410
Variable x : set
(*** Conj_add_SNo_Lev_bd__6__2 TMRTZ3HrGidEjbTRVuq1tVf4xjohsQ1gibQ bounty of about 25 bars ***)
L3411
Variable y : set
L3412
Variable z : set
L3413
Variable w : set
L3414
Hypothesis H0 : SNo x
L3415
Hypothesis H1 : ordinal (SNoLev x + SNoLev y)
L3416
Hypothesis H3 : z ∈ ordsucc (SNoLev (x + w))
L3417
Hypothesis H4 : ordinal z
L3418
Hypothesis H5 : Subq (SNoLev x + SNoLev y) z
L3419
Hypothesis H6 : Subq (SNoLev (x + w)) (SNoLev x + SNoLev w)
L3420
Hypothesis H7 : SNoLev x + SNoLev w ∈ SNoLev x + SNoLev y
L3421
Theorem. (Conj_add_SNo_Lev_bd__6__2)
Β¬ Subq (SNoLev x + SNoLev w) (SNoLev x + SNoLev y)
In Proofgold the corresponding term root is aafc31... and proposition id is 04cec9...
Proof:
Proof not loaded.
End of Section Conj_add_SNo_Lev_bd__6__2
Beginning of Section Conj_add_SNo_Lev_bd__7__7
L3427
Variable x : set
(*** Conj_add_SNo_Lev_bd__7__7 TMFy7YVW2vf3xvNmm1QA6bRRvSzYJJKZj5r bounty of about 25 bars ***)
L3428
Variable y : set
L3429
Variable z : set
L3430
Variable w : set
L3431
Hypothesis H0 : SNo x
L3432
Hypothesis H1 : SNo y
L3433
Hypothesis H2 : ordinal (SNoLev x + SNoLev y)
L3434
Hypothesis H3 : SNo w
L3435
Hypothesis H4 : SNoLev w ∈ SNoLev y
L3436
Hypothesis H5 : z ∈ ordsucc (SNoLev (x + w))
L3437
Hypothesis H6 : ordinal z
L3438
Hypothesis H8 : Subq (SNoLev (x + w)) (SNoLev x + SNoLev w)
L3439
Theorem. (Conj_add_SNo_Lev_bd__7__7)
¬ SNoLev x + SNoLev w ∈ SNoLev x + SNoLev y
In Proofgold the corresponding term root is 4ef372... and proposition id is ed6e9f...
Proof:
Proof not loaded.
End of Section Conj_add_SNo_Lev_bd__7__7
Beginning of Section Conj_add_SNo_Lev_bd__10__5
L3445
Variable x : set
(*** Conj_add_SNo_Lev_bd__10__5 TMHVYfyhxYKFVawbnbFsZ4HwQkTsi65xnCC bounty of about 25 bars ***)
L3446
Variable y : set
L3447
Variable z : set
L3448
Variable w : set
L3449
Hypothesis H0 : SNo y
L3450
Hypothesis H1 : SNo w
L3451
Hypothesis H2 : z ∈ ordsucc (SNoLev (w + y))
L3452
Hypothesis H3 : ordinal z
L3453
Hypothesis H4 : Subq (SNoLev x + SNoLev y) z
L3454
Hypothesis H6 : SNoLev w + SNoLev y ∈ SNoLev x + SNoLev y
L3455
Theorem. (Conj_add_SNo_Lev_bd__10__5)
Β¬ Subq z (SNoLev (w + y))
In Proofgold the corresponding term root is a552af... and proposition id is bae0de...
Proof:
Proof not loaded.
End of Section Conj_add_SNo_Lev_bd__10__5
Beginning of Section Conj_add_SNo_Lev_bd__11__2
L3461
Variable x : set
(*** Conj_add_SNo_Lev_bd__11__2 TMXgUeJGYffmMUWGm6xudY5g5voEb2CEKTL bounty of about 25 bars ***)
L3462
Variable y : set
L3463
Variable z : set
L3464
Variable w : set
L3465
Hypothesis H0 : SNo y
L3466
Hypothesis H1 : ordinal (SNoLev x + SNoLev y)
L3467
Hypothesis H3 : z ∈ ordsucc (SNoLev (w + y))
L3468
Hypothesis H4 : ordinal z
L3469
Hypothesis H5 : Subq (SNoLev x + SNoLev y) z
L3470
Hypothesis H6 : Subq (SNoLev (w + y)) (SNoLev w + SNoLev y)
L3471
Hypothesis H7 : SNoLev w + SNoLev y ∈ SNoLev x + SNoLev y
L3472
Theorem. (Conj_add_SNo_Lev_bd__11__2)
Β¬ Subq (SNoLev w + SNoLev y) (SNoLev x + SNoLev y)
In Proofgold the corresponding term root is 2c6c95... and proposition id is 82480a...
Proof:
Proof not loaded.
End of Section Conj_add_SNo_Lev_bd__11__2
Beginning of Section Conj_add_SNo_Lev_bd__12__6
L3478
Variable x : set
(*** Conj_add_SNo_Lev_bd__12__6 TMVCFjRzpyxJX4Ma39CJrKqoDKAEhAxmbtj bounty of about 25 bars ***)
L3479
Variable y : set
L3480
Variable z : set
L3481
Variable w : set
L3482
Hypothesis H0 : SNo x
L3483
Hypothesis H1 : SNo y
L3484
Hypothesis H2 : ordinal (SNoLev x + SNoLev y)
L3485
Hypothesis H3 : SNo w
L3486
Hypothesis H4 : SNoLev w ∈ SNoLev x
L3487
Hypothesis H5 : z ∈ ordsucc (SNoLev (w + y))
L3488
Hypothesis H7 : Subq (SNoLev x + SNoLev y) z
L3489
Hypothesis H8 : Subq (SNoLev (w + y)) (SNoLev w + SNoLev y)
L3490
Theorem. (Conj_add_SNo_Lev_bd__12__6)
¬ SNoLev w + SNoLev y ∈ SNoLev x + SNoLev y
In Proofgold the corresponding term root is 619007... and proposition id is 345e59...
Proof:
Proof not loaded.
End of Section Conj_add_SNo_Lev_bd__12__6
Beginning of Section Conj_add_SNo_Lev_bd__13__3
L3496
Variable x : set
(*** Conj_add_SNo_Lev_bd__13__3 TMXPkP3ALi7mCddXRumP7vezPZkG7LV6K1J bounty of about 25 bars ***)
L3497
Variable y : set
L3498
Variable z : set
L3499
Variable w : set
L3500
Hypothesis H0 : SNo x
L3501
Hypothesis H1 : SNo y
L3502
Hypothesis H2 : ordinal (SNoLev x + SNoLev y)
L3503
Hypothesis H4 : w ∈ SNoS_ (SNoLev x)
L3504
Hypothesis H5 : SNo w
L3505
Hypothesis H6 : SNoLev w ∈ SNoLev x
L3506
Hypothesis H7 : z ∈ ordsucc (SNoLev (w + y))
L3507
Hypothesis H8 : ordinal z
L3508
Hypothesis H9 : Subq (SNoLev x + SNoLev y) z
L3509
Theorem. (Conj_add_SNo_Lev_bd__13__3)
Β¬ Subq (SNoLev (w + y)) (SNoLev w + SNoLev y)
In Proofgold the corresponding term root is 667ef2... and proposition id is b84773...
Proof:
Proof not loaded.
End of Section Conj_add_SNo_Lev_bd__13__3
Beginning of Section Conj_add_SNo_Lev_bd__13__5
L3515
Variable x : set
(*** Conj_add_SNo_Lev_bd__13__5 TMW2nLeL2Rd146GsAgGY4WjrySEBhppfQ6G bounty of about 25 bars ***)
L3516
Variable y : set
L3517
Variable z : set
L3518
Variable w : set
L3519
Hypothesis H0 : SNo x
L3520
Hypothesis H1 : SNo y
L3521
Hypothesis H2 : ordinal (SNoLev x + SNoLev y)
L3522
Hypothesis H3 : (βˆ€u : set, u ∈ SNoS_ (SNoLev x) β†’ Subq (SNoLev (u + y)) (SNoLev u + SNoLev y))
L3523
Hypothesis H4 : w ∈ SNoS_ (SNoLev x)
L3524
Hypothesis H6 : SNoLev w ∈ SNoLev x
L3525
Hypothesis H7 : z ∈ ordsucc (SNoLev (w + y))
L3526
Hypothesis H8 : ordinal z
L3527
Hypothesis H9 : Subq (SNoLev x + SNoLev y) z
L3528
Theorem. (Conj_add_SNo_Lev_bd__13__5)
Β¬ Subq (SNoLev (w + y)) (SNoLev w + SNoLev y)
In Proofgold the corresponding term root is 942f3a... and proposition id is 86a991...
Proof:
Proof not loaded.
End of Section Conj_add_SNo_Lev_bd__13__5
Beginning of Section Conj_add_SNo_Lev_bd__13__6
L3534
Variable x : set
(*** Conj_add_SNo_Lev_bd__13__6 TMbJLfVumt2xXTEw8dH7j5gNxbSBHG8u4mS bounty of about 25 bars ***)
L3535
Variable y : set
L3536
Variable z : set
L3537
Variable w : set
L3538
Hypothesis H0 : SNo x
L3539
Hypothesis H1 : SNo y
L3540
Hypothesis H2 : ordinal (SNoLev x + SNoLev y)
L3541
Hypothesis H3 : (βˆ€u : set, u ∈ SNoS_ (SNoLev x) β†’ Subq (SNoLev (u + y)) (SNoLev u + SNoLev y))
L3542
Hypothesis H4 : w ∈ SNoS_ (SNoLev x)
L3543
Hypothesis H5 : SNo w
L3544
Hypothesis H7 : z ∈ ordsucc (SNoLev (w + y))
L3545
Hypothesis H8 : ordinal z
L3546
Hypothesis H9 : Subq (SNoLev x + SNoLev y) z
L3547
Theorem. (Conj_add_SNo_Lev_bd__13__6)
Β¬ Subq (SNoLev (w + y)) (SNoLev w + SNoLev y)
In Proofgold the corresponding term root is 0ecea1... and proposition id is 57bf76...
Proof:
Proof not loaded.
End of Section Conj_add_SNo_Lev_bd__13__6
Beginning of Section Conj_add_SNo_Lev_bd__13__7
L3553
Variable x : set
(*** Conj_add_SNo_Lev_bd__13__7 TMXCCfK7otff4E9jAQUHzmsLbPY9r6pLSP5 bounty of about 25 bars ***)
L3554
Variable y : set
L3555
Variable z : set
L3556
Variable w : set
L3557
Hypothesis H0 : SNo x
L3558
Hypothesis H1 : SNo y
L3559
Hypothesis H2 : ordinal (SNoLev x + SNoLev y)
L3560
Hypothesis H3 : (βˆ€u : set, u ∈ SNoS_ (SNoLev x) β†’ Subq (SNoLev (u + y)) (SNoLev u + SNoLev y))
L3561
Hypothesis H4 : w ∈ SNoS_ (SNoLev x)
L3562
Hypothesis H5 : SNo w
L3563
Hypothesis H6 : SNoLev w ∈ SNoLev x
L3564
Hypothesis H8 : ordinal z
L3565
Hypothesis H9 : Subq (SNoLev x + SNoLev y) z
L3566
Theorem. (Conj_add_SNo_Lev_bd__13__7)
Β¬ Subq (SNoLev (w + y)) (SNoLev w + SNoLev y)
In Proofgold the corresponding term root is 965c98... and proposition id is 8a6ac5...
Proof:
Proof not loaded.
End of Section Conj_add_SNo_Lev_bd__13__7
Beginning of Section Conj_add_SNo_Lev_bd__14__5
L3572
Variable x : set
(*** Conj_add_SNo_Lev_bd__14__5 TMLam8FtVoJ8UqUYZWgQtQesbKLeHCeQhBi bounty of about 25 bars ***)
L3573
Variable y : set
L3574
Variable z : set
L3575
Variable w : set
L3576
Hypothesis H0 : SNo x
L3577
Hypothesis H1 : SNo y
L3578
Hypothesis H2 : ordinal (SNoLev x + SNoLev y)
L3579
Hypothesis H3 : (βˆ€u : set, u ∈ SNoS_ (SNoLev x) β†’ Subq (SNoLev (u + y)) (SNoLev u + SNoLev y))
L3580
Hypothesis H4 : w ∈ SNoS_ (SNoLev x)
L3581
Hypothesis H6 : SNoLev w ∈ SNoLev x
L3582
Hypothesis H7 : z ∈ ordsucc (SNoLev (w + y))
L3583
Theorem. (Conj_add_SNo_Lev_bd__14__5)
ordinal z β†’ z ∈ SNoLev x + SNoLev y
In Proofgold the corresponding term root is 3a22f8... and proposition id is 931664...
Proof:
Proof not loaded.
End of Section Conj_add_SNo_Lev_bd__14__5
Beginning of Section Conj_add_SNo_Lev_bd__16__2
L3589
Variable x : set
(*** Conj_add_SNo_Lev_bd__16__2 TMRTZ3HrGidEjbTRVuq1tVf4xjohsQ1gibQ bounty of about 25 bars ***)
L3590
Variable y : set
L3591
Variable z : set
L3592
Variable w : set
L3593
Hypothesis H0 : SNo x
L3594
Hypothesis H1 : ordinal (SNoLev x + SNoLev y)
L3595
Hypothesis H3 : z ∈ ordsucc (SNoLev (x + w))
L3596
Hypothesis H4 : ordinal z
L3597
Hypothesis H5 : Subq (SNoLev x + SNoLev y) z
L3598
Hypothesis H6 : Subq (SNoLev (x + w)) (SNoLev x + SNoLev w)
L3599
Hypothesis H7 : SNoLev x + SNoLev w ∈ SNoLev x + SNoLev y
L3600
Theorem. (Conj_add_SNo_Lev_bd__16__2)
Β¬ Subq (SNoLev x + SNoLev w) (SNoLev x + SNoLev y)
In Proofgold the corresponding term root is aafc31... and proposition id is 04cec9...
Proof:
Proof not loaded.
End of Section Conj_add_SNo_Lev_bd__16__2
Beginning of Section Conj_add_SNo_Lev_bd__17__7
L3606
Variable x : set
(*** Conj_add_SNo_Lev_bd__17__7 TMFy7YVW2vf3xvNmm1QA6bRRvSzYJJKZj5r bounty of about 25 bars ***)
L3607
Variable y : set
L3608
Variable z : set
L3609
Variable w : set
L3610
Hypothesis H0 : SNo x
L3611
Hypothesis H1 : SNo y
L3612
Hypothesis H2 : ordinal (SNoLev x + SNoLev y)
L3613
Hypothesis H3 : SNo w
L3614
Hypothesis H4 : SNoLev w ∈ SNoLev y
L3615
Hypothesis H5 : z ∈ ordsucc (SNoLev (x + w))
L3616
Hypothesis H6 : ordinal z
L3617
Hypothesis H8 : Subq (SNoLev (x + w)) (SNoLev x + SNoLev w)
L3618
Theorem. (Conj_add_SNo_Lev_bd__17__7)
¬ SNoLev x + SNoLev w ∈ SNoLev x + SNoLev y
In Proofgold the corresponding term root is 4ef372... and proposition id is ed6e9f...
Proof:
Proof not loaded.
End of Section Conj_add_SNo_Lev_bd__17__7
Beginning of Section Conj_add_SNo_Lev_bd__20__5
L3624
Variable x : set
(*** Conj_add_SNo_Lev_bd__20__5 TMHVYfyhxYKFVawbnbFsZ4HwQkTsi65xnCC bounty of about 25 bars ***)
L3625
Variable y : set
L3626
Variable z : set
L3627
Variable w : set
L3628
Hypothesis H0 : SNo y
L3629
Hypothesis H1 : SNo w
L3630
Hypothesis H2 : z ∈ ordsucc (SNoLev (w + y))
L3631
Hypothesis H3 : ordinal z
L3632
Hypothesis H4 : Subq (SNoLev x + SNoLev y) z
L3633
Hypothesis H6 : SNoLev w + SNoLev y ∈ SNoLev x + SNoLev y
L3634
Theorem. (Conj_add_SNo_Lev_bd__20__5)
Β¬ Subq z (SNoLev (w + y))
In Proofgold the corresponding term root is a552af... and proposition id is bae0de...
Proof:
Proof not loaded.
End of Section Conj_add_SNo_Lev_bd__20__5
Beginning of Section Conj_add_SNo_Lev_bd__21__2
L3640
Variable x : set
(*** Conj_add_SNo_Lev_bd__21__2 TMXgUeJGYffmMUWGm6xudY5g5voEb2CEKTL bounty of about 25 bars ***)
L3641
Variable y : set
L3642
Variable z : set
L3643
Variable w : set
L3644
Hypothesis H0 : SNo y
L3645
Hypothesis H1 : ordinal (SNoLev x + SNoLev y)
L3646
Hypothesis H3 : z ∈ ordsucc (SNoLev (w + y))
L3647
Hypothesis H4 : ordinal z
L3648
Hypothesis H5 : Subq (SNoLev x + SNoLev y) z
L3649
Hypothesis H6 : Subq (SNoLev (w + y)) (SNoLev w + SNoLev y)
L3650
Hypothesis H7 : SNoLev w + SNoLev y ∈ SNoLev x + SNoLev y
L3651
Theorem. (Conj_add_SNo_Lev_bd__21__2)
Β¬ Subq (SNoLev w + SNoLev y) (SNoLev x + SNoLev y)
In Proofgold the corresponding term root is 2c6c95... and proposition id is 82480a...
Proof:
Proof not loaded.
End of Section Conj_add_SNo_Lev_bd__21__2
Beginning of Section Conj_add_SNo_Lev_bd__22__6
L3657
Variable x : set
(*** Conj_add_SNo_Lev_bd__22__6 TMVCFjRzpyxJX4Ma39CJrKqoDKAEhAxmbtj bounty of about 25 bars ***)
L3658
Variable y : set
L3659
Variable z : set
L3660
Variable w : set
L3661
Hypothesis H0 : SNo x
L3662
Hypothesis H1 : SNo y
L3663
Hypothesis H2 : ordinal (SNoLev x + SNoLev y)
L3664
Hypothesis H3 : SNo w
L3665
Hypothesis H4 : SNoLev w ∈ SNoLev x
L3666
Hypothesis H5 : z ∈ ordsucc (SNoLev (w + y))
L3667
Hypothesis H7 : Subq (SNoLev x + SNoLev y) z
L3668
Hypothesis H8 : Subq (SNoLev (w + y)) (SNoLev w + SNoLev y)
L3669
Theorem. (Conj_add_SNo_Lev_bd__22__6)
¬ SNoLev w + SNoLev y ∈ SNoLev x + SNoLev y
In Proofgold the corresponding term root is 619007... and proposition id is 345e59...
Proof:
Proof not loaded.
End of Section Conj_add_SNo_Lev_bd__22__6
Beginning of Section Conj_add_SNo_Lev_bd__23__3
L3675
Variable x : set
(*** Conj_add_SNo_Lev_bd__23__3 TMXPkP3ALi7mCddXRumP7vezPZkG7LV6K1J bounty of about 25 bars ***)
L3676
Variable y : set
L3677
Variable z : set
L3678
Variable w : set
L3679
Hypothesis H0 : SNo x
L3680
Hypothesis H1 : SNo y
L3681
Hypothesis H2 : ordinal (SNoLev x + SNoLev y)
L3682
Hypothesis H4 : w ∈ SNoS_ (SNoLev x)
L3683
Hypothesis H5 : SNo w
L3684
Hypothesis H6 : SNoLev w ∈ SNoLev x
L3685
Hypothesis H7 : z ∈ ordsucc (SNoLev (w + y))
L3686
Hypothesis H8 : ordinal z
L3687
Hypothesis H9 : Subq (SNoLev x + SNoLev y) z
L3688
Theorem. (Conj_add_SNo_Lev_bd__23__3)
Β¬ Subq (SNoLev (w + y)) (SNoLev w + SNoLev y)
In Proofgold the corresponding term root is 667ef2... and proposition id is b84773...
Proof:
Proof not loaded.
End of Section Conj_add_SNo_Lev_bd__23__3
Beginning of Section Conj_add_SNo_Lev_bd__23__5
L3694
Variable x : set
(*** Conj_add_SNo_Lev_bd__23__5 TMW2nLeL2Rd146GsAgGY4WjrySEBhppfQ6G bounty of about 25 bars ***)
L3695
Variable y : set
L3696
Variable z : set
L3697
Variable w : set
L3698
Hypothesis H0 : SNo x
L3699
Hypothesis H1 : SNo y
L3700
Hypothesis H2 : ordinal (SNoLev x + SNoLev y)
L3701
Hypothesis H3 : (βˆ€u : set, u ∈ SNoS_ (SNoLev x) β†’ Subq (SNoLev (u + y)) (SNoLev u + SNoLev y))
L3702
Hypothesis H4 : w ∈ SNoS_ (SNoLev x)
L3703
Hypothesis H6 : SNoLev w ∈ SNoLev x
L3704
Hypothesis H7 : z ∈ ordsucc (SNoLev (w + y))
L3705
Hypothesis H8 : ordinal z
L3706
Hypothesis H9 : Subq (SNoLev x + SNoLev y) z
L3707
Theorem. (Conj_add_SNo_Lev_bd__23__5)
Β¬ Subq (SNoLev (w + y)) (SNoLev w + SNoLev y)
In Proofgold the corresponding term root is 942f3a... and proposition id is 86a991...
Proof:
Proof not loaded.
End of Section Conj_add_SNo_Lev_bd__23__5
Beginning of Section Conj_add_SNo_Lev_bd__23__6
L3713
Variable x : set
(*** Conj_add_SNo_Lev_bd__23__6 TMbJLfVumt2xXTEw8dH7j5gNxbSBHG8u4mS bounty of about 25 bars ***)
L3714
Variable y : set
L3715
Variable z : set
L3716
Variable w : set
L3717
Hypothesis H0 : SNo x
L3718
Hypothesis H1 : SNo y
L3719
Hypothesis H2 : ordinal (SNoLev x + SNoLev y)
L3720
Hypothesis H3 : (βˆ€u : set, u ∈ SNoS_ (SNoLev x) β†’ Subq (SNoLev (u + y)) (SNoLev u + SNoLev y))
L3721
Hypothesis H4 : w ∈ SNoS_ (SNoLev x)
L3722
Hypothesis H5 : SNo w
L3723
Hypothesis H7 : z ∈ ordsucc (SNoLev (w + y))
L3724
Hypothesis H8 : ordinal z
L3725
Hypothesis H9 : Subq (SNoLev x + SNoLev y) z
L3726
Theorem. (Conj_add_SNo_Lev_bd__23__6)
Β¬ Subq (SNoLev (w + y)) (SNoLev w + SNoLev y)
In Proofgold the corresponding term root is 0ecea1... and proposition id is 57bf76...
Proof:
Proof not loaded.
End of Section Conj_add_SNo_Lev_bd__23__6
Beginning of Section Conj_add_SNo_Lev_bd__23__7
L3732
Variable x : set
(*** Conj_add_SNo_Lev_bd__23__7 TMXCCfK7otff4E9jAQUHzmsLbPY9r6pLSP5 bounty of about 25 bars ***)
L3733
Variable y : set
L3734
Variable z : set
L3735
Variable w : set
L3736
Hypothesis H0 : SNo x
L3737
Hypothesis H1 : SNo y
L3738
Hypothesis H2 : ordinal (SNoLev x + SNoLev y)
L3739
Hypothesis H3 : (βˆ€u : set, u ∈ SNoS_ (SNoLev x) β†’ Subq (SNoLev (u + y)) (SNoLev u + SNoLev y))
L3740
Hypothesis H4 : w ∈ SNoS_ (SNoLev x)
L3741
Hypothesis H5 : SNo w
L3742
Hypothesis H6 : SNoLev w ∈ SNoLev x
L3743
Hypothesis H8 : ordinal z
L3744
Hypothesis H9 : Subq (SNoLev x + SNoLev y) z
L3745
Theorem. (Conj_add_SNo_Lev_bd__23__7)
Β¬ Subq (SNoLev (w + y)) (SNoLev w + SNoLev y)
In Proofgold the corresponding term root is 965c98... and proposition id is 8a6ac5...
Proof:
Proof not loaded.
End of Section Conj_add_SNo_Lev_bd__23__7
Beginning of Section Conj_add_SNo_Lev_bd__24__5
L3751
Variable x : set
(*** Conj_add_SNo_Lev_bd__24__5 TMLam8FtVoJ8UqUYZWgQtQesbKLeHCeQhBi bounty of about 25 bars ***)
L3752
Variable y : set
L3753
Variable z : set
L3754
Variable w : set
L3755
Hypothesis H0 : SNo x
L3756
Hypothesis H1 : SNo y
L3757
Hypothesis H2 : ordinal (SNoLev x + SNoLev y)
L3758
Hypothesis H3 : (βˆ€u : set, u ∈ SNoS_ (SNoLev x) β†’ Subq (SNoLev (u + y)) (SNoLev u + SNoLev y))
L3759
Hypothesis H4 : w ∈ SNoS_ (SNoLev x)
L3760
Hypothesis H6 : SNoLev w ∈ SNoLev x
L3761
Hypothesis H7 : z ∈ ordsucc (SNoLev (w + y))
L3762
Theorem. (Conj_add_SNo_Lev_bd__24__5)
ordinal z β†’ z ∈ SNoLev x + SNoLev y
In Proofgold the corresponding term root is 3a22f8... and proposition id is 931664...
Proof:
Proof not loaded.
End of Section Conj_add_SNo_Lev_bd__24__5
Beginning of Section Conj_add_SNo_Lev_bd__29__1
L3768
Variable x : set
(*** Conj_add_SNo_Lev_bd__29__1 TMLrX1FqqLWS1fFPzPzads9SeKw15jcheat bounty of about 25 bars ***)
L3769
Variable y : set
L3770
Hypothesis H0 : SNo x
L3771
Hypothesis H2 : ordinal (SNoLev x + SNoLev y)
L3772
Hypothesis H3 : (βˆ€z : set, z ∈ SNoS_ (SNoLev x) β†’ Subq (SNoLev (z + y)) (SNoLev z + SNoLev y))
L3773
Hypothesis H4 : (βˆ€z : set, z ∈ SNoS_ (SNoLev y) β†’ Subq (SNoLev (x + z)) (SNoLev x + SNoLev z))
L3774
Hypothesis H5 : SNoLev (SNoCut (binunion (Repl (SNoL x) (Ξ»z : set β‡’ z + y)) (Repl (SNoL y) (add_SNo x))) (binunion (Repl (SNoR x) (Ξ»z : set β‡’ z + y)) (Repl (SNoR y) (add_SNo x)))) ∈ ordsucc (binunion (famunion (binunion (Repl (SNoL x) (Ξ»z : set β‡’ z + y)) (Repl (SNoL y) (add_SNo x))) (Ξ»z : set β‡’ ordsucc (SNoLev z))) (famunion (binunion (Repl (SNoR x) (Ξ»z : set β‡’ z + y)) (Repl (SNoR y) (add_SNo x))) (Ξ»z : set β‡’ ordsucc (SNoLev z))))
L3775
Hypothesis H6 : (βˆ€z : set, z ∈ Repl (SNoL x) (Ξ»w : set β‡’ w + y) β†’ (βˆ€p : set β†’ prop, (βˆ€w : set, w ∈ SNoS_ (SNoLev x) β†’ z = w + y β†’ SNo w β†’ SNoLev w ∈ SNoLev x β†’ w < x β†’ p (w + y)) β†’ p z))
L3776
Hypothesis H7 : (βˆ€z : set, z ∈ Repl (SNoL y) (add_SNo x) β†’ (βˆ€p : set β†’ prop, (βˆ€w : set, w ∈ SNoS_ (SNoLev y) β†’ z = x + w β†’ SNo w β†’ SNoLev w ∈ SNoLev y β†’ w < y β†’ p (x + w)) β†’ p z))
L3777
Hypothesis H8 : (βˆ€z : set, z ∈ Repl (SNoR x) (Ξ»w : set β‡’ w + y) β†’ (βˆ€p : set β†’ prop, (βˆ€w : set, w ∈ SNoS_ (SNoLev x) β†’ z = w + y β†’ SNo w β†’ SNoLev w ∈ SNoLev x β†’ x < w β†’ p (w + y)) β†’ p z))
L3778
Hypothesis H9 : (βˆ€z : set, z ∈ Repl (SNoR y) (add_SNo x) β†’ (βˆ€p : set β†’ prop, (βˆ€w : set, w ∈ SNoS_ (SNoLev y) β†’ z = x + w β†’ SNo w β†’ SNoLev w ∈ SNoLev y β†’ y < w β†’ p (x + w)) β†’ p z))
L3779
Hypothesis H10 : (βˆ€z : set, z ∈ Repl (SNoL x) (Ξ»w : set β‡’ w + y) β†’ SNo z)
L3780
Hypothesis H11 : (βˆ€z : set, z ∈ Repl (SNoL y) (add_SNo x) β†’ SNo z)
L3781
Theorem. (Conj_add_SNo_Lev_bd__29__1)
(βˆ€z : set, z ∈ Repl (SNoR x) (Ξ»w : set β‡’ w + y) β†’ SNo z) β†’ Subq (SNoLev (SNoCut (binunion (Repl (SNoL x) (Ξ»z : set β‡’ z + y)) (Repl (SNoL y) (add_SNo x))) (binunion (Repl (SNoR x) (Ξ»z : set β‡’ z + y)) (Repl (SNoR y) (add_SNo x))))) (SNoLev x + SNoLev y)
In Proofgold the corresponding term root is fbb395... and proposition id is 021de9...
Proof:
Proof not loaded.
End of Section Conj_add_SNo_Lev_bd__29__1
Beginning of Section Conj_add_SNo_Lev_bd__29__4
L3787
Variable x : set
(*** Conj_add_SNo_Lev_bd__29__4 TMSXhzeQQJdgvuVACNwLkFgbAje914asX6N bounty of about 25 bars ***)
L3788
Variable y : set
L3789
Hypothesis H0 : SNo x
L3790
Hypothesis H1 : SNo y
L3791
Hypothesis H2 : ordinal (SNoLev x + SNoLev y)
L3792
Hypothesis H3 : (βˆ€z : set, z ∈ SNoS_ (SNoLev x) β†’ Subq (SNoLev (z + y)) (SNoLev z + SNoLev y))
L3793
Hypothesis H5 : SNoLev (SNoCut (binunion (Repl (SNoL x) (Ξ»z : set β‡’ z + y)) (Repl (SNoL y) (add_SNo x))) (binunion (Repl (SNoR x) (Ξ»z : set β‡’ z + y)) (Repl (SNoR y) (add_SNo x)))) ∈ ordsucc (binunion (famunion (binunion (Repl (SNoL x) (Ξ»z : set β‡’ z + y)) (Repl (SNoL y) (add_SNo x))) (Ξ»z : set β‡’ ordsucc (SNoLev z))) (famunion (binunion (Repl (SNoR x) (Ξ»z : set β‡’ z + y)) (Repl (SNoR y) (add_SNo x))) (Ξ»z : set β‡’ ordsucc (SNoLev z))))
L3794
Hypothesis H6 : (βˆ€z : set, z ∈ Repl (SNoL x) (Ξ»w : set β‡’ w + y) β†’ (βˆ€p : set β†’ prop, (βˆ€w : set, w ∈ SNoS_ (SNoLev x) β†’ z = w + y β†’ SNo w β†’ SNoLev w ∈ SNoLev x β†’ w < x β†’ p (w + y)) β†’ p z))
L3795
Hypothesis H7 : (βˆ€z : set, z ∈ Repl (SNoL y) (add_SNo x) β†’ (βˆ€p : set β†’ prop, (βˆ€w : set, w ∈ SNoS_ (SNoLev y) β†’ z = x + w β†’ SNo w β†’ SNoLev w ∈ SNoLev y β†’ w < y β†’ p (x + w)) β†’ p z))
L3796
Hypothesis H8 : (βˆ€z : set, z ∈ Repl (SNoR x) (Ξ»w : set β‡’ w + y) β†’ (βˆ€p : set β†’ prop, (βˆ€w : set, w ∈ SNoS_ (SNoLev x) β†’ z = w + y β†’ SNo w β†’ SNoLev w ∈ SNoLev x β†’ x < w β†’ p (w + y)) β†’ p z))
L3797
Hypothesis H9 : (βˆ€z : set, z ∈ Repl (SNoR y) (add_SNo x) β†’ (βˆ€p : set β†’ prop, (βˆ€w : set, w ∈ SNoS_ (SNoLev y) β†’ z = x + w β†’ SNo w β†’ SNoLev w ∈ SNoLev y β†’ y < w β†’ p (x + w)) β†’ p z))
L3798
Hypothesis H10 : (βˆ€z : set, z ∈ Repl (SNoL x) (Ξ»w : set β‡’ w + y) β†’ SNo z)
L3799
Hypothesis H11 : (βˆ€z : set, z ∈ Repl (SNoL y) (add_SNo x) β†’ SNo z)
L3800
Theorem. (Conj_add_SNo_Lev_bd__29__4)
(βˆ€z : set, z ∈ Repl (SNoR x) (Ξ»w : set β‡’ w + y) β†’ SNo z) β†’ Subq (SNoLev (SNoCut (binunion (Repl (SNoL x) (Ξ»z : set β‡’ z + y)) (Repl (SNoL y) (add_SNo x))) (binunion (Repl (SNoR x) (Ξ»z : set β‡’ z + y)) (Repl (SNoR y) (add_SNo x))))) (SNoLev x + SNoLev y)
In Proofgold the corresponding term root is a470e2... and proposition id is 8f7365...
Proof:
Proof not loaded.
End of Section Conj_add_SNo_Lev_bd__29__4
Beginning of Section Conj_add_SNo_Lev_bd__29__5
L3806
Variable x : set
(*** Conj_add_SNo_Lev_bd__29__5 TMQXTk36PjeswuqCuKuaL9UtahR13RK4TjB bounty of about 25 bars ***)
L3807
Variable y : set
L3808
Hypothesis H0 : SNo x
L3809
Hypothesis H1 : SNo y
L3810
Hypothesis H2 : ordinal (SNoLev x + SNoLev y)
L3811
Hypothesis H3 : (βˆ€z : set, z ∈ SNoS_ (SNoLev x) β†’ Subq (SNoLev (z + y)) (SNoLev z + SNoLev y))
L3812
Hypothesis H4 : (βˆ€z : set, z ∈ SNoS_ (SNoLev y) β†’ Subq (SNoLev (x + z)) (SNoLev x + SNoLev z))
L3813
Hypothesis H6 : (βˆ€z : set, z ∈ Repl (SNoL x) (Ξ»w : set β‡’ w + y) β†’ (βˆ€p : set β†’ prop, (βˆ€w : set, w ∈ SNoS_ (SNoLev x) β†’ z = w + y β†’ SNo w β†’ SNoLev w ∈ SNoLev x β†’ w < x β†’ p (w + y)) β†’ p z))
L3814
Hypothesis H7 : (βˆ€z : set, z ∈ Repl (SNoL y) (add_SNo x) β†’ (βˆ€p : set β†’ prop, (βˆ€w : set, w ∈ SNoS_ (SNoLev y) β†’ z = x + w β†’ SNo w β†’ SNoLev w ∈ SNoLev y β†’ w < y β†’ p (x + w)) β†’ p z))
L3815
Hypothesis H8 : (βˆ€z : set, z ∈ Repl (SNoR x) (Ξ»w : set β‡’ w + y) β†’ (βˆ€p : set β†’ prop, (βˆ€w : set, w ∈ SNoS_ (SNoLev x) β†’ z = w + y β†’ SNo w β†’ SNoLev w ∈ SNoLev x β†’ x < w β†’ p (w + y)) β†’ p z))
L3816
Hypothesis H9 : (βˆ€z : set, z ∈ Repl (SNoR y) (add_SNo x) β†’ (βˆ€p : set β†’ prop, (βˆ€w : set, w ∈ SNoS_ (SNoLev y) β†’ z = x + w β†’ SNo w β†’ SNoLev w ∈ SNoLev y β†’ y < w β†’ p (x + w)) β†’ p z))
L3817
Hypothesis H10 : (βˆ€z : set, z ∈ Repl (SNoL x) (Ξ»w : set β‡’ w + y) β†’ SNo z)
L3818
Hypothesis H11 : (βˆ€z : set, z ∈ Repl (SNoL y) (add_SNo x) β†’ SNo z)
L3819
Theorem. (Conj_add_SNo_Lev_bd__29__5)
(βˆ€z : set, z ∈ Repl (SNoR x) (Ξ»w : set β‡’ w + y) β†’ SNo z) β†’ Subq (SNoLev (SNoCut (binunion (Repl (SNoL x) (Ξ»z : set β‡’ z + y)) (Repl (SNoL y) (add_SNo x))) (binunion (Repl (SNoR x) (Ξ»z : set β‡’ z + y)) (Repl (SNoR y) (add_SNo x))))) (SNoLev x + SNoLev y)
In Proofgold the corresponding term root is 9db3ae... and proposition id is e0aa39...
Proof:
Proof not loaded.
End of Section Conj_add_SNo_Lev_bd__29__5
Beginning of Section Conj_add_SNo_Lev_bd__34__6
L3825
Variable x : set
(*** Conj_add_SNo_Lev_bd__34__6 TMTSVhgQwhRwZ3p8XDLdqtkioQTu9dVgnwZ bounty of about 25 bars ***)
L3826
Variable y : set
L3827
Hypothesis H0 : SNo x
L3828
Hypothesis H1 : SNo y
L3829
Hypothesis H2 : ordinal (SNoLev x + SNoLev y)
L3830
Hypothesis H3 : (βˆ€z : set, z ∈ SNoS_ (SNoLev x) β†’ Subq (SNoLev (z + y)) (SNoLev z + SNoLev y))
L3831
Hypothesis H4 : (βˆ€z : set, z ∈ SNoS_ (SNoLev y) β†’ Subq (SNoLev (x + z)) (SNoLev x + SNoLev z))
L3832
Hypothesis H5 : SNoLev (SNoCut (binunion (Repl (SNoL x) (Ξ»z : set β‡’ z + y)) (Repl (SNoL y) (add_SNo x))) (binunion (Repl (SNoR x) (Ξ»z : set β‡’ z + y)) (Repl (SNoR y) (add_SNo x)))) ∈ ordsucc (binunion (famunion (binunion (Repl (SNoL x) (Ξ»z : set β‡’ z + y)) (Repl (SNoL y) (add_SNo x))) (Ξ»z : set β‡’ ordsucc (SNoLev z))) (famunion (binunion (Repl (SNoR x) (Ξ»z : set β‡’ z + y)) (Repl (SNoR y) (add_SNo x))) (Ξ»z : set β‡’ ordsucc (SNoLev z))))
L3833
Theorem. (Conj_add_SNo_Lev_bd__34__6)
(βˆ€z : set, z ∈ Repl (SNoL y) (add_SNo x) β†’ (βˆ€p : set β†’ prop, (βˆ€w : set, w ∈ SNoS_ (SNoLev y) β†’ z = x + w β†’ SNo w β†’ SNoLev w ∈ SNoLev y β†’ w < y β†’ p (x + w)) β†’ p z)) β†’ Subq (SNoLev (SNoCut (binunion (Repl (SNoL x) (Ξ»z : set β‡’ z + y)) (Repl (SNoL y) (add_SNo x))) (binunion (Repl (SNoR x) (Ξ»z : set β‡’ z + y)) (Repl (SNoR y) (add_SNo x))))) (SNoLev x + SNoLev y)
In Proofgold the corresponding term root is dd28d4... and proposition id is 8f007c...
Proof:
Proof not loaded.
End of Section Conj_add_SNo_Lev_bd__34__6
Beginning of Section Conj_add_SNo_Lev_bd__38__0
L3839
Variable x : set
(*** Conj_add_SNo_Lev_bd__38__0 TMYuMyHAiGLVkiryeWSrYc3pjFaLxbdxwc7 bounty of about 25 bars ***)
L3840
Variable y : set
L3841
Hypothesis H1 : SNo y
L3842
Theorem. (Conj_add_SNo_Lev_bd__38__0)
SNo (x + y) β†’ (βˆ€z : set, z ∈ SNoS_ (SNoLev x) β†’ Subq (SNoLev (z + y)) (SNoLev z + SNoLev y)) β†’ (βˆ€z : set, z ∈ SNoS_ (SNoLev y) β†’ Subq (SNoLev (x + z)) (SNoLev x + SNoLev z)) β†’ (βˆ€z : set, z ∈ SNoS_ (SNoLev x) β†’ (βˆ€w : set, w ∈ SNoS_ (SNoLev y) β†’ Subq (SNoLev (z + w)) (SNoLev z + SNoLev w))) β†’ Subq (SNoLev (x + y)) (SNoLev x + SNoLev y)
In Proofgold the corresponding term root is ab05dc... and proposition id is 47dd83...
Proof:
Proof not loaded.
End of Section Conj_add_SNo_Lev_bd__38__0
Beginning of Section Conj_add_SNo_SNoS_omega__1__2
L3848
Variable x : set
(*** Conj_add_SNo_SNoS_omega__1__2 TMHecR1j4ZYbMm1pvFmw1Bm8bxJUrEY5KgA bounty of about 25 bars ***)
L3849
Variable y : set
L3850
Hypothesis H0 : SNoLev x ∈ Ο‰
L3851
Hypothesis H1 : SNo x
L3852
Hypothesis H3 : SNo y
L3853
Theorem. (Conj_add_SNo_SNoS_omega__1__2)
ordinal (SNoLev (x + y)) β†’ SNoLev (x + y) ∈ Ο‰
In Proofgold the corresponding term root is 8b5245... and proposition id is c3218c...
Proof:
Proof not loaded.
End of Section Conj_add_SNo_SNoS_omega__1__2
Beginning of Section Conj_add_SNo_minus_Lt_lem__2__6
L3859
Variable x : set
(*** Conj_add_SNo_minus_Lt_lem__2__6 TMLkvCjKdTWPrJaMwtmGX7t1XXAsEUhaECE bounty of about 25 bars ***)
L3860
Variable y : set
L3861
Variable z : set
L3862
Variable w : set
L3863
Variable u : set
L3864
Variable v : set
L3865
Hypothesis H0 : SNo x
L3866
Hypothesis H1 : SNo y
L3867
Hypothesis H2 : SNo z
L3868
Hypothesis H3 : SNo w
L3869
Hypothesis H4 : SNo u
L3870
Hypothesis H5 : SNo v
L3871
Hypothesis H7 : SNo (- z)
L3872
Hypothesis H8 : SNo (- v)
L3873
Theorem. (Conj_add_SNo_minus_Lt_lem__2__6)
SNo (x + y) β†’ (x + y + - z) < w + u + - v
In Proofgold the corresponding term root is 69a631... and proposition id is 27e216...
Proof:
Proof not loaded.
End of Section Conj_add_SNo_minus_Lt_lem__2__6
Beginning of Section Conj_add_SNo_Lt_subprop3c__2__3
L3879
Variable x : set
(*** Conj_add_SNo_Lt_subprop3c__2__3 TMQaci4f6pqnx4w8NLzy5jwpmUCcAu6Aa3C bounty of about 25 bars ***)
L3880
Variable y : set
L3881
Variable z : set
L3882
Variable w : set
L3883
Variable u : set
L3884
Variable v : set
L3885
Hypothesis H0 : SNo x
L3886
Hypothesis H1 : SNo y
L3887
Hypothesis H2 : SNo z
L3888
Hypothesis H4 : SNo u
L3889
Hypothesis H5 : SNo v
L3890
Hypothesis H6 : (y + v) < u
L3891
Hypothesis H7 : (x + z) < v + w
L3892
Theorem. (Conj_add_SNo_Lt_subprop3c__2__3)
SNo (x + z) β†’ (x + y + z) < w + u
In Proofgold the corresponding term root is f816c7... and proposition id is d3774c...
Proof:
Proof not loaded.
End of Section Conj_add_SNo_Lt_subprop3c__2__3
Beginning of Section Conj_add_SNo_Lt_subprop3c__2__7
L3898
Variable x : set
(*** Conj_add_SNo_Lt_subprop3c__2__7 TMaot6HShaD8fYeXCtFqFkcZq8AkD1BQ7W6 bounty of about 25 bars ***)
L3899
Variable y : set
L3900
Variable z : set
L3901
Variable w : set
L3902
Variable u : set
L3903
Variable v : set
L3904
Hypothesis H0 : SNo x
L3905
Hypothesis H1 : SNo y
L3906
Hypothesis H2 : SNo z
L3907
Hypothesis H3 : SNo w
L3908
Hypothesis H4 : SNo u
L3909
Hypothesis H5 : SNo v
L3910
Hypothesis H6 : (y + v) < u
L3911
Theorem. (Conj_add_SNo_Lt_subprop3c__2__7)
SNo (x + z) β†’ (x + y + z) < w + u
In Proofgold the corresponding term root is c01095... and proposition id is 51d98e...
Proof:
Proof not loaded.
End of Section Conj_add_SNo_Lt_subprop3c__2__7
Beginning of Section Conj_add_SNo_Lt_subprop3c__3__0
L3917
Variable x : set
(*** Conj_add_SNo_Lt_subprop3c__3__0 TMUrc8oR5bhWTwTv55xSL2e2aMEfaTAmjCc bounty of about 25 bars ***)
L3918
Variable y : set
L3919
Variable z : set
L3920
Variable w : set
L3921
Variable u : set
L3922
Variable v : set
L3923
Variable x2 : set
L3924
Variable y2 : set
L3925
Hypothesis H1 : SNo y
L3926
Hypothesis H2 : SNo z
L3927
Hypothesis H3 : SNo w
L3928
Hypothesis H4 : SNo u
L3929
Hypothesis H5 : SNo v
L3930
Hypothesis H6 : SNo x2
L3931
Hypothesis H7 : SNo y2
L3932
Hypothesis H8 : (x + v) < x2 + y2
L3933
Hypothesis H9 : (y + y2) < u
L3934
Hypothesis H10 : (x2 + z) < w + v
L3935
Theorem. (Conj_add_SNo_Lt_subprop3c__3__0)
(x + z) < y2 + w β†’ (x + y + z) < w + u
In Proofgold the corresponding term root is 67ca67... and proposition id is c09040...
Proof:
Proof not loaded.
End of Section Conj_add_SNo_Lt_subprop3c__3__0
Beginning of Section Conj_add_SNo_Lt_subprop3c__3__6
L3941
Variable x : set
(*** Conj_add_SNo_Lt_subprop3c__3__6 TMQKTMTy1p4K93n7jXgxNHg26euZrRjZTQc bounty of about 25 bars ***)
L3942
Variable y : set
L3943
Variable z : set
L3944
Variable w : set
L3945
Variable u : set
L3946
Variable v : set
L3947
Variable x2 : set
L3948
Variable y2 : set
L3949
Hypothesis H0 : SNo x
L3950
Hypothesis H1 : SNo y
L3951
Hypothesis H2 : SNo z
L3952
Hypothesis H3 : SNo w
L3953
Hypothesis H4 : SNo u
L3954
Hypothesis H5 : SNo v
L3955
Hypothesis H7 : SNo y2
L3956
Hypothesis H8 : (x + v) < x2 + y2
L3957
Hypothesis H9 : (y + y2) < u
L3958
Hypothesis H10 : (x2 + z) < w + v
L3959
Theorem. (Conj_add_SNo_Lt_subprop3c__3__6)
(x + z) < y2 + w β†’ (x + y + z) < w + u
In Proofgold the corresponding term root is 63abd7... and proposition id is 09b2f3...
Proof:
Proof not loaded.
End of Section Conj_add_SNo_Lt_subprop3c__3__6
Beginning of Section Conj_add_SNo_Lt_subprop3d__2__4
L3965
Variable x : set
(*** Conj_add_SNo_Lt_subprop3d__2__4 TMYDaeETVd7deMPazHtzGPu4THoRLSTRuMW bounty of about 25 bars ***)
L3966
Variable y : set
L3967
Variable z : set
L3968
Variable w : set
L3969
Variable u : set
L3970
Variable v : set
L3971
Hypothesis H0 : SNo x
L3972
Hypothesis H1 : SNo y
L3973
Hypothesis H2 : SNo z
L3974
Hypothesis H3 : SNo w
L3975
Hypothesis H5 : SNo v
L3976
Hypothesis H6 : (x + u) < v + w
L3977
Theorem. (Conj_add_SNo_Lt_subprop3d__2__4)
x + y + z + u = (y + z) + x + u β†’ (x + y + z + u) < (y + z + w) + v
In Proofgold the corresponding term root is d8ee10... and proposition id is 3834a5...
Proof:
Proof not loaded.
End of Section Conj_add_SNo_Lt_subprop3d__2__4
Beginning of Section Conj_mul_SNo_eq__1__0
L3983
Variable x : set
(*** Conj_mul_SNo_eq__1__0 TMHNaUMQnp8mygqy5bkQwBG1Td5cZVPmmfL bounty of about 25 bars ***)
L3984
Variable y : set
L3985
Variable g : (set β†’ (set β†’ set))
L3986
Variable h : (set β†’ (set β†’ set))
L3987
Variable z : set
L3988
Variable w : set
L3989
Hypothesis H1 : z ∈ SNoS_ (SNoLev x)
L3990
Hypothesis H2 : SNo w
L3991
Hypothesis H3 : g z y = h z y
L3992
Hypothesis H4 : g x w = h x w
L3993
Theorem. (Conj_mul_SNo_eq__1__0)
g z w = h z w β†’ g z y + g x w + - (g z w) = h z y + h x w + - (h z w)
In Proofgold the corresponding term root is ab8d72... and proposition id is 36a11b...
Proof:
Proof not loaded.
End of Section Conj_mul_SNo_eq__1__0
Beginning of Section Conj_mul_SNo_eq__1__1
L3999
Variable x : set
(*** Conj_mul_SNo_eq__1__1 TMaU7woDD25pzpeo87GUsLTfT2MR71x4vAP bounty of about 25 bars ***)
L4000
Variable y : set
L4001
Variable g : (set β†’ (set β†’ set))
L4002
Variable h : (set β†’ (set β†’ set))
L4003
Variable z : set
L4004
Variable w : set
L4005
Hypothesis H0 : (βˆ€u : set, u ∈ SNoS_ (SNoLev x) β†’ (βˆ€v : set, SNo v β†’ g u v = h u v))
L4006
Hypothesis H2 : SNo w
L4007
Hypothesis H3 : g z y = h z y
L4008
Hypothesis H4 : g x w = h x w
L4009
Theorem. (Conj_mul_SNo_eq__1__1)
g z w = h z w β†’ g z y + g x w + - (g z w) = h z y + h x w + - (h z w)
In Proofgold the corresponding term root is 41a020... and proposition id is 712f00...
Proof:
Proof not loaded.
End of Section Conj_mul_SNo_eq__1__1
Beginning of Section Conj_mul_SNo_eq__1__2
L4015
Variable x : set
(*** Conj_mul_SNo_eq__1__2 TML44XGPM1TnRxBFEotSt3DiX9pCv8byiYt bounty of about 25 bars ***)
L4016
Variable y : set
L4017
Variable g : (set β†’ (set β†’ set))
L4018
Variable h : (set β†’ (set β†’ set))
L4019
Variable z : set
L4020
Variable w : set
L4021
Hypothesis H0 : (βˆ€u : set, u ∈ SNoS_ (SNoLev x) β†’ (βˆ€v : set, SNo v β†’ g u v = h u v))
L4022
Hypothesis H1 : z ∈ SNoS_ (SNoLev x)
L4023
Hypothesis H3 : g z y = h z y
L4024
Hypothesis H4 : g x w = h x w
L4025
Theorem. (Conj_mul_SNo_eq__1__2)
g z w = h z w β†’ g z y + g x w + - (g z w) = h z y + h x w + - (h z w)
In Proofgold the corresponding term root is fca8d6... and proposition id is 68a430...
Proof:
Proof not loaded.
End of Section Conj_mul_SNo_eq__1__2
Beginning of Section Conj_mul_SNo_eq__2__3
L4031
Variable x : set
(*** Conj_mul_SNo_eq__2__3 TMYevhbTJiUkHxoZCg315NG4SJi3UUacxR5 bounty of about 25 bars ***)
L4032
Variable y : set
L4033
Variable g : (set β†’ (set β†’ set))
L4034
Variable h : (set β†’ (set β†’ set))
L4035
Variable z : set
L4036
Variable w : set
L4037
Hypothesis H0 : (βˆ€u : set, u ∈ SNoS_ (SNoLev x) β†’ (βˆ€v : set, SNo v β†’ g u v = h u v))
L4038
Hypothesis H1 : (βˆ€u : set, u ∈ SNoS_ (SNoLev y) β†’ g x u = h x u)
L4039
Hypothesis H2 : z ∈ SNoS_ (SNoLev x)
L4040
Hypothesis H4 : SNo w
L4041
Hypothesis H5 : g z y = h z y
L4042
Theorem. (Conj_mul_SNo_eq__2__3)
g x w = h x w β†’ g z y + g x w + - (g z w) = h z y + h x w + - (h z w)
In Proofgold the corresponding term root is 68613e... and proposition id is e4f91c...
Proof:
Proof not loaded.
End of Section Conj_mul_SNo_eq__2__3
Beginning of Section Conj_mul_SNo_eq__3__3
L4048
Variable x : set
(*** Conj_mul_SNo_eq__3__3 TMc9x9bnxHCRdv9pdrLT23SFMWU723q3CF5 bounty of about 25 bars ***)
L4049
Variable y : set
L4050
Variable g : (set β†’ (set β†’ set))
L4051
Variable h : (set β†’ (set β†’ set))
L4052
Variable z : set
L4053
Variable w : set
L4054
Hypothesis H0 : SNo y
L4055
Hypothesis H1 : (βˆ€u : set, u ∈ SNoS_ (SNoLev x) β†’ (βˆ€v : set, SNo v β†’ g u v = h u v))
L4056
Hypothesis H2 : (βˆ€u : set, u ∈ SNoS_ (SNoLev y) β†’ g x u = h x u)
L4057
Hypothesis H4 : w ∈ SNoS_ (SNoLev y)
L4058
Hypothesis H5 : SNo w
L4059
Theorem. (Conj_mul_SNo_eq__3__3)
g z y = h z y β†’ g z y + g x w + - (g z w) = h z y + h x w + - (h z w)
In Proofgold the corresponding term root is 7dc197... and proposition id is b02b25...
Proof:
Proof not loaded.
End of Section Conj_mul_SNo_eq__3__3
Beginning of Section Conj_mul_SNo_eq__3__4
L4065
Variable x : set
(*** Conj_mul_SNo_eq__3__4 TMbKdCNpgb8ceK8K44wes4z26jE2eV5bUXK bounty of about 25 bars ***)
L4066
Variable y : set
L4067
Variable g : (set β†’ (set β†’ set))
L4068
Variable h : (set β†’ (set β†’ set))
L4069
Variable z : set
L4070
Variable w : set
L4071
Hypothesis H0 : SNo y
L4072
Hypothesis H1 : (βˆ€u : set, u ∈ SNoS_ (SNoLev x) β†’ (βˆ€v : set, SNo v β†’ g u v = h u v))
L4073
Hypothesis H2 : (βˆ€u : set, u ∈ SNoS_ (SNoLev y) β†’ g x u = h x u)
L4074
Hypothesis H3 : z ∈ SNoS_ (SNoLev x)
L4075
Hypothesis H5 : SNo w
L4076
Theorem. (Conj_mul_SNo_eq__3__4)
g z y = h z y β†’ g z y + g x w + - (g z w) = h z y + h x w + - (h z w)
In Proofgold the corresponding term root is 392627... and proposition id is 44641d...
Proof:
Proof not loaded.
End of Section Conj_mul_SNo_eq__3__4
Beginning of Section Conj_mul_SNo_eq__4__1
L4082
Variable x : set
(*** Conj_mul_SNo_eq__4__1 TMauMmtuJKRSxbZsSigSs3qQzCRTArRKSzx bounty of about 25 bars ***)
L4083
Variable y : set
L4084
Variable g : (set β†’ (set β†’ set))
L4085
Variable h : (set β†’ (set β†’ set))
L4086
Variable z : set
L4087
Variable w : set
L4088
Hypothesis H0 : SNo y
L4089
Hypothesis H2 : (βˆ€u : set, u ∈ SNoS_ (SNoLev y) β†’ g x u = h x u)
L4090
Hypothesis H3 : w ∈ SNoL y
L4091
Hypothesis H4 : z ∈ SNoS_ (SNoLev x)
L4092
Hypothesis H5 : w ∈ SNoS_ (SNoLev y)
L4093
Theorem. (Conj_mul_SNo_eq__4__1)
SNo w β†’ g z y + g x w + - (g z w) = h z y + h x w + - (h z w)
In Proofgold the corresponding term root is 9c249a... and proposition id is 51b9a9...
Proof:
Proof not loaded.
End of Section Conj_mul_SNo_eq__4__1
Beginning of Section Conj_mul_SNo_eq__4__3
L4099
Variable x : set
(*** Conj_mul_SNo_eq__4__3 TMM9PLC4canhoRGg8wSstnVhUP3kwndnHts bounty of about 25 bars ***)
L4100
Variable y : set
L4101
Variable g : (set β†’ (set β†’ set))
L4102
Variable h : (set β†’ (set β†’ set))
L4103
Variable z : set
L4104
Variable w : set
L4105
Hypothesis H0 : SNo y
L4106
Hypothesis H1 : (βˆ€u : set, u ∈ SNoS_ (SNoLev x) β†’ (βˆ€v : set, SNo v β†’ g u v = h u v))
L4107
Hypothesis H2 : (βˆ€u : set, u ∈ SNoS_ (SNoLev y) β†’ g x u = h x u)
L4108
Hypothesis H4 : z ∈ SNoS_ (SNoLev x)
L4109
Hypothesis H5 : w ∈ SNoS_ (SNoLev y)
L4110
Theorem. (Conj_mul_SNo_eq__4__3)
SNo w β†’ g z y + g x w + - (g z w) = h z y + h x w + - (h z w)
In Proofgold the corresponding term root is 7b822c... and proposition id is 3816fc...
Proof:
Proof not loaded.
End of Section Conj_mul_SNo_eq__4__3
Beginning of Section Conj_mul_SNo_eq__5__1
L4116
Variable x : set
(*** Conj_mul_SNo_eq__5__1 TMbMyv2xSKFXdojrpPHAhGYYnxTvNRzKX1W bounty of about 25 bars ***)
L4117
Variable y : set
L4118
Variable g : (set β†’ (set β†’ set))
L4119
Variable h : (set β†’ (set β†’ set))
L4120
Variable z : set
L4121
Variable w : set
L4122
Hypothesis H0 : SNo y
L4123
Hypothesis H2 : (βˆ€u : set, u ∈ SNoS_ (SNoLev y) β†’ g x u = h x u)
L4124
Hypothesis H3 : w ∈ SNoL y
L4125
Hypothesis H4 : z ∈ SNoS_ (SNoLev x)
L4126
Theorem. (Conj_mul_SNo_eq__5__1)
w ∈ SNoS_ (SNoLev y) β†’ g z y + g x w + - (g z w) = h z y + h x w + - (h z w)
In Proofgold the corresponding term root is d0c5ad... and proposition id is ed0c24...
Proof:
Proof not loaded.
End of Section Conj_mul_SNo_eq__5__1
Beginning of Section Conj_mul_SNo_eq__6__0
L4132
Variable x : set
(*** Conj_mul_SNo_eq__6__0 TMTv14qJ7Jd2ukEaCcgcTkSCrLZto6mzqXU bounty of about 25 bars ***)
L4133
Variable y : set
L4134
Variable g : (set β†’ (set β†’ set))
L4135
Variable h : (set β†’ (set β†’ set))
L4136
Variable z : set
L4137
Variable w : set
L4138
Hypothesis H1 : SNo y
L4139
Hypothesis H2 : (βˆ€u : set, u ∈ SNoS_ (SNoLev x) β†’ (βˆ€v : set, SNo v β†’ g u v = h u v))
L4140
Hypothesis H3 : (βˆ€u : set, u ∈ SNoS_ (SNoLev y) β†’ g x u = h x u)
L4141
Hypothesis H4 : z ∈ SNoL x
L4142
Hypothesis H5 : w ∈ SNoL y
L4143
Theorem. (Conj_mul_SNo_eq__6__0)
z ∈ SNoS_ (SNoLev x) β†’ g z y + g x w + - (g z w) = h z y + h x w + - (h z w)
In Proofgold the corresponding term root is 4ab470... and proposition id is 7d6b56...
Proof:
Proof not loaded.
End of Section Conj_mul_SNo_eq__6__0
Beginning of Section Conj_mul_SNo_eq__7__0
L4149
Variable x : set
(*** Conj_mul_SNo_eq__7__0 TMHNaUMQnp8mygqy5bkQwBG1Td5cZVPmmfL bounty of about 25 bars ***)
L4150
Variable y : set
L4151
Variable g : (set β†’ (set β†’ set))
L4152
Variable h : (set β†’ (set β†’ set))
L4153
Variable z : set
L4154
Variable w : set
L4155
Hypothesis H1 : z ∈ SNoS_ (SNoLev x)
L4156
Hypothesis H2 : SNo w
L4157
Hypothesis H3 : g z y = h z y
L4158
Hypothesis H4 : g x w = h x w
L4159
Theorem. (Conj_mul_SNo_eq__7__0)
g z w = h z w β†’ g z y + g x w + - (g z w) = h z y + h x w + - (h z w)
In Proofgold the corresponding term root is ab8d72... and proposition id is 36a11b...
Proof:
Proof not loaded.
End of Section Conj_mul_SNo_eq__7__0
Beginning of Section Conj_mul_SNo_eq__7__1
L4165
Variable x : set
(*** Conj_mul_SNo_eq__7__1 TMaU7woDD25pzpeo87GUsLTfT2MR71x4vAP bounty of about 25 bars ***)
L4166
Variable y : set
L4167
Variable g : (set β†’ (set β†’ set))
L4168
Variable h : (set β†’ (set β†’ set))
L4169
Variable z : set
L4170
Variable w : set
L4171
Hypothesis H0 : (βˆ€u : set, u ∈ SNoS_ (SNoLev x) β†’ (βˆ€v : set, SNo v β†’ g u v = h u v))
L4172
Hypothesis H2 : SNo w
L4173
Hypothesis H3 : g z y = h z y
L4174
Hypothesis H4 : g x w = h x w
L4175
Theorem. (Conj_mul_SNo_eq__7__1)
g z w = h z w β†’ g z y + g x w + - (g z w) = h z y + h x w + - (h z w)
In Proofgold the corresponding term root is 41a020... and proposition id is 712f00...
Proof:
Proof not loaded.
End of Section Conj_mul_SNo_eq__7__1
Beginning of Section Conj_mul_SNo_eq__7__2
L4181
Variable x : set
(*** Conj_mul_SNo_eq__7__2 TML44XGPM1TnRxBFEotSt3DiX9pCv8byiYt bounty of about 25 bars ***)
L4182
Variable y : set
L4183
Variable g : (set β†’ (set β†’ set))
L4184
Variable h : (set β†’ (set β†’ set))
L4185
Variable z : set
L4186
Variable w : set
L4187
Hypothesis H0 : (βˆ€u : set, u ∈ SNoS_ (SNoLev x) β†’ (βˆ€v : set, SNo v β†’ g u v = h u v))
L4188
Hypothesis H1 : z ∈ SNoS_ (SNoLev x)
L4189
Hypothesis H3 : g z y = h z y
L4190
Hypothesis H4 : g x w = h x w
L4191
Theorem. (Conj_mul_SNo_eq__7__2)
g z w = h z w β†’ g z y + g x w + - (g z w) = h z y + h x w + - (h z w)
In Proofgold the corresponding term root is fca8d6... and proposition id is 68a430...
Proof:
Proof not loaded.
End of Section Conj_mul_SNo_eq__7__2
Beginning of Section Conj_mul_SNo_eq__8__3
L4197
Variable x : set
(*** Conj_mul_SNo_eq__8__3 TMYevhbTJiUkHxoZCg315NG4SJi3UUacxR5 bounty of about 25 bars ***)
L4198
Variable y : set
L4199
Variable g : (set β†’ (set β†’ set))
L4200
Variable h : (set β†’ (set β†’ set))
L4201
Variable z : set
L4202
Variable w : set
L4203
Hypothesis H0 : (βˆ€u : set, u ∈ SNoS_ (SNoLev x) β†’ (βˆ€v : set, SNo v β†’ g u v = h u v))
L4204
Hypothesis H1 : (βˆ€u : set, u ∈ SNoS_ (SNoLev y) β†’ g x u = h x u)
L4205
Hypothesis H2 : z ∈ SNoS_ (SNoLev x)
L4206
Hypothesis H4 : SNo w
L4207
Hypothesis H5 : g z y = h z y
L4208
Theorem. (Conj_mul_SNo_eq__8__3)
g x w = h x w β†’ g z y + g x w + - (g z w) = h z y + h x w + - (h z w)
In Proofgold the corresponding term root is 68613e... and proposition id is e4f91c...
Proof:
Proof not loaded.
End of Section Conj_mul_SNo_eq__8__3
Beginning of Section Conj_mul_SNo_eq__9__3
L4214
Variable x : set
(*** Conj_mul_SNo_eq__9__3 TMc9x9bnxHCRdv9pdrLT23SFMWU723q3CF5 bounty of about 25 bars ***)
L4215
Variable y : set
L4216
Variable g : (set β†’ (set β†’ set))
L4217
Variable h : (set β†’ (set β†’ set))
L4218
Variable z : set
L4219
Variable w : set
L4220
Hypothesis H0 : SNo y
L4221
Hypothesis H1 : (βˆ€u : set, u ∈ SNoS_ (SNoLev x) β†’ (βˆ€v : set, SNo v β†’ g u v = h u v))
L4222
Hypothesis H2 : (βˆ€u : set, u ∈ SNoS_ (SNoLev y) β†’ g x u = h x u)
L4223
Hypothesis H4 : w ∈ SNoS_ (SNoLev y)
L4224
Hypothesis H5 : SNo w
L4225
Theorem. (Conj_mul_SNo_eq__9__3)
g z y = h z y β†’ g z y + g x w + - (g z w) = h z y + h x w + - (h z w)
In Proofgold the corresponding term root is 7dc197... and proposition id is b02b25...
Proof:
Proof not loaded.
End of Section Conj_mul_SNo_eq__9__3
Beginning of Section Conj_mul_SNo_eq__9__4
L4231
Variable x : set
(*** Conj_mul_SNo_eq__9__4 TMbKdCNpgb8ceK8K44wes4z26jE2eV5bUXK bounty of about 25 bars ***)
L4232
Variable y : set
L4233
Variable g : (set β†’ (set β†’ set))
L4234
Variable h : (set β†’ (set β†’ set))
L4235
Variable z : set
L4236
Variable w : set
L4237
Hypothesis H0 : SNo y
L4238
Hypothesis H1 : (βˆ€u : set, u ∈ SNoS_ (SNoLev x) β†’ (βˆ€v : set, SNo v β†’ g u v = h u v))
L4239
Hypothesis H2 : (βˆ€u : set, u ∈ SNoS_ (SNoLev y) β†’ g x u = h x u)
L4240
Hypothesis H3 : z ∈ SNoS_ (SNoLev x)
L4241
Hypothesis H5 : SNo w
L4242
Theorem. (Conj_mul_SNo_eq__9__4)
g z y = h z y β†’ g z y + g x w + - (g z w) = h z y + h x w + - (h z w)
In Proofgold the corresponding term root is 392627... and proposition id is 44641d...
Proof:
Proof not loaded.
End of Section Conj_mul_SNo_eq__9__4
Beginning of Section Conj_mul_SNo_eq__10__1
L4248
Variable x : set
(*** Conj_mul_SNo_eq__10__1 TMMYeriypZ4Zd8fgKdEqbdRRZqqUQHWn8Nv bounty of about 25 bars ***)
L4249
Variable y : set
L4250
Variable g : (set β†’ (set β†’ set))
L4251
Variable h : (set β†’ (set β†’ set))
L4252
Variable z : set
L4253
Variable w : set
L4254
Hypothesis H0 : SNo y
L4255
Hypothesis H2 : (βˆ€u : set, u ∈ SNoS_ (SNoLev y) β†’ g x u = h x u)
L4256
Hypothesis H3 : w ∈ SNoR y
L4257
Hypothesis H4 : z ∈ SNoS_ (SNoLev x)
L4258
Hypothesis H5 : w ∈ SNoS_ (SNoLev y)
L4259
Theorem. (Conj_mul_SNo_eq__10__1)
SNo w β†’ g z y + g x w + - (g z w) = h z y + h x w + - (h z w)
In Proofgold the corresponding term root is 6dc17a... and proposition id is c5b4b5...
Proof:
Proof not loaded.
End of Section Conj_mul_SNo_eq__10__1
Beginning of Section Conj_mul_SNo_eq__10__3
L4265
Variable x : set
(*** Conj_mul_SNo_eq__10__3 TMM9PLC4canhoRGg8wSstnVhUP3kwndnHts bounty of about 25 bars ***)
L4266
Variable y : set
L4267
Variable g : (set β†’ (set β†’ set))
L4268
Variable h : (set β†’ (set β†’ set))
L4269
Variable z : set
L4270
Variable w : set
L4271
Hypothesis H0 : SNo y
L4272
Hypothesis H1 : (βˆ€u : set, u ∈ SNoS_ (SNoLev x) β†’ (βˆ€v : set, SNo v β†’ g u v = h u v))
L4273
Hypothesis H2 : (βˆ€u : set, u ∈ SNoS_ (SNoLev y) β†’ g x u = h x u)
L4274
Hypothesis H4 : z ∈ SNoS_ (SNoLev x)
L4275
Hypothesis H5 : w ∈ SNoS_ (SNoLev y)
L4276
Theorem. (Conj_mul_SNo_eq__10__3)
SNo w β†’ g z y + g x w + - (g z w) = h z y + h x w + - (h z w)
In Proofgold the corresponding term root is 7b822c... and proposition id is 3816fc...
Proof:
Proof not loaded.
End of Section Conj_mul_SNo_eq__10__3
Beginning of Section Conj_mul_SNo_eq__12__4
L4282
Variable x : set
(*** Conj_mul_SNo_eq__12__4 TMM7v72BuE8LgBLome7o2XeS32ZynDx9b1u bounty of about 25 bars ***)
L4283
Variable y : set
L4284
Variable g : (set β†’ (set β†’ set))
L4285
Variable h : (set β†’ (set β†’ set))
L4286
Variable z : set
L4287
Variable w : set
L4288
Hypothesis H0 : SNo x
L4289
Hypothesis H1 : SNo y
L4290
Hypothesis H2 : (βˆ€u : set, u ∈ SNoS_ (SNoLev x) β†’ (βˆ€v : set, SNo v β†’ g u v = h u v))
L4291
Hypothesis H3 : (βˆ€u : set, u ∈ SNoS_ (SNoLev y) β†’ g x u = h x u)
L4292
Hypothesis H5 : w ∈ SNoR y
L4293
Theorem. (Conj_mul_SNo_eq__12__4)
z ∈ SNoS_ (SNoLev x) β†’ g z y + g x w + - (g z w) = h z y + h x w + - (h z w)
In Proofgold the corresponding term root is e7a1ec... and proposition id is a34877...
Proof:
Proof not loaded.
End of Section Conj_mul_SNo_eq__12__4
Beginning of Section Conj_mul_SNo_eq__13__0
L4299
Variable x : set
(*** Conj_mul_SNo_eq__13__0 TMHNaUMQnp8mygqy5bkQwBG1Td5cZVPmmfL bounty of about 25 bars ***)
L4300
Variable y : set
L4301
Variable g : (set β†’ (set β†’ set))
L4302
Variable h : (set β†’ (set β†’ set))
L4303
Variable z : set
L4304
Variable w : set
L4305
Hypothesis H1 : z ∈ SNoS_ (SNoLev x)
L4306
Hypothesis H2 : SNo w
L4307
Hypothesis H3 : g z y = h z y
L4308
Hypothesis H4 : g x w = h x w
L4309
Theorem. (Conj_mul_SNo_eq__13__0)
g z w = h z w β†’ g z y + g x w + - (g z w) = h z y + h x w + - (h z w)
In Proofgold the corresponding term root is ab8d72... and proposition id is 36a11b...
Proof:
Proof not loaded.
End of Section Conj_mul_SNo_eq__13__0
Beginning of Section Conj_mul_SNo_eq__13__1
L4315
Variable x : set
(*** Conj_mul_SNo_eq__13__1 TMaU7woDD25pzpeo87GUsLTfT2MR71x4vAP bounty of about 25 bars ***)
L4316
Variable y : set
L4317
Variable g : (set β†’ (set β†’ set))
L4318
Variable h : (set β†’ (set β†’ set))
L4319
Variable z : set
L4320
Variable w : set
L4321
Hypothesis H0 : (βˆ€u : set, u ∈ SNoS_ (SNoLev x) β†’ (βˆ€v : set, SNo v β†’ g u v = h u v))
L4322
Hypothesis H2 : SNo w
L4323
Hypothesis H3 : g z y = h z y
L4324
Hypothesis H4 : g x w = h x w
L4325
Theorem. (Conj_mul_SNo_eq__13__1)
g z w = h z w β†’ g z y + g x w + - (g z w) = h z y + h x w + - (h z w)
In Proofgold the corresponding term root is 41a020... and proposition id is 712f00...
Proof:
Proof not loaded.
End of Section Conj_mul_SNo_eq__13__1
Beginning of Section Conj_mul_SNo_eq__13__2
L4331
Variable x : set
(*** Conj_mul_SNo_eq__13__2 TML44XGPM1TnRxBFEotSt3DiX9pCv8byiYt bounty of about 25 bars ***)
L4332
Variable y : set
L4333
Variable g : (set β†’ (set β†’ set))
L4334
Variable h : (set β†’ (set β†’ set))
L4335
Variable z : set
L4336
Variable w : set
L4337
Hypothesis H0 : (βˆ€u : set, u ∈ SNoS_ (SNoLev x) β†’ (βˆ€v : set, SNo v β†’ g u v = h u v))
L4338
Hypothesis H1 : z ∈ SNoS_ (SNoLev x)
L4339
Hypothesis H3 : g z y = h z y
L4340
Hypothesis H4 : g x w = h x w
L4341
Theorem. (Conj_mul_SNo_eq__13__2)
g z w = h z w β†’ g z y + g x w + - (g z w) = h z y + h x w + - (h z w)
In Proofgold the corresponding term root is fca8d6... and proposition id is 68a430...
Proof:
Proof not loaded.
End of Section Conj_mul_SNo_eq__13__2
Beginning of Section Conj_mul_SNo_eq__14__3
L4347
Variable x : set
(*** Conj_mul_SNo_eq__14__3 TMYevhbTJiUkHxoZCg315NG4SJi3UUacxR5 bounty of about 25 bars ***)
L4348
Variable y : set
L4349
Variable g : (set β†’ (set β†’ set))
L4350
Variable h : (set β†’ (set β†’ set))
L4351
Variable z : set
L4352
Variable w : set
L4353
Hypothesis H0 : (βˆ€u : set, u ∈ SNoS_ (SNoLev x) β†’ (βˆ€v : set, SNo v β†’ g u v = h u v))
L4354
Hypothesis H1 : (βˆ€u : set, u ∈ SNoS_ (SNoLev y) β†’ g x u = h x u)
L4355
Hypothesis H2 : z ∈ SNoS_ (SNoLev x)
L4356
Hypothesis H4 : SNo w
L4357
Hypothesis H5 : g z y = h z y
L4358
Theorem. (Conj_mul_SNo_eq__14__3)
g x w = h x w β†’ g z y + g x w + - (g z w) = h z y + h x w + - (h z w)
In Proofgold the corresponding term root is 68613e... and proposition id is e4f91c...
Proof:
Proof not loaded.
End of Section Conj_mul_SNo_eq__14__3
Beginning of Section Conj_mul_SNo_eq__15__3
L4364
Variable x : set
(*** Conj_mul_SNo_eq__15__3 TMc9x9bnxHCRdv9pdrLT23SFMWU723q3CF5 bounty of about 25 bars ***)
L4365
Variable y : set
L4366
Variable g : (set β†’ (set β†’ set))
L4367
Variable h : (set β†’ (set β†’ set))
L4368
Variable z : set
L4369
Variable w : set
L4370
Hypothesis H0 : SNo y
L4371
Hypothesis H1 : (βˆ€u : set, u ∈ SNoS_ (SNoLev x) β†’ (βˆ€v : set, SNo v β†’ g u v = h u v))
L4372
Hypothesis H2 : (βˆ€u : set, u ∈ SNoS_ (SNoLev y) β†’ g x u = h x u)
L4373
Hypothesis H4 : w ∈ SNoS_ (SNoLev y)
L4374
Hypothesis H5 : SNo w
L4375
Theorem. (Conj_mul_SNo_eq__15__3)
g z y = h z y β†’ g z y + g x w + - (g z w) = h z y + h x w + - (h z w)
In Proofgold the corresponding term root is 7dc197... and proposition id is b02b25...
Proof:
Proof not loaded.
End of Section Conj_mul_SNo_eq__15__3
Beginning of Section Conj_mul_SNo_eq__15__4
L4381
Variable x : set
(*** Conj_mul_SNo_eq__15__4 TMbKdCNpgb8ceK8K44wes4z26jE2eV5bUXK bounty of about 25 bars ***)
L4382
Variable y : set
L4383
Variable g : (set β†’ (set β†’ set))
L4384
Variable h : (set β†’ (set β†’ set))
L4385
Variable z : set
L4386
Variable w : set
L4387
Hypothesis H0 : SNo y
L4388
Hypothesis H1 : (βˆ€u : set, u ∈ SNoS_ (SNoLev x) β†’ (βˆ€v : set, SNo v β†’ g u v = h u v))
L4389
Hypothesis H2 : (βˆ€u : set, u ∈ SNoS_ (SNoLev y) β†’ g x u = h x u)
L4390
Hypothesis H3 : z ∈ SNoS_ (SNoLev x)
L4391
Hypothesis H5 : SNo w
L4392
Theorem. (Conj_mul_SNo_eq__15__4)
g z y = h z y β†’ g z y + g x w + - (g z w) = h z y + h x w + - (h z w)
In Proofgold the corresponding term root is 392627... and proposition id is 44641d...
Proof:
Proof not loaded.
End of Section Conj_mul_SNo_eq__15__4
Beginning of Section Conj_mul_SNo_eq__16__1
L4398
Variable x : set
(*** Conj_mul_SNo_eq__16__1 TMMYeriypZ4Zd8fgKdEqbdRRZqqUQHWn8Nv bounty of about 25 bars ***)
L4399
Variable y : set
L4400
Variable g : (set β†’ (set β†’ set))
L4401
Variable h : (set β†’ (set β†’ set))
L4402
Variable z : set
L4403
Variable w : set
L4404
Hypothesis H0 : SNo y
L4405
Hypothesis H2 : (βˆ€u : set, u ∈ SNoS_ (SNoLev y) β†’ g x u = h x u)
L4406
Hypothesis H3 : w ∈ SNoR y
L4407
Hypothesis H4 : z ∈ SNoS_ (SNoLev x)
L4408
Hypothesis H5 : w ∈ SNoS_ (SNoLev y)
L4409
Theorem. (Conj_mul_SNo_eq__16__1)
SNo w β†’ g z y + g x w + - (g z w) = h z y + h x w + - (h z w)
In Proofgold the corresponding term root is 6dc17a... and proposition id is c5b4b5...
Proof:
Proof not loaded.
End of Section Conj_mul_SNo_eq__16__1
Beginning of Section Conj_mul_SNo_eq__16__3
L4415
Variable x : set
(*** Conj_mul_SNo_eq__16__3 TMM9PLC4canhoRGg8wSstnVhUP3kwndnHts bounty of about 25 bars ***)
L4416
Variable y : set
L4417
Variable g : (set β†’ (set β†’ set))
L4418
Variable h : (set β†’ (set β†’ set))
L4419
Variable z : set
L4420
Variable w : set
L4421
Hypothesis H0 : SNo y
L4422
Hypothesis H1 : (βˆ€u : set, u ∈ SNoS_ (SNoLev x) β†’ (βˆ€v : set, SNo v β†’ g u v = h u v))
L4423
Hypothesis H2 : (βˆ€u : set, u ∈ SNoS_ (SNoLev y) β†’ g x u = h x u)
L4424
Hypothesis H4 : z ∈ SNoS_ (SNoLev x)
L4425
Hypothesis H5 : w ∈ SNoS_ (SNoLev y)
L4426
Theorem. (Conj_mul_SNo_eq__16__3)
SNo w β†’ g z y + g x w + - (g z w) = h z y + h x w + - (h z w)
In Proofgold the corresponding term root is 7b822c... and proposition id is 3816fc...
Proof:
Proof not loaded.
End of Section Conj_mul_SNo_eq__16__3
Beginning of Section Conj_mul_SNo_eq__18__4
L4432
Variable x : set
(*** Conj_mul_SNo_eq__18__4 TMM7v72BuE8LgBLome7o2XeS32ZynDx9b1u bounty of about 25 bars ***)
L4433
Variable y : set
L4434
Variable g : (set β†’ (set β†’ set))
L4435
Variable h : (set β†’ (set β†’ set))
L4436
Variable z : set
L4437
Variable w : set
L4438
Hypothesis H0 : SNo x
L4439
Hypothesis H1 : SNo y
L4440
Hypothesis H2 : (βˆ€u : set, u ∈ SNoS_ (SNoLev x) β†’ (βˆ€v : set, SNo v β†’ g u v = h u v))
L4441
Hypothesis H3 : (βˆ€u : set, u ∈ SNoS_ (SNoLev y) β†’ g x u = h x u)
L4442
Hypothesis H5 : w ∈ SNoR y
L4443
Theorem. (Conj_mul_SNo_eq__18__4)
z ∈ SNoS_ (SNoLev x) β†’ g z y + g x w + - (g z w) = h z y + h x w + - (h z w)
In Proofgold the corresponding term root is e7a1ec... and proposition id is a34877...
Proof:
Proof not loaded.
End of Section Conj_mul_SNo_eq__18__4
Beginning of Section Conj_mul_SNo_eq__19__0
L4449
Variable x : set
(*** Conj_mul_SNo_eq__19__0 TMHNaUMQnp8mygqy5bkQwBG1Td5cZVPmmfL bounty of about 25 bars ***)
L4450
Variable y : set
L4451
Variable g : (set β†’ (set β†’ set))
L4452
Variable h : (set β†’ (set β†’ set))
L4453
Variable z : set
L4454
Variable w : set
L4455
Hypothesis H1 : z ∈ SNoS_ (SNoLev x)
L4456
Hypothesis H2 : SNo w
L4457
Hypothesis H3 : g z y = h z y
L4458
Hypothesis H4 : g x w = h x w
L4459
Theorem. (Conj_mul_SNo_eq__19__0)
g z w = h z w β†’ g z y + g x w + - (g z w) = h z y + h x w + - (h z w)
In Proofgold the corresponding term root is ab8d72... and proposition id is 36a11b...
Proof:
Proof not loaded.
End of Section Conj_mul_SNo_eq__19__0
Beginning of Section Conj_mul_SNo_eq__19__1
L4465
Variable x : set
(*** Conj_mul_SNo_eq__19__1 TMaU7woDD25pzpeo87GUsLTfT2MR71x4vAP bounty of about 25 bars ***)
L4466
Variable y : set
L4467
Variable g : (set β†’ (set β†’ set))
L4468
Variable h : (set β†’ (set β†’ set))
L4469
Variable z : set
L4470
Variable w : set
L4471
Hypothesis H0 : (βˆ€u : set, u ∈ SNoS_ (SNoLev x) β†’ (βˆ€v : set, SNo v β†’ g u v = h u v))
L4472
Hypothesis H2 : SNo w
L4473
Hypothesis H3 : g z y = h z y
L4474
Hypothesis H4 : g x w = h x w
L4475
Theorem. (Conj_mul_SNo_eq__19__1)
g z w = h z w β†’ g z y + g x w + - (g z w) = h z y + h x w + - (h z w)
In Proofgold the corresponding term root is 41a020... and proposition id is 712f00...
Proof:
Proof not loaded.
End of Section Conj_mul_SNo_eq__19__1
Beginning of Section Conj_mul_SNo_eq__19__2
L4481
Variable x : set
(*** Conj_mul_SNo_eq__19__2 TML44XGPM1TnRxBFEotSt3DiX9pCv8byiYt bounty of about 25 bars ***)
L4482
Variable y : set
L4483
Variable g : (set β†’ (set β†’ set))
L4484
Variable h : (set β†’ (set β†’ set))
L4485
Variable z : set
L4486
Variable w : set
L4487
Hypothesis H0 : (βˆ€u : set, u ∈ SNoS_ (SNoLev x) β†’ (βˆ€v : set, SNo v β†’ g u v = h u v))
L4488
Hypothesis H1 : z ∈ SNoS_ (SNoLev x)
L4489
Hypothesis H3 : g z y = h z y
L4490
Hypothesis H4 : g x w = h x w
L4491
Theorem. (Conj_mul_SNo_eq__19__2)
g z w = h z w β†’ g z y + g x w + - (g z w) = h z y + h x w + - (h z w)
In Proofgold the corresponding term root is fca8d6... and proposition id is 68a430...
Proof:
Proof not loaded.
End of Section Conj_mul_SNo_eq__19__2
Beginning of Section Conj_mul_SNo_eq__20__3
L4497
Variable x : set
(*** Conj_mul_SNo_eq__20__3 TMYevhbTJiUkHxoZCg315NG4SJi3UUacxR5 bounty of about 25 bars ***)
L4498
Variable y : set
L4499
Variable g : (set β†’ (set β†’ set))
L4500
Variable h : (set β†’ (set β†’ set))
L4501
Variable z : set
L4502
Variable w : set
L4503
Hypothesis H0 : (βˆ€u : set, u ∈ SNoS_ (SNoLev x) β†’ (βˆ€v : set, SNo v β†’ g u v = h u v))
L4504
Hypothesis H1 : (βˆ€u : set, u ∈ SNoS_ (SNoLev y) β†’ g x u = h x u)
L4505
Hypothesis H2 : z ∈ SNoS_ (SNoLev x)
L4506
Hypothesis H4 : SNo w
L4507
Hypothesis H5 : g z y = h z y
L4508
Theorem. (Conj_mul_SNo_eq__20__3)
g x w = h x w β†’ g z y + g x w + - (g z w) = h z y + h x w + - (h z w)
In Proofgold the corresponding term root is 68613e... and proposition id is e4f91c...
Proof:
Proof not loaded.
End of Section Conj_mul_SNo_eq__20__3
Beginning of Section Conj_mul_SNo_eq__21__3
L4514
Variable x : set
(*** Conj_mul_SNo_eq__21__3 TMc9x9bnxHCRdv9pdrLT23SFMWU723q3CF5 bounty of about 25 bars ***)
L4515
Variable y : set
L4516
Variable g : (set β†’ (set β†’ set))
L4517
Variable h : (set β†’ (set β†’ set))
L4518
Variable z : set
L4519
Variable w : set
L4520
Hypothesis H0 : SNo y
L4521
Hypothesis H1 : (βˆ€u : set, u ∈ SNoS_ (SNoLev x) β†’ (βˆ€v : set, SNo v β†’ g u v = h u v))
L4522
Hypothesis H2 : (βˆ€u : set, u ∈ SNoS_ (SNoLev y) β†’ g x u = h x u)
L4523
Hypothesis H4 : w ∈ SNoS_ (SNoLev y)
L4524
Hypothesis H5 : SNo w
L4525
Theorem. (Conj_mul_SNo_eq__21__3)
g z y = h z y β†’ g z y + g x w + - (g z w) = h z y + h x w + - (h z w)
In Proofgold the corresponding term root is 7dc197... and proposition id is b02b25...
Proof:
Proof not loaded.
End of Section Conj_mul_SNo_eq__21__3
Beginning of Section Conj_mul_SNo_eq__21__4
L4531
Variable x : set
(*** Conj_mul_SNo_eq__21__4 TMbKdCNpgb8ceK8K44wes4z26jE2eV5bUXK bounty of about 25 bars ***)
L4532
Variable y : set
L4533
Variable g : (set β†’ (set β†’ set))
L4534
Variable h : (set β†’ (set β†’ set))
L4535
Variable z : set
L4536
Variable w : set
L4537
Hypothesis H0 : SNo y
L4538
Hypothesis H1 : (βˆ€u : set, u ∈ SNoS_ (SNoLev x) β†’ (βˆ€v : set, SNo v β†’ g u v = h u v))
L4539
Hypothesis H2 : (βˆ€u : set, u ∈ SNoS_ (SNoLev y) β†’ g x u = h x u)
L4540
Hypothesis H3 : z ∈ SNoS_ (SNoLev x)
L4541
Hypothesis H5 : SNo w
L4542
Theorem. (Conj_mul_SNo_eq__21__4)
g z y = h z y β†’ g z y + g x w + - (g z w) = h z y + h x w + - (h z w)
In Proofgold the corresponding term root is 392627... and proposition id is 44641d...
Proof:
Proof not loaded.
End of Section Conj_mul_SNo_eq__21__4
Beginning of Section Conj_mul_SNo_eq__22__1
L4548
Variable x : set
(*** Conj_mul_SNo_eq__22__1 TMauMmtuJKRSxbZsSigSs3qQzCRTArRKSzx bounty of about 25 bars ***)
L4549
Variable y : set
L4550
Variable g : (set β†’ (set β†’ set))
L4551
Variable h : (set β†’ (set β†’ set))
L4552
Variable z : set
L4553
Variable w : set
L4554
Hypothesis H0 : SNo y
L4555
Hypothesis H2 : (βˆ€u : set, u ∈ SNoS_ (SNoLev y) β†’ g x u = h x u)
L4556
Hypothesis H3 : w ∈ SNoL y
L4557
Hypothesis H4 : z ∈ SNoS_ (SNoLev x)
L4558
Hypothesis H5 : w ∈ SNoS_ (SNoLev y)
L4559
Theorem. (Conj_mul_SNo_eq__22__1)
SNo w β†’ g z y + g x w + - (g z w) = h z y + h x w + - (h z w)
In Proofgold the corresponding term root is 9c249a... and proposition id is 51b9a9...
Proof:
Proof not loaded.
End of Section Conj_mul_SNo_eq__22__1
Beginning of Section Conj_mul_SNo_eq__22__3
L4565
Variable x : set
(*** Conj_mul_SNo_eq__22__3 TMM9PLC4canhoRGg8wSstnVhUP3kwndnHts bounty of about 25 bars ***)
L4566
Variable y : set
L4567
Variable g : (set β†’ (set β†’ set))
L4568
Variable h : (set β†’ (set β†’ set))
L4569
Variable z : set
L4570
Variable w : set
L4571
Hypothesis H0 : SNo y
L4572
Hypothesis H1 : (βˆ€u : set, u ∈ SNoS_ (SNoLev x) β†’ (βˆ€v : set, SNo v β†’ g u v = h u v))
L4573
Hypothesis H2 : (βˆ€u : set, u ∈ SNoS_ (SNoLev y) β†’ g x u = h x u)
L4574
Hypothesis H4 : z ∈ SNoS_ (SNoLev x)
L4575
Hypothesis H5 : w ∈ SNoS_ (SNoLev y)
L4576
Theorem. (Conj_mul_SNo_eq__22__3)
SNo w β†’ g z y + g x w + - (g z w) = h z y + h x w + - (h z w)
In Proofgold the corresponding term root is 7b822c... and proposition id is 3816fc...
Proof:
Proof not loaded.
End of Section Conj_mul_SNo_eq__22__3
Beginning of Section Conj_mul_SNo_eq__23__1
L4582
Variable x : set
(*** Conj_mul_SNo_eq__23__1 TMbMyv2xSKFXdojrpPHAhGYYnxTvNRzKX1W bounty of about 25 bars ***)
L4583
Variable y : set
L4584
Variable g : (set β†’ (set β†’ set))
L4585
Variable h : (set β†’ (set β†’ set))
L4586
Variable z : set
L4587
Variable w : set
L4588
Hypothesis H0 : SNo y
L4589
Hypothesis H2 : (βˆ€u : set, u ∈ SNoS_ (SNoLev y) β†’ g x u = h x u)
L4590
Hypothesis H3 : w ∈ SNoL y
L4591
Hypothesis H4 : z ∈ SNoS_ (SNoLev x)
L4592
Theorem. (Conj_mul_SNo_eq__23__1)
w ∈ SNoS_ (SNoLev y) β†’ g z y + g x w + - (g z w) = h z y + h x w + - (h z w)
In Proofgold the corresponding term root is d0c5ad... and proposition id is ed0c24...
Proof:
Proof not loaded.
End of Section Conj_mul_SNo_eq__23__1
Beginning of Section Conj_mul_SNo_eq__24__3
L4598
Variable x : set
(*** Conj_mul_SNo_eq__24__3 TMHm1Yw84cnyNYCAX3wytiGYt3WyS6TD8jc bounty of about 25 bars ***)
L4599
Variable y : set
L4600
Variable g : (set β†’ (set β†’ set))
L4601
Variable h : (set β†’ (set β†’ set))
L4602
Variable z : set
L4603
Variable w : set
L4604
Hypothesis H0 : SNo x
L4605
Hypothesis H1 : SNo y
L4606
Hypothesis H2 : (βˆ€u : set, u ∈ SNoS_ (SNoLev x) β†’ (βˆ€v : set, SNo v β†’ g u v = h u v))
L4607
Hypothesis H4 : z ∈ SNoR x
L4608
Hypothesis H5 : w ∈ SNoL y
L4609
Theorem. (Conj_mul_SNo_eq__24__3)
z ∈ SNoS_ (SNoLev x) β†’ g z y + g x w + - (g z w) = h z y + h x w + - (h z w)
In Proofgold the corresponding term root is 12d7cd... and proposition id is 5b4815...
Proof:
Proof not loaded.
End of Section Conj_mul_SNo_eq__24__3
Beginning of Section Conj_mul_SNo_eq__25__0
L4615
Variable x : set
(*** Conj_mul_SNo_eq__25__0 TMSVjHnyGhnvGyUsVy7a8bmeVh1RXJVvgjg bounty of about 25 bars ***)
L4616
Variable y : set
L4617
Variable g : (set β†’ (set β†’ set))
L4618
Variable h : (set β†’ (set β†’ set))
L4619
Hypothesis H1 : SNo y
L4620
Hypothesis H2 : (βˆ€z : set, z ∈ SNoS_ (SNoLev x) β†’ (βˆ€w : set, SNo w β†’ g z w = h z w))
L4621
Hypothesis H3 : (βˆ€z : set, z ∈ SNoS_ (SNoLev y) β†’ g x z = h x z)
L4622
Hypothesis H4 : Repl (setprod (SNoL x) (SNoL y)) (Ξ»z : set β‡’ g (ap z Empty) y + g x (ap z (ordsucc Empty)) + - (g (ap z Empty) (ap z (ordsucc Empty)))) = Repl (setprod (SNoL x) (SNoL y)) (Ξ»z : set β‡’ h (ap z Empty) y + h x (ap z (ordsucc Empty)) + - (h (ap z Empty) (ap z (ordsucc Empty))))
L4623
Hypothesis H5 : Repl (setprod (SNoR x) (SNoR y)) (Ξ»z : set β‡’ g (ap z Empty) y + g x (ap z (ordsucc Empty)) + - (g (ap z Empty) (ap z (ordsucc Empty)))) = Repl (setprod (SNoR x) (SNoR y)) (Ξ»z : set β‡’ h (ap z Empty) y + h x (ap z (ordsucc Empty)) + - (h (ap z Empty) (ap z (ordsucc Empty))))
L4624
Hypothesis H6 : Repl (setprod (SNoL x) (SNoR y)) (Ξ»z : set β‡’ g (ap z Empty) y + g x (ap z (ordsucc Empty)) + - (g (ap z Empty) (ap z (ordsucc Empty)))) = Repl (setprod (SNoL x) (SNoR y)) (Ξ»z : set β‡’ h (ap z Empty) y + h x (ap z (ordsucc Empty)) + - (h (ap z Empty) (ap z (ordsucc Empty))))
L4625
Theorem. (Conj_mul_SNo_eq__25__0)
Repl (setprod (SNoR x) (SNoL y)) (Ξ»z : set β‡’ g (ap z Empty) y + g x (ap z (ordsucc Empty)) + - (g (ap z Empty) (ap z (ordsucc Empty)))) = Repl (setprod (SNoR x) (SNoL y)) (Ξ»z : set β‡’ h (ap z Empty) y + h x (ap z (ordsucc Empty)) + - (h (ap z Empty) (ap z (ordsucc Empty)))) β†’ SNoCut (binunion (Repl (setprod (SNoL x) (SNoL y)) (Ξ»z : set β‡’ g (ap z Empty) y + g x (ap z (ordsucc Empty)) + - (g (ap z Empty) (ap z (ordsucc Empty))))) (Repl (setprod (SNoR x) (SNoR y)) (Ξ»z : set β‡’ g (ap z Empty) y + g x (ap z (ordsucc Empty)) + - (g (ap z Empty) (ap z (ordsucc Empty)))))) (binunion (Repl (setprod (SNoL x) (SNoR y)) (Ξ»z : set β‡’ g (ap z Empty) y + g x (ap z (ordsucc Empty)) + - (g (ap z Empty) (ap z (ordsucc Empty))))) (Repl (setprod (SNoR x) (SNoL y)) (Ξ»z : set β‡’ g (ap z Empty) y + g x (ap z (ordsucc Empty)) + - (g (ap z Empty) (ap z (ordsucc Empty)))))) = SNoCut (binunion (Repl (setprod (SNoL x) (SNoL y)) (Ξ»z : set β‡’ h (ap z Empty) y + h x (ap z (ordsucc Empty)) + - (h (ap z Empty) (ap z (ordsucc Empty))))) (Repl (setprod (SNoR x) (SNoR y)) (Ξ»z : set β‡’ h (ap z Empty) y + h x (ap z (ordsucc Empty)) + - (h (ap z Empty) (ap z (ordsucc Empty)))))) (binunion (Repl (setprod (SNoL x) (SNoR y)) (Ξ»z : set β‡’ h (ap z Empty) y + h x (ap z (ordsucc Empty)) + - (h (ap z Empty) (ap z (ordsucc Empty))))) (Repl (setprod (SNoR x) (SNoL y)) (Ξ»z : set β‡’ h (ap z Empty) y + h x (ap z (ordsucc Empty)) + - (h (ap z Empty) (ap z (ordsucc Empty))))))
In Proofgold the corresponding term root is 113d31... and proposition id is 7e49d5...
Proof:
Proof not loaded.
End of Section Conj_mul_SNo_eq__25__0
Beginning of Section Conj_mul_SNo_eq__25__3
L4631
Variable x : set
(*** Conj_mul_SNo_eq__25__3 TMPpyQvo4dmoKVv3eMdMETbbwQS3VoRXNPq bounty of about 25 bars ***)
L4632
Variable y : set
L4633
Variable g : (set β†’ (set β†’ set))
L4634
Variable h : (set β†’ (set β†’ set))
L4635
Hypothesis H0 : SNo x
L4636
Hypothesis H1 : SNo y
L4637
Hypothesis H2 : (βˆ€z : set, z ∈ SNoS_ (SNoLev x) β†’ (βˆ€w : set, SNo w β†’ g z w = h z w))
L4638
Hypothesis H4 : Repl (setprod (SNoL x) (SNoL y)) (Ξ»z : set β‡’ g (ap z Empty) y + g x (ap z (ordsucc Empty)) + - (g (ap z Empty) (ap z (ordsucc Empty)))) = Repl (setprod (SNoL x) (SNoL y)) (Ξ»z : set β‡’ h (ap z Empty) y + h x (ap z (ordsucc Empty)) + - (h (ap z Empty) (ap z (ordsucc Empty))))
L4639
Hypothesis H5 : Repl (setprod (SNoR x) (SNoR y)) (Ξ»z : set β‡’ g (ap z Empty) y + g x (ap z (ordsucc Empty)) + - (g (ap z Empty) (ap z (ordsucc Empty)))) = Repl (setprod (SNoR x) (SNoR y)) (Ξ»z : set β‡’ h (ap z Empty) y + h x (ap z (ordsucc Empty)) + - (h (ap z Empty) (ap z (ordsucc Empty))))
L4640
Hypothesis H6 : Repl (setprod (SNoL x) (SNoR y)) (Ξ»z : set β‡’ g (ap z Empty) y + g x (ap z (ordsucc Empty)) + - (g (ap z Empty) (ap z (ordsucc Empty)))) = Repl (setprod (SNoL x) (SNoR y)) (Ξ»z : set β‡’ h (ap z Empty) y + h x (ap z (ordsucc Empty)) + - (h (ap z Empty) (ap z (ordsucc Empty))))
L4641
Theorem. (Conj_mul_SNo_eq__25__3)
Repl (setprod (SNoR x) (SNoL y)) (Ξ»z : set β‡’ g (ap z Empty) y + g x (ap z (ordsucc Empty)) + - (g (ap z Empty) (ap z (ordsucc Empty)))) = Repl (setprod (SNoR x) (SNoL y)) (Ξ»z : set β‡’ h (ap z Empty) y + h x (ap z (ordsucc Empty)) + - (h (ap z Empty) (ap z (ordsucc Empty)))) β†’ SNoCut (binunion (Repl (setprod (SNoL x) (SNoL y)) (Ξ»z : set β‡’ g (ap z Empty) y + g x (ap z (ordsucc Empty)) + - (g (ap z Empty) (ap z (ordsucc Empty))))) (Repl (setprod (SNoR x) (SNoR y)) (Ξ»z : set β‡’ g (ap z Empty) y + g x (ap z (ordsucc Empty)) + - (g (ap z Empty) (ap z (ordsucc Empty)))))) (binunion (Repl (setprod (SNoL x) (SNoR y)) (Ξ»z : set β‡’ g (ap z Empty) y + g x (ap z (ordsucc Empty)) + - (g (ap z Empty) (ap z (ordsucc Empty))))) (Repl (setprod (SNoR x) (SNoL y)) (Ξ»z : set β‡’ g (ap z Empty) y + g x (ap z (ordsucc Empty)) + - (g (ap z Empty) (ap z (ordsucc Empty)))))) = SNoCut (binunion (Repl (setprod (SNoL x) (SNoL y)) (Ξ»z : set β‡’ h (ap z Empty) y + h x (ap z (ordsucc Empty)) + - (h (ap z Empty) (ap z (ordsucc Empty))))) (Repl (setprod (SNoR x) (SNoR y)) (Ξ»z : set β‡’ h (ap z Empty) y + h x (ap z (ordsucc Empty)) + - (h (ap z Empty) (ap z (ordsucc Empty)))))) (binunion (Repl (setprod (SNoL x) (SNoR y)) (Ξ»z : set β‡’ h (ap z Empty) y + h x (ap z (ordsucc Empty)) + - (h (ap z Empty) (ap z (ordsucc Empty))))) (Repl (setprod (SNoR x) (SNoL y)) (Ξ»z : set β‡’ h (ap z Empty) y + h x (ap z (ordsucc Empty)) + - (h (ap z Empty) (ap z (ordsucc Empty))))))
In Proofgold the corresponding term root is aae1c7... and proposition id is 7cb2ea...
Proof:
Proof not loaded.
End of Section Conj_mul_SNo_eq__25__3
Beginning of Section Conj_mul_SNo_eq__25__4
L4647
Variable x : set
(*** Conj_mul_SNo_eq__25__4 TMLFUH5mm6w6g7asL61fiVzmsX3gaVcvcd7 bounty of about 25 bars ***)
L4648
Variable y : set
L4649
Variable g : (set β†’ (set β†’ set))
L4650
Variable h : (set β†’ (set β†’ set))
L4651
Hypothesis H0 : SNo x
L4652
Hypothesis H1 : SNo y
L4653
Hypothesis H2 : (βˆ€z : set, z ∈ SNoS_ (SNoLev x) β†’ (βˆ€w : set, SNo w β†’ g z w = h z w))
L4654
Hypothesis H3 : (βˆ€z : set, z ∈ SNoS_ (SNoLev y) β†’ g x z = h x z)
L4655
Hypothesis H5 : Repl (setprod (SNoR x) (SNoR y)) (Ξ»z : set β‡’ g (ap z Empty) y + g x (ap z (ordsucc Empty)) + - (g (ap z Empty) (ap z (ordsucc Empty)))) = Repl (setprod (SNoR x) (SNoR y)) (Ξ»z : set β‡’ h (ap z Empty) y + h x (ap z (ordsucc Empty)) + - (h (ap z Empty) (ap z (ordsucc Empty))))
L4656
Hypothesis H6 : Repl (setprod (SNoL x) (SNoR y)) (Ξ»z : set β‡’ g (ap z Empty) y + g x (ap z (ordsucc Empty)) + - (g (ap z Empty) (ap z (ordsucc Empty)))) = Repl (setprod (SNoL x) (SNoR y)) (Ξ»z : set β‡’ h (ap z Empty) y + h x (ap z (ordsucc Empty)) + - (h (ap z Empty) (ap z (ordsucc Empty))))
L4657
Theorem. (Conj_mul_SNo_eq__25__4)
Repl (setprod (SNoR x) (SNoL y)) (Ξ»z : set β‡’ g (ap z Empty) y + g x (ap z (ordsucc Empty)) + - (g (ap z Empty) (ap z (ordsucc Empty)))) = Repl (setprod (SNoR x) (SNoL y)) (Ξ»z : set β‡’ h (ap z Empty) y + h x (ap z (ordsucc Empty)) + - (h (ap z Empty) (ap z (ordsucc Empty)))) β†’ SNoCut (binunion (Repl (setprod (SNoL x) (SNoL y)) (Ξ»z : set β‡’ g (ap z Empty) y + g x (ap z (ordsucc Empty)) + - (g (ap z Empty) (ap z (ordsucc Empty))))) (Repl (setprod (SNoR x) (SNoR y)) (Ξ»z : set β‡’ g (ap z Empty) y + g x (ap z (ordsucc Empty)) + - (g (ap z Empty) (ap z (ordsucc Empty)))))) (binunion (Repl (setprod (SNoL x) (SNoR y)) (Ξ»z : set β‡’ g (ap z Empty) y + g x (ap z (ordsucc Empty)) + - (g (ap z Empty) (ap z (ordsucc Empty))))) (Repl (setprod (SNoR x) (SNoL y)) (Ξ»z : set β‡’ g (ap z Empty) y + g x (ap z (ordsucc Empty)) + - (g (ap z Empty) (ap z (ordsucc Empty)))))) = SNoCut (binunion (Repl (setprod (SNoL x) (SNoL y)) (Ξ»z : set β‡’ h (ap z Empty) y + h x (ap z (ordsucc Empty)) + - (h (ap z Empty) (ap z (ordsucc Empty))))) (Repl (setprod (SNoR x) (SNoR y)) (Ξ»z : set β‡’ h (ap z Empty) y + h x (ap z (ordsucc Empty)) + - (h (ap z Empty) (ap z (ordsucc Empty)))))) (binunion (Repl (setprod (SNoL x) (SNoR y)) (Ξ»z : set β‡’ h (ap z Empty) y + h x (ap z (ordsucc Empty)) + - (h (ap z Empty) (ap z (ordsucc Empty))))) (Repl (setprod (SNoR x) (SNoL y)) (Ξ»z : set β‡’ h (ap z Empty) y + h x (ap z (ordsucc Empty)) + - (h (ap z Empty) (ap z (ordsucc Empty))))))
In Proofgold the corresponding term root is a66779... and proposition id is a2c7f1...
Proof:
Proof not loaded.
End of Section Conj_mul_SNo_eq__25__4
Beginning of Section Conj_mul_SNo_eq__26__3
L4663
Variable x : set
(*** Conj_mul_SNo_eq__26__3 TMFHNBzkKbpRnRahKMHtwK7yTzoHXFXsA5a bounty of about 25 bars ***)
L4664
Variable y : set
L4665
Variable g : (set β†’ (set β†’ set))
L4666
Variable h : (set β†’ (set β†’ set))
L4667
Hypothesis H0 : SNo x
L4668
Hypothesis H1 : SNo y
L4669
Hypothesis H2 : (βˆ€z : set, z ∈ SNoS_ (SNoLev x) β†’ (βˆ€w : set, SNo w β†’ g z w = h z w))
L4670
Hypothesis H4 : Repl (setprod (SNoL x) (SNoL y)) (Ξ»z : set β‡’ g (ap z Empty) y + g x (ap z (ordsucc Empty)) + - (g (ap z Empty) (ap z (ordsucc Empty)))) = Repl (setprod (SNoL x) (SNoL y)) (Ξ»z : set β‡’ h (ap z Empty) y + h x (ap z (ordsucc Empty)) + - (h (ap z Empty) (ap z (ordsucc Empty))))
L4671
Hypothesis H5 : Repl (setprod (SNoR x) (SNoR y)) (Ξ»z : set β‡’ g (ap z Empty) y + g x (ap z (ordsucc Empty)) + - (g (ap z Empty) (ap z (ordsucc Empty)))) = Repl (setprod (SNoR x) (SNoR y)) (Ξ»z : set β‡’ h (ap z Empty) y + h x (ap z (ordsucc Empty)) + - (h (ap z Empty) (ap z (ordsucc Empty))))
L4672
Theorem. (Conj_mul_SNo_eq__26__3)
Repl (setprod (SNoL x) (SNoR y)) (Ξ»z : set β‡’ g (ap z Empty) y + g x (ap z (ordsucc Empty)) + - (g (ap z Empty) (ap z (ordsucc Empty)))) = Repl (setprod (SNoL x) (SNoR y)) (Ξ»z : set β‡’ h (ap z Empty) y + h x (ap z (ordsucc Empty)) + - (h (ap z Empty) (ap z (ordsucc Empty)))) β†’ SNoCut (binunion (Repl (setprod (SNoL x) (SNoL y)) (Ξ»z : set β‡’ g (ap z Empty) y + g x (ap z (ordsucc Empty)) + - (g (ap z Empty) (ap z (ordsucc Empty))))) (Repl (setprod (SNoR x) (SNoR y)) (Ξ»z : set β‡’ g (ap z Empty) y + g x (ap z (ordsucc Empty)) + - (g (ap z Empty) (ap z (ordsucc Empty)))))) (binunion (Repl (setprod (SNoL x) (SNoR y)) (Ξ»z : set β‡’ g (ap z Empty) y + g x (ap z (ordsucc Empty)) + - (g (ap z Empty) (ap z (ordsucc Empty))))) (Repl (setprod (SNoR x) (SNoL y)) (Ξ»z : set β‡’ g (ap z Empty) y + g x (ap z (ordsucc Empty)) + - (g (ap z Empty) (ap z (ordsucc Empty)))))) = SNoCut (binunion (Repl (setprod (SNoL x) (SNoL y)) (Ξ»z : set β‡’ h (ap z Empty) y + h x (ap z (ordsucc Empty)) + - (h (ap z Empty) (ap z (ordsucc Empty))))) (Repl (setprod (SNoR x) (SNoR y)) (Ξ»z : set β‡’ h (ap z Empty) y + h x (ap z (ordsucc Empty)) + - (h (ap z Empty) (ap z (ordsucc Empty)))))) (binunion (Repl (setprod (SNoL x) (SNoR y)) (Ξ»z : set β‡’ h (ap z Empty) y + h x (ap z (ordsucc Empty)) + - (h (ap z Empty) (ap z (ordsucc Empty))))) (Repl (setprod (SNoR x) (SNoL y)) (Ξ»z : set β‡’ h (ap z Empty) y + h x (ap z (ordsucc Empty)) + - (h (ap z Empty) (ap z (ordsucc Empty))))))
In Proofgold the corresponding term root is b5d892... and proposition id is 0a8aa0...
Proof:
Proof not loaded.
End of Section Conj_mul_SNo_eq__26__3
Beginning of Section Conj_mul_SNo_eq__27__2
L4678
Variable x : set
(*** Conj_mul_SNo_eq__27__2 TMd6GTSPub86kSBSYDMKkhw1WjnDKssSgBt bounty of about 25 bars ***)
L4679
Variable y : set
L4680
Variable g : (set β†’ (set β†’ set))
L4681
Variable h : (set β†’ (set β†’ set))
L4682
Hypothesis H0 : SNo x
L4683
Hypothesis H1 : SNo y
L4684
Hypothesis H3 : (βˆ€z : set, z ∈ SNoS_ (SNoLev y) β†’ g x z = h x z)
L4685
Hypothesis H4 : Repl (setprod (SNoL x) (SNoL y)) (Ξ»z : set β‡’ g (ap z Empty) y + g x (ap z (ordsucc Empty)) + - (g (ap z Empty) (ap z (ordsucc Empty)))) = Repl (setprod (SNoL x) (SNoL y)) (Ξ»z : set β‡’ h (ap z Empty) y + h x (ap z (ordsucc Empty)) + - (h (ap z Empty) (ap z (ordsucc Empty))))
L4686
Theorem. (Conj_mul_SNo_eq__27__2)
Repl (setprod (SNoR x) (SNoR y)) (Ξ»z : set β‡’ g (ap z Empty) y + g x (ap z (ordsucc Empty)) + - (g (ap z Empty) (ap z (ordsucc Empty)))) = Repl (setprod (SNoR x) (SNoR y)) (Ξ»z : set β‡’ h (ap z Empty) y + h x (ap z (ordsucc Empty)) + - (h (ap z Empty) (ap z (ordsucc Empty)))) β†’ SNoCut (binunion (Repl (setprod (SNoL x) (SNoL y)) (Ξ»z : set β‡’ g (ap z Empty) y + g x (ap z (ordsucc Empty)) + - (g (ap z Empty) (ap z (ordsucc Empty))))) (Repl (setprod (SNoR x) (SNoR y)) (Ξ»z : set β‡’ g (ap z Empty) y + g x (ap z (ordsucc Empty)) + - (g (ap z Empty) (ap z (ordsucc Empty)))))) (binunion (Repl (setprod (SNoL x) (SNoR y)) (Ξ»z : set β‡’ g (ap z Empty) y + g x (ap z (ordsucc Empty)) + - (g (ap z Empty) (ap z (ordsucc Empty))))) (Repl (setprod (SNoR x) (SNoL y)) (Ξ»z : set β‡’ g (ap z Empty) y + g x (ap z (ordsucc Empty)) + - (g (ap z Empty) (ap z (ordsucc Empty)))))) = SNoCut (binunion (Repl (setprod (SNoL x) (SNoL y)) (Ξ»z : set β‡’ h (ap z Empty) y + h x (ap z (ordsucc Empty)) + - (h (ap z Empty) (ap z (ordsucc Empty))))) (Repl (setprod (SNoR x) (SNoR y)) (Ξ»z : set β‡’ h (ap z Empty) y + h x (ap z (ordsucc Empty)) + - (h (ap z Empty) (ap z (ordsucc Empty)))))) (binunion (Repl (setprod (SNoL x) (SNoR y)) (Ξ»z : set β‡’ h (ap z Empty) y + h x (ap z (ordsucc Empty)) + - (h (ap z Empty) (ap z (ordsucc Empty))))) (Repl (setprod (SNoR x) (SNoL y)) (Ξ»z : set β‡’ h (ap z Empty) y + h x (ap z (ordsucc Empty)) + - (h (ap z Empty) (ap z (ordsucc Empty))))))
In Proofgold the corresponding term root is b90f5b... and proposition id is 25244c...
Proof:
Proof not loaded.
End of Section Conj_mul_SNo_eq__27__2
Beginning of Section Conj_mul_SNo_prop_1__2__0
L4692
Variable x : set
(*** Conj_mul_SNo_prop_1__2__0 TMVrz6btCAk4frmaKByxBXAZeVa5DqQjEUR bounty of about 25 bars ***)
L4693
Variable y : set
L4694
Variable z : set
L4695
Variable w : set
L4696
Variable u : set
L4697
Variable v : set
L4698
Hypothesis H1 : SNo (w * v)
L4699
Hypothesis H2 : SNo (z * y)
L4700
Hypothesis H3 : SNo (x * u)
L4701
Hypothesis H4 : SNo (w * y)
L4702
Hypothesis H5 : SNo (x * v)
L4703
Hypothesis H6 : (z * y + x * u + w * v) < w * y + x * v + z * u
L4704
Theorem. (Conj_mul_SNo_prop_1__2__0)
(z * y + x * u + - (z * u)) + z * u + w * v = z * y + x * u + w * v β†’ ((z * y + x * u + - (z * u)) + z * u + w * v) < (w * y + x * v + - (w * v)) + z * u + w * v
In Proofgold the corresponding term root is 31968f... and proposition id is 72a1fe...
Proof:
Proof not loaded.
End of Section Conj_mul_SNo_prop_1__2__0
Beginning of Section Conj_mul_SNo_prop_1__3__6
L4710
Variable x : set
(*** Conj_mul_SNo_prop_1__3__6 TML5g8m58GqjEw5fqVcfNiMr5zfT16iJzMX bounty of about 25 bars ***)
L4711
Variable y : set
L4712
Variable z : set
L4713
Variable w : set
L4714
Variable u : set
L4715
Variable v : set
L4716
Variable x2 : set
L4717
Variable y2 : set
L4718
Hypothesis H0 : (βˆ€z2 : set, z2 ∈ SNoS_ (SNoLev x) β†’ (βˆ€w2 : set, SNo w2 β†’ (βˆ€P : prop, (SNo (z2 * w2) β†’ (βˆ€u2 : set, u2 ∈ SNoL z2 β†’ (βˆ€v2 : set, v2 ∈ SNoL w2 β†’ (u2 * w2 + z2 * v2) < z2 * w2 + u2 * v2)) β†’ (βˆ€u2 : set, u2 ∈ SNoR z2 β†’ (βˆ€v2 : set, v2 ∈ SNoR w2 β†’ (u2 * w2 + z2 * v2) < z2 * w2 + u2 * v2)) β†’ (βˆ€u2 : set, u2 ∈ SNoL z2 β†’ (βˆ€v2 : set, v2 ∈ SNoR w2 β†’ (z2 * w2 + u2 * v2) < u2 * w2 + z2 * v2)) β†’ (βˆ€u2 : set, u2 ∈ SNoR z2 β†’ (βˆ€v2 : set, v2 ∈ SNoL w2 β†’ (z2 * w2 + u2 * v2) < u2 * w2 + z2 * v2)) β†’ P) β†’ P)))
L4719
Hypothesis H1 : v ∈ SNoS_ (SNoLev x)
L4720
Hypothesis H2 : SNo x2
L4721
Hypothesis H3 : SNo (u * x2)
L4722
Hypothesis H4 : SNo (v * y2)
L4723
Hypothesis H5 : SNo (u * y)
L4724
Hypothesis H7 : SNo (v * y)
L4725
Hypothesis H8 : SNo (x * y2)
L4726
Hypothesis H9 : SNo (u * y2)
L4727
Theorem. (Conj_mul_SNo_prop_1__3__6)
SNo (v * x2) β†’ (βˆ€P : prop, (SNo (u * y) β†’ SNo (x * x2) β†’ SNo (u * x2) β†’ SNo (v * y) β†’ SNo (x * y2) β†’ SNo (v * y2) β†’ SNo (u * y2) β†’ SNo (v * x2) β†’ (z = u * y + x * x2 + - (u * x2) β†’ w = v * y + x * y2 + - (v * y2) β†’ (u * y + x * x2 + v * y2) < v * y + x * y2 + u * x2 β†’ z < w) β†’ P) β†’ P)
In Proofgold the corresponding term root is 51fc50... and proposition id is ab596a...
Proof:
Proof not loaded.
End of Section Conj_mul_SNo_prop_1__3__6
Beginning of Section Conj_mul_SNo_prop_1__6__1
L4733
Variable x : set
(*** Conj_mul_SNo_prop_1__6__1 TMJj8RpLFJETYBj2N7Dqxo1PE8ZeuSKwzwu bounty of about 25 bars ***)
L4734
Variable y : set
L4735
Variable z : set
L4736
Variable w : set
L4737
Variable u : set
L4738
Variable v : set
L4739
Variable x2 : set
L4740
Variable y2 : set
L4741
Hypothesis H0 : (βˆ€z2 : set, z2 ∈ SNoS_ (SNoLev x) β†’ (βˆ€w2 : set, SNo w2 β†’ (βˆ€P : prop, (SNo (z2 * w2) β†’ (βˆ€u2 : set, u2 ∈ SNoL z2 β†’ (βˆ€v2 : set, v2 ∈ SNoL w2 β†’ (u2 * w2 + z2 * v2) < z2 * w2 + u2 * v2)) β†’ (βˆ€u2 : set, u2 ∈ SNoR z2 β†’ (βˆ€v2 : set, v2 ∈ SNoR w2 β†’ (u2 * w2 + z2 * v2) < z2 * w2 + u2 * v2)) β†’ (βˆ€u2 : set, u2 ∈ SNoL z2 β†’ (βˆ€v2 : set, v2 ∈ SNoR w2 β†’ (z2 * w2 + u2 * v2) < u2 * w2 + z2 * v2)) β†’ (βˆ€u2 : set, u2 ∈ SNoR z2 β†’ (βˆ€v2 : set, v2 ∈ SNoL w2 β†’ (z2 * w2 + u2 * v2) < u2 * w2 + z2 * v2)) β†’ P) β†’ P)))
L4742
Hypothesis H2 : (βˆ€z2 : set, z2 ∈ SNoS_ (SNoLev y) β†’ (βˆ€P : prop, (SNo (x * z2) β†’ (βˆ€w2 : set, w2 ∈ SNoL x β†’ (βˆ€u2 : set, u2 ∈ SNoL z2 β†’ (w2 * z2 + x * u2) < x * z2 + w2 * u2)) β†’ (βˆ€w2 : set, w2 ∈ SNoR x β†’ (βˆ€u2 : set, u2 ∈ SNoR z2 β†’ (w2 * z2 + x * u2) < x * z2 + w2 * u2)) β†’ (βˆ€w2 : set, w2 ∈ SNoL x β†’ (βˆ€u2 : set, u2 ∈ SNoR z2 β†’ (x * z2 + w2 * u2) < w2 * z2 + x * u2)) β†’ (βˆ€w2 : set, w2 ∈ SNoR x β†’ (βˆ€u2 : set, u2 ∈ SNoL z2 β†’ (x * z2 + w2 * u2) < w2 * z2 + x * u2)) β†’ P) β†’ P))
L4743
Hypothesis H3 : u ∈ SNoS_ (SNoLev x)
L4744
Hypothesis H4 : v ∈ SNoS_ (SNoLev x)
L4745
Hypothesis H5 : y2 ∈ SNoS_ (SNoLev y)
L4746
Hypothesis H6 : SNo x2
L4747
Hypothesis H7 : SNo y2
L4748
Hypothesis H8 : SNo (u * x2)
L4749
Hypothesis H9 : SNo (v * y2)
L4750
Hypothesis H10 : SNo (u * y)
L4751
Hypothesis H11 : SNo (x * x2)
L4752
Theorem. (Conj_mul_SNo_prop_1__6__1)
SNo (v * y) β†’ (βˆ€P : prop, (SNo (u * y) β†’ SNo (x * x2) β†’ SNo (u * x2) β†’ SNo (v * y) β†’ SNo (x * y2) β†’ SNo (v * y2) β†’ SNo (u * y2) β†’ SNo (v * x2) β†’ (z = u * y + x * x2 + - (u * x2) β†’ w = v * y + x * y2 + - (v * y2) β†’ (u * y + x * x2 + v * y2) < v * y + x * y2 + u * x2 β†’ z < w) β†’ P) β†’ P)
In Proofgold the corresponding term root is 103e1f... and proposition id is 64b86f...
Proof:
Proof not loaded.
End of Section Conj_mul_SNo_prop_1__6__1
Beginning of Section Conj_mul_SNo_prop_1__7__11
L4758
Variable x : set
(*** Conj_mul_SNo_prop_1__7__11 TMGvacqUPn3QdeinwN9k3WHnmtydV9cg8s3 bounty of about 25 bars ***)
L4759
Variable y : set
L4760
Variable z : set
L4761
Variable w : set
L4762
Variable u : set
L4763
Variable v : set
L4764
Variable x2 : set
L4765
Variable y2 : set
L4766
Hypothesis H0 : (βˆ€z2 : set, z2 ∈ SNoS_ (SNoLev x) β†’ (βˆ€w2 : set, SNo w2 β†’ (βˆ€P : prop, (SNo (z2 * w2) β†’ (βˆ€u2 : set, u2 ∈ SNoL z2 β†’ (βˆ€v2 : set, v2 ∈ SNoL w2 β†’ (u2 * w2 + z2 * v2) < z2 * w2 + u2 * v2)) β†’ (βˆ€u2 : set, u2 ∈ SNoR z2 β†’ (βˆ€v2 : set, v2 ∈ SNoR w2 β†’ (u2 * w2 + z2 * v2) < z2 * w2 + u2 * v2)) β†’ (βˆ€u2 : set, u2 ∈ SNoL z2 β†’ (βˆ€v2 : set, v2 ∈ SNoR w2 β†’ (z2 * w2 + u2 * v2) < u2 * w2 + z2 * v2)) β†’ (βˆ€u2 : set, u2 ∈ SNoR z2 β†’ (βˆ€v2 : set, v2 ∈ SNoL w2 β†’ (z2 * w2 + u2 * v2) < u2 * w2 + z2 * v2)) β†’ P) β†’ P)))
L4767
Hypothesis H1 : SNo y
L4768
Hypothesis H2 : (βˆ€z2 : set, z2 ∈ SNoS_ (SNoLev y) β†’ (βˆ€P : prop, (SNo (x * z2) β†’ (βˆ€w2 : set, w2 ∈ SNoL x β†’ (βˆ€u2 : set, u2 ∈ SNoL z2 β†’ (w2 * z2 + x * u2) < x * z2 + w2 * u2)) β†’ (βˆ€w2 : set, w2 ∈ SNoR x β†’ (βˆ€u2 : set, u2 ∈ SNoR z2 β†’ (w2 * z2 + x * u2) < x * z2 + w2 * u2)) β†’ (βˆ€w2 : set, w2 ∈ SNoL x β†’ (βˆ€u2 : set, u2 ∈ SNoR z2 β†’ (x * z2 + w2 * u2) < w2 * z2 + x * u2)) β†’ (βˆ€w2 : set, w2 ∈ SNoR x β†’ (βˆ€u2 : set, u2 ∈ SNoL z2 β†’ (x * z2 + w2 * u2) < w2 * z2 + x * u2)) β†’ P) β†’ P))
L4769
Hypothesis H3 : u ∈ SNoS_ (SNoLev x)
L4770
Hypothesis H4 : v ∈ SNoS_ (SNoLev x)
L4771
Hypothesis H5 : x2 ∈ SNoS_ (SNoLev y)
L4772
Hypothesis H6 : y2 ∈ SNoS_ (SNoLev y)
L4773
Hypothesis H7 : SNo x2
L4774
Hypothesis H8 : SNo y2
L4775
Hypothesis H9 : SNo (u * x2)
L4776
Hypothesis H10 : SNo (v * y2)
L4777
Theorem. (Conj_mul_SNo_prop_1__7__11)
SNo (x * x2) β†’ (βˆ€P : prop, (SNo (u * y) β†’ SNo (x * x2) β†’ SNo (u * x2) β†’ SNo (v * y) β†’ SNo (x * y2) β†’ SNo (v * y2) β†’ SNo (u * y2) β†’ SNo (v * x2) β†’ (z = u * y + x * x2 + - (u * x2) β†’ w = v * y + x * y2 + - (v * y2) β†’ (u * y + x * x2 + v * y2) < v * y + x * y2 + u * x2 β†’ z < w) β†’ P) β†’ P)
In Proofgold the corresponding term root is dda3fc... and proposition id is 4dda75...
Proof:
Proof not loaded.
End of Section Conj_mul_SNo_prop_1__7__11
Beginning of Section Conj_mul_SNo_prop_1__8__5
L4783
Variable x : set
(*** Conj_mul_SNo_prop_1__8__5 TMKjpLMGuhE4aLTe9gb214P12yjLkWWY2KR bounty of about 25 bars ***)
L4784
Variable y : set
L4785
Variable z : set
L4786
Variable w : set
L4787
Variable u : set
L4788
Variable v : set
L4789
Variable x2 : set
L4790
Variable y2 : set
L4791
Hypothesis H0 : (βˆ€z2 : set, z2 ∈ SNoS_ (SNoLev x) β†’ (βˆ€w2 : set, SNo w2 β†’ (βˆ€P : prop, (SNo (z2 * w2) β†’ (βˆ€u2 : set, u2 ∈ SNoL z2 β†’ (βˆ€v2 : set, v2 ∈ SNoL w2 β†’ (u2 * w2 + z2 * v2) < z2 * w2 + u2 * v2)) β†’ (βˆ€u2 : set, u2 ∈ SNoR z2 β†’ (βˆ€v2 : set, v2 ∈ SNoR w2 β†’ (u2 * w2 + z2 * v2) < z2 * w2 + u2 * v2)) β†’ (βˆ€u2 : set, u2 ∈ SNoL z2 β†’ (βˆ€v2 : set, v2 ∈ SNoR w2 β†’ (z2 * w2 + u2 * v2) < u2 * w2 + z2 * v2)) β†’ (βˆ€u2 : set, u2 ∈ SNoR z2 β†’ (βˆ€v2 : set, v2 ∈ SNoL w2 β†’ (z2 * w2 + u2 * v2) < u2 * w2 + z2 * v2)) β†’ P) β†’ P)))
L4792
Hypothesis H1 : SNo y
L4793
Hypothesis H2 : (βˆ€z2 : set, z2 ∈ SNoS_ (SNoLev y) β†’ (βˆ€P : prop, (SNo (x * z2) β†’ (βˆ€w2 : set, w2 ∈ SNoL x β†’ (βˆ€u2 : set, u2 ∈ SNoL z2 β†’ (w2 * z2 + x * u2) < x * z2 + w2 * u2)) β†’ (βˆ€w2 : set, w2 ∈ SNoR x β†’ (βˆ€u2 : set, u2 ∈ SNoR z2 β†’ (w2 * z2 + x * u2) < x * z2 + w2 * u2)) β†’ (βˆ€w2 : set, w2 ∈ SNoL x β†’ (βˆ€u2 : set, u2 ∈ SNoR z2 β†’ (x * z2 + w2 * u2) < w2 * z2 + x * u2)) β†’ (βˆ€w2 : set, w2 ∈ SNoR x β†’ (βˆ€u2 : set, u2 ∈ SNoL z2 β†’ (x * z2 + w2 * u2) < w2 * z2 + x * u2)) β†’ P) β†’ P))
L4794
Hypothesis H3 : u ∈ SNoS_ (SNoLev x)
L4795
Hypothesis H4 : v ∈ SNoS_ (SNoLev x)
L4796
Hypothesis H6 : y2 ∈ SNoS_ (SNoLev y)
L4797
Hypothesis H7 : SNo x2
L4798
Hypothesis H8 : SNo y2
L4799
Hypothesis H9 : SNo (u * x2)
L4800
Hypothesis H10 : SNo (v * y2)
L4801
Theorem. (Conj_mul_SNo_prop_1__8__5)
SNo (u * y) β†’ (βˆ€P : prop, (SNo (u * y) β†’ SNo (x * x2) β†’ SNo (u * x2) β†’ SNo (v * y) β†’ SNo (x * y2) β†’ SNo (v * y2) β†’ SNo (u * y2) β†’ SNo (v * x2) β†’ (z = u * y + x * x2 + - (u * x2) β†’ w = v * y + x * y2 + - (v * y2) β†’ (u * y + x * x2 + v * y2) < v * y + x * y2 + u * x2 β†’ z < w) β†’ P) β†’ P)
In Proofgold the corresponding term root is 7c5d35... and proposition id is 3f798f...
Proof:
Proof not loaded.
End of Section Conj_mul_SNo_prop_1__8__5
Beginning of Section Conj_mul_SNo_prop_1__9__1
L4807
Variable x : set
(*** Conj_mul_SNo_prop_1__9__1 TMMxdg4CN8UzgoS1utojiQaJDUfRsckhB3c bounty of about 25 bars ***)
L4808
Variable y : set
L4809
Variable z : set
L4810
Variable w : set
L4811
Variable u : set
L4812
Variable v : set
L4813
Variable x2 : set
L4814
Variable y2 : set
L4815
Hypothesis H0 : (βˆ€z2 : set, z2 ∈ SNoS_ (SNoLev x) β†’ (βˆ€w2 : set, SNo w2 β†’ (βˆ€P : prop, (SNo (z2 * w2) β†’ (βˆ€u2 : set, u2 ∈ SNoL z2 β†’ (βˆ€v2 : set, v2 ∈ SNoL w2 β†’ (u2 * w2 + z2 * v2) < z2 * w2 + u2 * v2)) β†’ (βˆ€u2 : set, u2 ∈ SNoR z2 β†’ (βˆ€v2 : set, v2 ∈ SNoR w2 β†’ (u2 * w2 + z2 * v2) < z2 * w2 + u2 * v2)) β†’ (βˆ€u2 : set, u2 ∈ SNoL z2 β†’ (βˆ€v2 : set, v2 ∈ SNoR w2 β†’ (z2 * w2 + u2 * v2) < u2 * w2 + z2 * v2)) β†’ (βˆ€u2 : set, u2 ∈ SNoR z2 β†’ (βˆ€v2 : set, v2 ∈ SNoL w2 β†’ (z2 * w2 + u2 * v2) < u2 * w2 + z2 * v2)) β†’ P) β†’ P)))
L4816
Hypothesis H2 : (βˆ€z2 : set, z2 ∈ SNoS_ (SNoLev y) β†’ (βˆ€P : prop, (SNo (x * z2) β†’ (βˆ€w2 : set, w2 ∈ SNoL x β†’ (βˆ€u2 : set, u2 ∈ SNoL z2 β†’ (w2 * z2 + x * u2) < x * z2 + w2 * u2)) β†’ (βˆ€w2 : set, w2 ∈ SNoR x β†’ (βˆ€u2 : set, u2 ∈ SNoR z2 β†’ (w2 * z2 + x * u2) < x * z2 + w2 * u2)) β†’ (βˆ€w2 : set, w2 ∈ SNoL x β†’ (βˆ€u2 : set, u2 ∈ SNoR z2 β†’ (x * z2 + w2 * u2) < w2 * z2 + x * u2)) β†’ (βˆ€w2 : set, w2 ∈ SNoR x β†’ (βˆ€u2 : set, u2 ∈ SNoL z2 β†’ (x * z2 + w2 * u2) < w2 * z2 + x * u2)) β†’ P) β†’ P))
L4817
Hypothesis H3 : u ∈ SNoS_ (SNoLev x)
L4818
Hypothesis H4 : v ∈ SNoS_ (SNoLev x)
L4819
Hypothesis H5 : x2 ∈ SNoS_ (SNoLev y)
L4820
Hypothesis H6 : y2 ∈ SNoS_ (SNoLev y)
L4821
Hypothesis H7 : SNo x2
L4822
Hypothesis H8 : SNo y2
L4823
Hypothesis H9 : SNo (u * x2)
L4824
Theorem. (Conj_mul_SNo_prop_1__9__1)
SNo (v * y2) β†’ (βˆ€P : prop, (SNo (u * y) β†’ SNo (x * x2) β†’ SNo (u * x2) β†’ SNo (v * y) β†’ SNo (x * y2) β†’ SNo (v * y2) β†’ SNo (u * y2) β†’ SNo (v * x2) β†’ (z = u * y + x * x2 + - (u * x2) β†’ w = v * y + x * y2 + - (v * y2) β†’ (u * y + x * x2 + v * y2) < v * y + x * y2 + u * x2 β†’ z < w) β†’ P) β†’ P)
In Proofgold the corresponding term root is 132005... and proposition id is 7bc439...
Proof:
Proof not loaded.
End of Section Conj_mul_SNo_prop_1__9__1
Beginning of Section Conj_mul_SNo_prop_1__9__6
L4830
Variable x : set
(*** Conj_mul_SNo_prop_1__9__6 TMaZ2Pm7ChBMXhMnqAte7CAQq6t4DWRwuFT bounty of about 25 bars ***)
L4831
Variable y : set
L4832
Variable z : set
L4833
Variable w : set
L4834
Variable u : set
L4835
Variable v : set
L4836
Variable x2 : set
L4837
Variable y2 : set
L4838
Hypothesis H0 : (βˆ€z2 : set, z2 ∈ SNoS_ (SNoLev x) β†’ (βˆ€w2 : set, SNo w2 β†’ (βˆ€P : prop, (SNo (z2 * w2) β†’ (βˆ€u2 : set, u2 ∈ SNoL z2 β†’ (βˆ€v2 : set, v2 ∈ SNoL w2 β†’ (u2 * w2 + z2 * v2) < z2 * w2 + u2 * v2)) β†’ (βˆ€u2 : set, u2 ∈ SNoR z2 β†’ (βˆ€v2 : set, v2 ∈ SNoR w2 β†’ (u2 * w2 + z2 * v2) < z2 * w2 + u2 * v2)) β†’ (βˆ€u2 : set, u2 ∈ SNoL z2 β†’ (βˆ€v2 : set, v2 ∈ SNoR w2 β†’ (z2 * w2 + u2 * v2) < u2 * w2 + z2 * v2)) β†’ (βˆ€u2 : set, u2 ∈ SNoR z2 β†’ (βˆ€v2 : set, v2 ∈ SNoL w2 β†’ (z2 * w2 + u2 * v2) < u2 * w2 + z2 * v2)) β†’ P) β†’ P)))
L4839
Hypothesis H1 : SNo y
L4840
Hypothesis H2 : (βˆ€z2 : set, z2 ∈ SNoS_ (SNoLev y) β†’ (βˆ€P : prop, (SNo (x * z2) β†’ (βˆ€w2 : set, w2 ∈ SNoL x β†’ (βˆ€u2 : set, u2 ∈ SNoL z2 β†’ (w2 * z2 + x * u2) < x * z2 + w2 * u2)) β†’ (βˆ€w2 : set, w2 ∈ SNoR x β†’ (βˆ€u2 : set, u2 ∈ SNoR z2 β†’ (w2 * z2 + x * u2) < x * z2 + w2 * u2)) β†’ (βˆ€w2 : set, w2 ∈ SNoL x β†’ (βˆ€u2 : set, u2 ∈ SNoR z2 β†’ (x * z2 + w2 * u2) < w2 * z2 + x * u2)) β†’ (βˆ€w2 : set, w2 ∈ SNoR x β†’ (βˆ€u2 : set, u2 ∈ SNoL z2 β†’ (x * z2 + w2 * u2) < w2 * z2 + x * u2)) β†’ P) β†’ P))
L4841
Hypothesis H3 : u ∈ SNoS_ (SNoLev x)
L4842
Hypothesis H4 : v ∈ SNoS_ (SNoLev x)
L4843
Hypothesis H5 : x2 ∈ SNoS_ (SNoLev y)
L4844
Hypothesis H7 : SNo x2
L4845
Hypothesis H8 : SNo y2
L4846
Hypothesis H9 : SNo (u * x2)
L4847
Theorem. (Conj_mul_SNo_prop_1__9__6)
SNo (v * y2) β†’ (βˆ€P : prop, (SNo (u * y) β†’ SNo (x * x2) β†’ SNo (u * x2) β†’ SNo (v * y) β†’ SNo (x * y2) β†’ SNo (v * y2) β†’ SNo (u * y2) β†’ SNo (v * x2) β†’ (z = u * y + x * x2 + - (u * x2) β†’ w = v * y + x * y2 + - (v * y2) β†’ (u * y + x * x2 + v * y2) < v * y + x * y2 + u * x2 β†’ z < w) β†’ P) β†’ P)
In Proofgold the corresponding term root is d320bc... and proposition id is 05787c...
Proof:
Proof not loaded.
End of Section Conj_mul_SNo_prop_1__9__6
Beginning of Section Conj_mul_SNo_prop_1__13__4
L4853
Variable x : set
(*** Conj_mul_SNo_prop_1__13__4 TMK7gT5Nt7Q8RXz7BYodQXSz1P9USKnFQ5b bounty of about 25 bars ***)
L4854
Variable y : set
L4855
Variable z : set
L4856
Variable w : set
L4857
Variable u : set
L4858
Variable v : set
L4859
Hypothesis H0 : SNo y
L4860
Hypothesis H1 : (βˆ€x2 : set, x2 ∈ SNoS_ (SNoLev y) β†’ (βˆ€P : prop, (SNo (x * x2) β†’ (βˆ€y2 : set, y2 ∈ SNoL x β†’ (βˆ€z2 : set, z2 ∈ SNoL x2 β†’ (y2 * x2 + x * z2) < x * x2 + y2 * z2)) β†’ (βˆ€y2 : set, y2 ∈ SNoR x β†’ (βˆ€z2 : set, z2 ∈ SNoR x2 β†’ (y2 * x2 + x * z2) < x * x2 + y2 * z2)) β†’ (βˆ€y2 : set, y2 ∈ SNoL x β†’ (βˆ€z2 : set, z2 ∈ SNoR x2 β†’ (x * x2 + y2 * z2) < y2 * x2 + x * z2)) β†’ (βˆ€y2 : set, y2 ∈ SNoR x β†’ (βˆ€z2 : set, z2 ∈ SNoL x2 β†’ (x * x2 + y2 * z2) < y2 * x2 + x * z2)) β†’ P) β†’ P))
L4861
Hypothesis H2 : z ∈ SNoR y
L4862
Hypothesis H3 : w ∈ SNoL y
L4863
Hypothesis H5 : SNo (x * w)
L4864
Hypothesis H6 : u ∈ SNoR x
L4865
Hypothesis H7 : SNo (u * z)
L4866
Hypothesis H8 : SNo (u * w)
L4867
Hypothesis H9 : v ∈ SNoL z
L4868
Hypothesis H10 : v ∈ SNoR w
L4869
Hypothesis H11 : v ∈ SNoS_ (SNoLev y)
L4870
Hypothesis H12 : SNo (u * v)
L4871
Theorem. (Conj_mul_SNo_prop_1__13__4)
SNo (x * v) β†’ (x * z + u * w) < u * z + x * w
In Proofgold the corresponding term root is 84ae85... and proposition id is beb486...
Proof:
Proof not loaded.
End of Section Conj_mul_SNo_prop_1__13__4
Beginning of Section Conj_mul_SNo_prop_1__13__5
L4877
Variable x : set
(*** Conj_mul_SNo_prop_1__13__5 TMVyzndZqEtDSJBiHqqK1rsEtpANkCTDKpk bounty of about 25 bars ***)
L4878
Variable y : set
L4879
Variable z : set
L4880
Variable w : set
L4881
Variable u : set
L4882
Variable v : set
L4883
Hypothesis H0 : SNo y
L4884
Hypothesis H1 : (βˆ€x2 : set, x2 ∈ SNoS_ (SNoLev y) β†’ (βˆ€P : prop, (SNo (x * x2) β†’ (βˆ€y2 : set, y2 ∈ SNoL x β†’ (βˆ€z2 : set, z2 ∈ SNoL x2 β†’ (y2 * x2 + x * z2) < x * x2 + y2 * z2)) β†’ (βˆ€y2 : set, y2 ∈ SNoR x β†’ (βˆ€z2 : set, z2 ∈ SNoR x2 β†’ (y2 * x2 + x * z2) < x * x2 + y2 * z2)) β†’ (βˆ€y2 : set, y2 ∈ SNoL x β†’ (βˆ€z2 : set, z2 ∈ SNoR x2 β†’ (x * x2 + y2 * z2) < y2 * x2 + x * z2)) β†’ (βˆ€y2 : set, y2 ∈ SNoR x β†’ (βˆ€z2 : set, z2 ∈ SNoL x2 β†’ (x * x2 + y2 * z2) < y2 * x2 + x * z2)) β†’ P) β†’ P))
L4885
Hypothesis H2 : z ∈ SNoR y
L4886
Hypothesis H3 : w ∈ SNoL y
L4887
Hypothesis H4 : SNo (x * z)
L4888
Hypothesis H6 : u ∈ SNoR x
L4889
Hypothesis H7 : SNo (u * z)
L4890
Hypothesis H8 : SNo (u * w)
L4891
Hypothesis H9 : v ∈ SNoL z
L4892
Hypothesis H10 : v ∈ SNoR w
L4893
Hypothesis H11 : v ∈ SNoS_ (SNoLev y)
L4894
Hypothesis H12 : SNo (u * v)
L4895
Theorem. (Conj_mul_SNo_prop_1__13__5)
SNo (x * v) β†’ (x * z + u * w) < u * z + x * w
In Proofgold the corresponding term root is 29827e... and proposition id is c77ccc...
Proof:
Proof not loaded.
End of Section Conj_mul_SNo_prop_1__13__5
Beginning of Section Conj_mul_SNo_prop_1__17__11
L4901
Variable x : set
(*** Conj_mul_SNo_prop_1__17__11 TMJdiXoTF8q2Ctv3kA6kpGsvR3XVKnSJRro bounty of about 25 bars ***)
L4902
Variable y : set
L4903
Variable z : set
L4904
Variable w : set
L4905
Variable u : set
L4906
Hypothesis H0 : SNo y
L4907
Hypothesis H1 : (βˆ€v : set, v ∈ SNoS_ (SNoLev y) β†’ (βˆ€P : prop, (SNo (x * v) β†’ (βˆ€x2 : set, x2 ∈ SNoL x β†’ (βˆ€y2 : set, y2 ∈ SNoL v β†’ (x2 * v + x * y2) < x * v + x2 * y2)) β†’ (βˆ€x2 : set, x2 ∈ SNoR x β†’ (βˆ€y2 : set, y2 ∈ SNoR v β†’ (x2 * v + x * y2) < x * v + x2 * y2)) β†’ (βˆ€x2 : set, x2 ∈ SNoL x β†’ (βˆ€y2 : set, y2 ∈ SNoR v β†’ (x * v + x2 * y2) < x2 * v + x * y2)) β†’ (βˆ€x2 : set, x2 ∈ SNoR x β†’ (βˆ€y2 : set, y2 ∈ SNoL v β†’ (x * v + x2 * y2) < x2 * v + x * y2)) β†’ P) β†’ P))
L4908
Hypothesis H2 : (βˆ€v : set, v ∈ SNoR x β†’ (βˆ€x2 : set, SNo x2 β†’ SNo (v * x2)))
L4909
Hypothesis H3 : z ∈ SNoR y
L4910
Hypothesis H4 : SNo z
L4911
Hypothesis H5 : SNoLev z ∈ SNoLev y
L4912
Hypothesis H6 : w ∈ SNoL y
L4913
Hypothesis H7 : SNo w
L4914
Hypothesis H8 : SNo (x * z)
L4915
Hypothesis H9 : SNo (x * w)
L4916
Hypothesis H10 : w < z
L4917
Theorem. (Conj_mul_SNo_prop_1__17__11)
SNo (u * z) β†’ (x * z + u * w) < u * z + x * w
In Proofgold the corresponding term root is fbd2d7... and proposition id is 9a770c...
Proof:
Proof not loaded.
End of Section Conj_mul_SNo_prop_1__17__11
Beginning of Section Conj_mul_SNo_prop_1__18__5
L4923
Variable x : set
(*** Conj_mul_SNo_prop_1__18__5 TMUBBZ4iytUoskdnE2gNzcT97cfufmZkbUb bounty of about 25 bars ***)
L4924
Variable y : set
L4925
Variable z : set
L4926
Variable w : set
L4927
Variable u : set
L4928
Variable v : set
L4929
Variable x2 : set
L4930
Hypothesis H0 : SNo x
L4931
Hypothesis H1 : (βˆ€y2 : set, y2 ∈ SNoS_ (SNoLev x) β†’ (βˆ€z2 : set, SNo z2 β†’ (βˆ€P : prop, (SNo (y2 * z2) β†’ (βˆ€w2 : set, w2 ∈ SNoL y2 β†’ (βˆ€u2 : set, u2 ∈ SNoL z2 β†’ (w2 * z2 + y2 * u2) < y2 * z2 + w2 * u2)) β†’ (βˆ€w2 : set, w2 ∈ SNoR y2 β†’ (βˆ€u2 : set, u2 ∈ SNoR z2 β†’ (w2 * z2 + y2 * u2) < y2 * z2 + w2 * u2)) β†’ (βˆ€w2 : set, w2 ∈ SNoL y2 β†’ (βˆ€u2 : set, u2 ∈ SNoR z2 β†’ (y2 * z2 + w2 * u2) < w2 * z2 + y2 * u2)) β†’ (βˆ€w2 : set, w2 ∈ SNoR y2 β†’ (βˆ€u2 : set, u2 ∈ SNoL z2 β†’ (y2 * z2 + w2 * u2) < w2 * z2 + y2 * u2)) β†’ P) β†’ P)))
L4932
Hypothesis H2 : SNo y
L4933
Hypothesis H3 : (βˆ€y2 : set, y2 ∈ SNoR x β†’ (βˆ€z2 : set, SNo z2 β†’ SNo (y2 * z2)))
L4934
Hypothesis H4 : (βˆ€y2 : set, y2 ∈ SNoR x β†’ SNo (y2 * y))
L4935
Hypothesis H6 : v ∈ SNoL y
L4936
Hypothesis H7 : SNo v
L4937
Hypothesis H8 : SNo (z * y)
L4938
Hypothesis H9 : SNo (x * w)
L4939
Hypothesis H10 : SNo (z * w)
L4940
Hypothesis H11 : SNo (u * y)
L4941
Hypothesis H12 : SNo (x * v)
L4942
Hypothesis H13 : SNo (u * v)
L4943
Hypothesis H14 : SNo (z * v)
L4944
Hypothesis H15 : (x * w + z * v) < z * w + x * v
L4945
Hypothesis H16 : x2 ∈ SNoL u
L4946
Hypothesis H17 : x2 ∈ SNoR x
L4947
Hypothesis H18 : (z * y + x2 * v) < x2 * y + z * v
L4948
Theorem. (Conj_mul_SNo_prop_1__18__5)
(x2 * y + u * v) < u * y + x2 * v β†’ (z * y + x * w + u * v) < u * y + x * v + z * w
In Proofgold the corresponding term root is 2134e1... and proposition id is 09e92e...
Proof:
Proof not loaded.
End of Section Conj_mul_SNo_prop_1__18__5
Beginning of Section Conj_mul_SNo_prop_1__19__5
L4954
Variable x : set
(*** Conj_mul_SNo_prop_1__19__5 TMMpG3fBq2rPV6if2kMt7fJaog1qMqGXcNL bounty of about 25 bars ***)
L4955
Variable y : set
L4956
Variable z : set
L4957
Variable w : set
L4958
Variable u : set
L4959
Variable v : set
L4960
Variable x2 : set
L4961
Hypothesis H0 : SNo x
L4962
Hypothesis H1 : (βˆ€y2 : set, y2 ∈ SNoS_ (SNoLev x) β†’ (βˆ€z2 : set, SNo z2 β†’ (βˆ€P : prop, (SNo (y2 * z2) β†’ (βˆ€w2 : set, w2 ∈ SNoL y2 β†’ (βˆ€u2 : set, u2 ∈ SNoL z2 β†’ (w2 * z2 + y2 * u2) < y2 * z2 + w2 * u2)) β†’ (βˆ€w2 : set, w2 ∈ SNoR y2 β†’ (βˆ€u2 : set, u2 ∈ SNoR z2 β†’ (w2 * z2 + y2 * u2) < y2 * z2 + w2 * u2)) β†’ (βˆ€w2 : set, w2 ∈ SNoL y2 β†’ (βˆ€u2 : set, u2 ∈ SNoR z2 β†’ (y2 * z2 + w2 * u2) < w2 * z2 + y2 * u2)) β†’ (βˆ€w2 : set, w2 ∈ SNoR y2 β†’ (βˆ€u2 : set, u2 ∈ SNoL z2 β†’ (y2 * z2 + w2 * u2) < w2 * z2 + y2 * u2)) β†’ P) β†’ P)))
L4963
Hypothesis H2 : SNo y
L4964
Hypothesis H3 : (βˆ€y2 : set, y2 ∈ SNoR x β†’ (βˆ€z2 : set, SNo z2 β†’ SNo (y2 * z2)))
L4965
Hypothesis H4 : (βˆ€y2 : set, y2 ∈ SNoR x β†’ SNo (y2 * y))
L4966
Hypothesis H6 : u ∈ SNoR x
L4967
Hypothesis H7 : v ∈ SNoL y
L4968
Hypothesis H8 : SNo v
L4969
Hypothesis H9 : SNo (z * y)
L4970
Hypothesis H10 : SNo (x * w)
L4971
Hypothesis H11 : SNo (z * w)
L4972
Hypothesis H12 : SNo (u * y)
L4973
Hypothesis H13 : SNo (x * v)
L4974
Hypothesis H14 : SNo (u * v)
L4975
Hypothesis H15 : SNo (z * v)
L4976
Hypothesis H16 : (x * w + z * v) < z * w + x * v
L4977
Hypothesis H17 : x2 ∈ SNoR z
L4978
Hypothesis H18 : x2 ∈ SNoL u
L4979
Hypothesis H19 : x2 ∈ SNoR x
L4980
Theorem. (Conj_mul_SNo_prop_1__19__5)
(z * y + x2 * v) < x2 * y + z * v β†’ (z * y + x * w + u * v) < u * y + x * v + z * w
In Proofgold the corresponding term root is 062fb3... and proposition id is babdb7...
Proof:
Proof not loaded.
End of Section Conj_mul_SNo_prop_1__19__5
Beginning of Section Conj_mul_SNo_prop_1__20__0
L4986
Variable x : set
(*** Conj_mul_SNo_prop_1__20__0 TMVPHdyk9jpfya3CYgH2QKb6qjfxdnbfWyd bounty of about 25 bars ***)
L4987
Variable y : set
L4988
Variable z : set
L4989
Variable w : set
L4990
Variable u : set
L4991
Variable v : set
L4992
Variable x2 : set
L4993
Hypothesis H1 : (βˆ€y2 : set, y2 ∈ SNoS_ (SNoLev x) β†’ (βˆ€z2 : set, SNo z2 β†’ (βˆ€P : prop, (SNo (y2 * z2) β†’ (βˆ€w2 : set, w2 ∈ SNoL y2 β†’ (βˆ€u2 : set, u2 ∈ SNoL z2 β†’ (w2 * z2 + y2 * u2) < y2 * z2 + w2 * u2)) β†’ (βˆ€w2 : set, w2 ∈ SNoR y2 β†’ (βˆ€u2 : set, u2 ∈ SNoR z2 β†’ (w2 * z2 + y2 * u2) < y2 * z2 + w2 * u2)) β†’ (βˆ€w2 : set, w2 ∈ SNoL y2 β†’ (βˆ€u2 : set, u2 ∈ SNoR z2 β†’ (y2 * z2 + w2 * u2) < w2 * z2 + y2 * u2)) β†’ (βˆ€w2 : set, w2 ∈ SNoR y2 β†’ (βˆ€u2 : set, u2 ∈ SNoL z2 β†’ (y2 * z2 + w2 * u2) < w2 * z2 + y2 * u2)) β†’ P) β†’ P)))
L4994
Hypothesis H2 : SNo y
L4995
Hypothesis H3 : (βˆ€y2 : set, y2 ∈ SNoR x β†’ (βˆ€z2 : set, SNo z2 β†’ SNo (y2 * z2)))
L4996
Hypothesis H4 : (βˆ€y2 : set, y2 ∈ SNoR x β†’ SNo (y2 * y))
L4997
Hypothesis H5 : z ∈ SNoR x
L4998
Hypothesis H6 : SNo z
L4999
Hypothesis H7 : SNoLev z ∈ SNoLev x
L5000
Hypothesis H8 : x < z
L5001
Hypothesis H9 : u ∈ SNoR x
L5002
Hypothesis H10 : v ∈ SNoL y
L5003
Hypothesis H11 : SNo v
L5004
Hypothesis H12 : SNo (z * y)
L5005
Hypothesis H13 : SNo (x * w)
L5006
Hypothesis H14 : SNo (z * w)
L5007
Hypothesis H15 : SNo (u * y)
L5008
Hypothesis H16 : SNo (x * v)
L5009
Hypothesis H17 : SNo (u * v)
L5010
Hypothesis H18 : SNo (z * v)
L5011
Hypothesis H19 : (x * w + z * v) < z * w + x * v
L5012
Hypothesis H20 : x2 ∈ SNoR z
L5013
Hypothesis H21 : x2 ∈ SNoL u
L5014
Hypothesis H22 : SNo x2
L5015
Hypothesis H23 : SNoLev x2 ∈ SNoLev z
L5016
Hypothesis H24 : z < x2
L5017
Theorem. (Conj_mul_SNo_prop_1__20__0)
x2 ∈ SNoR x β†’ (z * y + x * w + u * v) < u * y + x * v + z * w
In Proofgold the corresponding term root is 784e0c... and proposition id is 99b85a...
Proof:
Proof not loaded.
End of Section Conj_mul_SNo_prop_1__20__0
Beginning of Section Conj_mul_SNo_prop_1__20__19
L5023
Variable x : set
(*** Conj_mul_SNo_prop_1__20__19 TMYTkKd8JDKsJd4wkca1HaRz1JFEZAZt25q bounty of about 25 bars ***)
L5024
Variable y : set
L5025
Variable z : set
L5026
Variable w : set
L5027
Variable u : set
L5028
Variable v : set
L5029
Variable x2 : set
L5030
Hypothesis H0 : SNo x
L5031
Hypothesis H1 : (βˆ€y2 : set, y2 ∈ SNoS_ (SNoLev x) β†’ (βˆ€z2 : set, SNo z2 β†’ (βˆ€P : prop, (SNo (y2 * z2) β†’ (βˆ€w2 : set, w2 ∈ SNoL y2 β†’ (βˆ€u2 : set, u2 ∈ SNoL z2 β†’ (w2 * z2 + y2 * u2) < y2 * z2 + w2 * u2)) β†’ (βˆ€w2 : set, w2 ∈ SNoR y2 β†’ (βˆ€u2 : set, u2 ∈ SNoR z2 β†’ (w2 * z2 + y2 * u2) < y2 * z2 + w2 * u2)) β†’ (βˆ€w2 : set, w2 ∈ SNoL y2 β†’ (βˆ€u2 : set, u2 ∈ SNoR z2 β†’ (y2 * z2 + w2 * u2) < w2 * z2 + y2 * u2)) β†’ (βˆ€w2 : set, w2 ∈ SNoR y2 β†’ (βˆ€u2 : set, u2 ∈ SNoL z2 β†’ (y2 * z2 + w2 * u2) < w2 * z2 + y2 * u2)) β†’ P) β†’ P)))
L5032
Hypothesis H2 : SNo y
L5033
Hypothesis H3 : (βˆ€y2 : set, y2 ∈ SNoR x β†’ (βˆ€z2 : set, SNo z2 β†’ SNo (y2 * z2)))
L5034
Hypothesis H4 : (βˆ€y2 : set, y2 ∈ SNoR x β†’ SNo (y2 * y))
L5035
Hypothesis H5 : z ∈ SNoR x
L5036
Hypothesis H6 : SNo z
L5037
Hypothesis H7 : SNoLev z ∈ SNoLev x
L5038
Hypothesis H8 : x < z
L5039
Hypothesis H9 : u ∈ SNoR x
L5040
Hypothesis H10 : v ∈ SNoL y
L5041
Hypothesis H11 : SNo v
L5042
Hypothesis H12 : SNo (z * y)
L5043
Hypothesis H13 : SNo (x * w)
L5044
Hypothesis H14 : SNo (z * w)
L5045
Hypothesis H15 : SNo (u * y)
L5046
Hypothesis H16 : SNo (x * v)
L5047
Hypothesis H17 : SNo (u * v)
L5048
Hypothesis H18 : SNo (z * v)
L5049
Hypothesis H20 : x2 ∈ SNoR z
L5050
Hypothesis H21 : x2 ∈ SNoL u
L5051
Hypothesis H22 : SNo x2
L5052
Hypothesis H23 : SNoLev x2 ∈ SNoLev z
L5053
Hypothesis H24 : z < x2
L5054
Theorem. (Conj_mul_SNo_prop_1__20__19)
x2 ∈ SNoR x β†’ (z * y + x * w + u * v) < u * y + x * v + z * w
In Proofgold the corresponding term root is e4e961... and proposition id is f6c9bd...
Proof:
Proof not loaded.
End of Section Conj_mul_SNo_prop_1__20__19
Beginning of Section Conj_mul_SNo_prop_1__20__24
L5060
Variable x : set
(*** Conj_mul_SNo_prop_1__20__24 TMUZozXVomEwAATkkEWDczRzpaVctYnSmZw bounty of about 25 bars ***)
L5061
Variable y : set
L5062
Variable z : set
L5063
Variable w : set
L5064
Variable u : set
L5065
Variable v : set
L5066
Variable x2 : set
L5067
Hypothesis H0 : SNo x
L5068
Hypothesis H1 : (βˆ€y2 : set, y2 ∈ SNoS_ (SNoLev x) β†’ (βˆ€z2 : set, SNo z2 β†’ (βˆ€P : prop, (SNo (y2 * z2) β†’ (βˆ€w2 : set, w2 ∈ SNoL y2 β†’ (βˆ€u2 : set, u2 ∈ SNoL z2 β†’ (w2 * z2 + y2 * u2) < y2 * z2 + w2 * u2)) β†’ (βˆ€w2 : set, w2 ∈ SNoR y2 β†’ (βˆ€u2 : set, u2 ∈ SNoR z2 β†’ (w2 * z2 + y2 * u2) < y2 * z2 + w2 * u2)) β†’ (βˆ€w2 : set, w2 ∈ SNoL y2 β†’ (βˆ€u2 : set, u2 ∈ SNoR z2 β†’ (y2 * z2 + w2 * u2) < w2 * z2 + y2 * u2)) β†’ (βˆ€w2 : set, w2 ∈ SNoR y2 β†’ (βˆ€u2 : set, u2 ∈ SNoL z2 β†’ (y2 * z2 + w2 * u2) < w2 * z2 + y2 * u2)) β†’ P) β†’ P)))
L5069
Hypothesis H2 : SNo y
L5070
Hypothesis H3 : (βˆ€y2 : set, y2 ∈ SNoR x β†’ (βˆ€z2 : set, SNo z2 β†’ SNo (y2 * z2)))
L5071
Hypothesis H4 : (βˆ€y2 : set, y2 ∈ SNoR x β†’ SNo (y2 * y))
L5072
Hypothesis H5 : z ∈ SNoR x
L5073
Hypothesis H6 : SNo z
L5074
Hypothesis H7 : SNoLev z ∈ SNoLev x
L5075
Hypothesis H8 : x < z
L5076
Hypothesis H9 : u ∈ SNoR x
L5077
Hypothesis H10 : v ∈ SNoL y
L5078
Hypothesis H11 : SNo v
L5079
Hypothesis H12 : SNo (z * y)
L5080
Hypothesis H13 : SNo (x * w)
L5081
Hypothesis H14 : SNo (z * w)
L5082
Hypothesis H15 : SNo (u * y)
L5083
Hypothesis H16 : SNo (x * v)
L5084
Hypothesis H17 : SNo (u * v)
L5085
Hypothesis H18 : SNo (z * v)
L5086
Hypothesis H19 : (x * w + z * v) < z * w + x * v
L5087
Hypothesis H20 : x2 ∈ SNoR z
L5088
Hypothesis H21 : x2 ∈ SNoL u
L5089
Hypothesis H22 : SNo x2
L5090
Hypothesis H23 : SNoLev x2 ∈ SNoLev z
L5091
Theorem. (Conj_mul_SNo_prop_1__20__24)
x2 ∈ SNoR x β†’ (z * y + x * w + u * v) < u * y + x * v + z * w
In Proofgold the corresponding term root is 441125... and proposition id is ff0c0a...
Proof:
Proof not loaded.
End of Section Conj_mul_SNo_prop_1__20__24
Beginning of Section Conj_mul_SNo_prop_1__23__4
L5097
Variable x : set
(*** Conj_mul_SNo_prop_1__23__4 TMKMjywM9qUXFY45cw55JwB9wmETHUXjGKs bounty of about 25 bars ***)
L5098
Variable y : set
L5099
Variable z : set
L5100
Variable w : set
L5101
Variable u : set
L5102
Variable v : set
L5103
Variable x2 : set
L5104
Hypothesis H0 : SNo x
L5105
Hypothesis H1 : (βˆ€y2 : set, y2 ∈ SNoS_ (SNoLev x) β†’ (βˆ€z2 : set, SNo z2 β†’ (βˆ€P : prop, (SNo (y2 * z2) β†’ (βˆ€w2 : set, w2 ∈ SNoL y2 β†’ (βˆ€u2 : set, u2 ∈ SNoL z2 β†’ (w2 * z2 + y2 * u2) < y2 * z2 + w2 * u2)) β†’ (βˆ€w2 : set, w2 ∈ SNoR y2 β†’ (βˆ€u2 : set, u2 ∈ SNoR z2 β†’ (w2 * z2 + y2 * u2) < y2 * z2 + w2 * u2)) β†’ (βˆ€w2 : set, w2 ∈ SNoL y2 β†’ (βˆ€u2 : set, u2 ∈ SNoR z2 β†’ (y2 * z2 + w2 * u2) < w2 * z2 + y2 * u2)) β†’ (βˆ€w2 : set, w2 ∈ SNoR y2 β†’ (βˆ€u2 : set, u2 ∈ SNoL z2 β†’ (y2 * z2 + w2 * u2) < w2 * z2 + y2 * u2)) β†’ P) β†’ P)))
L5106
Hypothesis H2 : SNo y
L5107
Hypothesis H3 : (βˆ€y2 : set, y2 ∈ SNoR x β†’ (βˆ€z2 : set, SNo z2 β†’ SNo (y2 * z2)))
L5108
Hypothesis H5 : z ∈ SNoR x
L5109
Hypothesis H6 : w ∈ SNoR y
L5110
Hypothesis H7 : SNo w
L5111
Hypothesis H8 : u ∈ SNoR x
L5112
Hypothesis H9 : SNo u
L5113
Hypothesis H10 : SNoLev u ∈ SNoLev x
L5114
Hypothesis H11 : x < u
L5115
Hypothesis H12 : SNo (z * y)
L5116
Hypothesis H13 : SNo (x * w)
L5117
Hypothesis H14 : SNo (z * w)
L5118
Hypothesis H15 : SNo (u * y)
L5119
Hypothesis H16 : SNo (x * v)
L5120
Hypothesis H17 : SNo (u * v)
L5121
Hypothesis H18 : SNo (u * w)
L5122
Hypothesis H19 : (x * w + u * v) < u * w + x * v
L5123
Hypothesis H20 : x2 ∈ SNoL z
L5124
Hypothesis H21 : x2 ∈ SNoR u
L5125
Hypothesis H22 : SNo x2
L5126
Hypothesis H23 : SNoLev x2 ∈ SNoLev u
L5127
Hypothesis H24 : u < x2
L5128
Theorem. (Conj_mul_SNo_prop_1__23__4)
x2 ∈ SNoR x β†’ (z * y + x * w + u * v) < u * y + x * v + z * w
In Proofgold the corresponding term root is 23b478... and proposition id is 793f23...
Proof:
Proof not loaded.
End of Section Conj_mul_SNo_prop_1__23__4
Beginning of Section Conj_mul_SNo_prop_1__25__24
L5134
Variable x : set
(*** Conj_mul_SNo_prop_1__25__24 TMRifuMyoJrgRcxL3ceZ4FMNeySvSHFKE9h bounty of about 25 bars ***)
L5135
Variable y : set
L5136
Variable z : set
L5137
Variable w : set
L5138
Variable u : set
L5139
Variable v : set
L5140
Hypothesis H0 : SNo x
L5141
Hypothesis H1 : (βˆ€x2 : set, x2 ∈ SNoS_ (SNoLev x) β†’ (βˆ€y2 : set, SNo y2 β†’ (βˆ€P : prop, (SNo (x2 * y2) β†’ (βˆ€z2 : set, z2 ∈ SNoL x2 β†’ (βˆ€w2 : set, w2 ∈ SNoL y2 β†’ (z2 * y2 + x2 * w2) < x2 * y2 + z2 * w2)) β†’ (βˆ€z2 : set, z2 ∈ SNoR x2 β†’ (βˆ€w2 : set, w2 ∈ SNoR y2 β†’ (z2 * y2 + x2 * w2) < x2 * y2 + z2 * w2)) β†’ (βˆ€z2 : set, z2 ∈ SNoL x2 β†’ (βˆ€w2 : set, w2 ∈ SNoR y2 β†’ (x2 * y2 + z2 * w2) < z2 * y2 + x2 * w2)) β†’ (βˆ€z2 : set, z2 ∈ SNoR x2 β†’ (βˆ€w2 : set, w2 ∈ SNoL y2 β†’ (x2 * y2 + z2 * w2) < z2 * y2 + x2 * w2)) β†’ P) β†’ P)))
L5142
Hypothesis H2 : SNo y
L5143
Hypothesis H3 : (βˆ€x2 : set, x2 ∈ SNoR x β†’ (βˆ€y2 : set, SNo y2 β†’ SNo (x2 * y2)))
L5144
Hypothesis H4 : (βˆ€x2 : set, x2 ∈ SNoR x β†’ SNo (x2 * y))
L5145
Hypothesis H5 : z ∈ SNoR x
L5146
Hypothesis H6 : w ∈ SNoR y
L5147
Hypothesis H7 : SNo z
L5148
Hypothesis H8 : SNoLev z ∈ SNoLev x
L5149
Hypothesis H9 : x < z
L5150
Hypothesis H10 : SNo w
L5151
Hypothesis H11 : u ∈ SNoR x
L5152
Hypothesis H12 : v ∈ SNoL y
L5153
Hypothesis H13 : SNo u
L5154
Hypothesis H14 : SNoLev u ∈ SNoLev x
L5155
Hypothesis H15 : x < u
L5156
Hypothesis H16 : SNo v
L5157
Hypothesis H17 : SNo (z * y)
L5158
Hypothesis H18 : SNo (x * w)
L5159
Hypothesis H19 : SNo (z * w)
L5160
Hypothesis H20 : SNo (u * y)
L5161
Hypothesis H21 : SNo (x * v)
L5162
Hypothesis H22 : SNo (u * v)
L5163
Hypothesis H23 : SNo (z * v)
L5164
Hypothesis H25 : (x * w + z * v) < z * w + x * v
L5165
Hypothesis H26 : (x * w + u * v) < u * w + x * v
L5166
Theorem. (Conj_mul_SNo_prop_1__25__24)
((z * y + u * v) < u * y + z * v β†’ (z * y + x * w + u * v) < u * y + x * v + z * w) β†’ (z * y + x * w + u * v) < u * y + x * v + z * w
In Proofgold the corresponding term root is af6e22... and proposition id is baf852...
Proof:
Proof not loaded.
End of Section Conj_mul_SNo_prop_1__25__24
Beginning of Section Conj_mul_SNo_prop_1__26__15
L5172
Variable x : set
(*** Conj_mul_SNo_prop_1__26__15 TMGiBqnCadKN41Du1yReNN3piRmxHw1py8d bounty of about 25 bars ***)
L5173
Variable y : set
L5174
Variable z : set
L5175
Variable w : set
L5176
Variable u : set
L5177
Variable v : set
L5178
Hypothesis H0 : SNo x
L5179
Hypothesis H1 : (βˆ€x2 : set, x2 ∈ SNoS_ (SNoLev x) β†’ (βˆ€y2 : set, SNo y2 β†’ (βˆ€P : prop, (SNo (x2 * y2) β†’ (βˆ€z2 : set, z2 ∈ SNoL x2 β†’ (βˆ€w2 : set, w2 ∈ SNoL y2 β†’ (z2 * y2 + x2 * w2) < x2 * y2 + z2 * w2)) β†’ (βˆ€z2 : set, z2 ∈ SNoR x2 β†’ (βˆ€w2 : set, w2 ∈ SNoR y2 β†’ (z2 * y2 + x2 * w2) < x2 * y2 + z2 * w2)) β†’ (βˆ€z2 : set, z2 ∈ SNoL x2 β†’ (βˆ€w2 : set, w2 ∈ SNoR y2 β†’ (x2 * y2 + z2 * w2) < z2 * y2 + x2 * w2)) β†’ (βˆ€z2 : set, z2 ∈ SNoR x2 β†’ (βˆ€w2 : set, w2 ∈ SNoL y2 β†’ (x2 * y2 + z2 * w2) < z2 * y2 + x2 * w2)) β†’ P) β†’ P)))
L5180
Hypothesis H2 : SNo y
L5181
Hypothesis H3 : (βˆ€x2 : set, x2 ∈ SNoR x β†’ (βˆ€y2 : set, SNo y2 β†’ SNo (x2 * y2)))
L5182
Hypothesis H4 : (βˆ€x2 : set, x2 ∈ SNoR x β†’ SNo (x2 * y))
L5183
Hypothesis H5 : z ∈ SNoR x
L5184
Hypothesis H6 : w ∈ SNoR y
L5185
Hypothesis H7 : SNo z
L5186
Hypothesis H8 : SNoLev z ∈ SNoLev x
L5187
Hypothesis H9 : x < z
L5188
Hypothesis H10 : SNo w
L5189
Hypothesis H11 : u ∈ SNoR x
L5190
Hypothesis H12 : v ∈ SNoL y
L5191
Hypothesis H13 : SNo u
L5192
Hypothesis H14 : SNoLev u ∈ SNoLev x
L5193
Hypothesis H16 : SNo v
L5194
Hypothesis H17 : SNo (z * y)
L5195
Hypothesis H18 : SNo (x * w)
L5196
Hypothesis H19 : SNo (z * w)
L5197
Hypothesis H20 : SNo (u * y)
L5198
Hypothesis H21 : SNo (x * v)
L5199
Hypothesis H22 : SNo (u * v)
L5200
Hypothesis H23 : SNo (z * v)
L5201
Hypothesis H24 : SNo (u * w)
L5202
Hypothesis H25 : (βˆ€x2 : set, x2 ∈ SNoR x β†’ (x * w + x2 * v) < x2 * w + x * v)
L5203
Hypothesis H26 : (x * w + z * v) < z * w + x * v
L5204
Theorem. (Conj_mul_SNo_prop_1__26__15)
(x * w + u * v) < u * w + x * v β†’ (z * y + x * w + u * v) < u * y + x * v + z * w
In Proofgold the corresponding term root is 94150b... and proposition id is a8650e...
Proof:
Proof not loaded.
End of Section Conj_mul_SNo_prop_1__26__15
Beginning of Section Conj_mul_SNo_prop_1__27__8
L5210
Variable x : set
(*** Conj_mul_SNo_prop_1__27__8 TMTswzTWRarKb5uYw6NmLnyxAMnCPgiHcqY bounty of about 25 bars ***)
L5211
Variable y : set
L5212
Variable z : set
L5213
Variable w : set
L5214
Variable u : set
L5215
Variable v : set
L5216
Hypothesis H0 : SNo x
L5217
Hypothesis H1 : (βˆ€x2 : set, x2 ∈ SNoS_ (SNoLev x) β†’ (βˆ€y2 : set, SNo y2 β†’ (βˆ€P : prop, (SNo (x2 * y2) β†’ (βˆ€z2 : set, z2 ∈ SNoL x2 β†’ (βˆ€w2 : set, w2 ∈ SNoL y2 β†’ (z2 * y2 + x2 * w2) < x2 * y2 + z2 * w2)) β†’ (βˆ€z2 : set, z2 ∈ SNoR x2 β†’ (βˆ€w2 : set, w2 ∈ SNoR y2 β†’ (z2 * y2 + x2 * w2) < x2 * y2 + z2 * w2)) β†’ (βˆ€z2 : set, z2 ∈ SNoL x2 β†’ (βˆ€w2 : set, w2 ∈ SNoR y2 β†’ (x2 * y2 + z2 * w2) < z2 * y2 + x2 * w2)) β†’ (βˆ€z2 : set, z2 ∈ SNoR x2 β†’ (βˆ€w2 : set, w2 ∈ SNoL y2 β†’ (x2 * y2 + z2 * w2) < z2 * y2 + x2 * w2)) β†’ P) β†’ P)))
L5218
Hypothesis H2 : SNo y
L5219
Hypothesis H3 : (βˆ€x2 : set, x2 ∈ SNoR x β†’ (βˆ€y2 : set, SNo y2 β†’ SNo (x2 * y2)))
L5220
Hypothesis H4 : (βˆ€x2 : set, x2 ∈ SNoR x β†’ SNo (x2 * y))
L5221
Hypothesis H5 : z ∈ SNoR x
L5222
Hypothesis H6 : w ∈ SNoR y
L5223
Hypothesis H7 : SNo z
L5224
Hypothesis H9 : x < z
L5225
Hypothesis H10 : SNo w
L5226
Hypothesis H11 : u ∈ SNoR x
L5227
Hypothesis H12 : v ∈ SNoL y
L5228
Hypothesis H13 : SNo u
L5229
Hypothesis H14 : SNoLev u ∈ SNoLev x
L5230
Hypothesis H15 : x < u
L5231
Hypothesis H16 : SNo v
L5232
Hypothesis H17 : SNo (z * y)
L5233
Hypothesis H18 : SNo (x * w)
L5234
Hypothesis H19 : SNo (z * w)
L5235
Hypothesis H20 : SNo (u * y)
L5236
Hypothesis H21 : SNo (x * v)
L5237
Hypothesis H22 : SNo (u * v)
L5238
Hypothesis H23 : SNo (z * v)
L5239
Hypothesis H24 : SNo (u * w)
L5240
Hypothesis H25 : (βˆ€x2 : set, x2 ∈ SNoR x β†’ (x * w + x2 * v) < x2 * w + x * v)
L5241
Theorem. (Conj_mul_SNo_prop_1__27__8)
(x * w + z * v) < z * w + x * v β†’ (z * y + x * w + u * v) < u * y + x * v + z * w
In Proofgold the corresponding term root is 751f85... and proposition id is d9868f...
Proof:
Proof not loaded.
End of Section Conj_mul_SNo_prop_1__27__8
Beginning of Section Conj_mul_SNo_prop_1__28__0
L5247
Variable x : set
(*** Conj_mul_SNo_prop_1__28__0 TMKbE1bdjRbvCFLPcGZFzuMmTYQ9gTAv7tS bounty of about 25 bars ***)
L5248
Variable y : set
L5249
Variable z : set
L5250
Variable w : set
L5251
Variable u : set
L5252
Variable v : set
L5253
Hypothesis H1 : (βˆ€x2 : set, x2 ∈ SNoS_ (SNoLev x) β†’ (βˆ€y2 : set, SNo y2 β†’ (βˆ€P : prop, (SNo (x2 * y2) β†’ (βˆ€z2 : set, z2 ∈ SNoL x2 β†’ (βˆ€w2 : set, w2 ∈ SNoL y2 β†’ (z2 * y2 + x2 * w2) < x2 * y2 + z2 * w2)) β†’ (βˆ€z2 : set, z2 ∈ SNoR x2 β†’ (βˆ€w2 : set, w2 ∈ SNoR y2 β†’ (z2 * y2 + x2 * w2) < x2 * y2 + z2 * w2)) β†’ (βˆ€z2 : set, z2 ∈ SNoL x2 β†’ (βˆ€w2 : set, w2 ∈ SNoR y2 β†’ (x2 * y2 + z2 * w2) < z2 * y2 + x2 * w2)) β†’ (βˆ€z2 : set, z2 ∈ SNoR x2 β†’ (βˆ€w2 : set, w2 ∈ SNoL y2 β†’ (x2 * y2 + z2 * w2) < z2 * y2 + x2 * w2)) β†’ P) β†’ P)))
L5254
Hypothesis H2 : SNo y
L5255
Hypothesis H3 : (βˆ€x2 : set, x2 ∈ SNoS_ (SNoLev y) β†’ (βˆ€P : prop, (SNo (x * x2) β†’ (βˆ€y2 : set, y2 ∈ SNoL x β†’ (βˆ€z2 : set, z2 ∈ SNoL x2 β†’ (y2 * x2 + x * z2) < x * x2 + y2 * z2)) β†’ (βˆ€y2 : set, y2 ∈ SNoR x β†’ (βˆ€z2 : set, z2 ∈ SNoR x2 β†’ (y2 * x2 + x * z2) < x * x2 + y2 * z2)) β†’ (βˆ€y2 : set, y2 ∈ SNoL x β†’ (βˆ€z2 : set, z2 ∈ SNoR x2 β†’ (x * x2 + y2 * z2) < y2 * x2 + x * z2)) β†’ (βˆ€y2 : set, y2 ∈ SNoR x β†’ (βˆ€z2 : set, z2 ∈ SNoL x2 β†’ (x * x2 + y2 * z2) < y2 * x2 + x * z2)) β†’ P) β†’ P))
L5256
Hypothesis H4 : (βˆ€x2 : set, x2 ∈ SNoR x β†’ (βˆ€y2 : set, SNo y2 β†’ SNo (x2 * y2)))
L5257
Hypothesis H5 : (βˆ€x2 : set, x2 ∈ SNoR x β†’ SNo (x2 * y))
L5258
Hypothesis H6 : z ∈ SNoR x
L5259
Hypothesis H7 : w ∈ SNoR y
L5260
Hypothesis H8 : SNo z
L5261
Hypothesis H9 : SNoLev z ∈ SNoLev x
L5262
Hypothesis H10 : x < z
L5263
Hypothesis H11 : SNo w
L5264
Hypothesis H12 : SNoLev w ∈ SNoLev y
L5265
Hypothesis H13 : u ∈ SNoR x
L5266
Hypothesis H14 : v ∈ SNoL y
L5267
Hypothesis H15 : SNo u
L5268
Hypothesis H16 : SNoLev u ∈ SNoLev x
L5269
Hypothesis H17 : x < u
L5270
Hypothesis H18 : SNo v
L5271
Hypothesis H19 : SNo (z * y)
L5272
Hypothesis H20 : SNo (x * w)
L5273
Hypothesis H21 : SNo (z * w)
L5274
Hypothesis H22 : SNo (u * y)
L5275
Hypothesis H23 : SNo (x * v)
L5276
Hypothesis H24 : SNo (u * v)
L5277
Hypothesis H25 : SNo (z * v)
L5278
Hypothesis H26 : SNo (u * w)
L5279
Hypothesis H27 : v < w
L5280
Theorem. (Conj_mul_SNo_prop_1__28__0)
(βˆ€x2 : set, x2 ∈ SNoR x β†’ (x * w + x2 * v) < x2 * w + x * v) β†’ (z * y + x * w + u * v) < u * y + x * v + z * w
In Proofgold the corresponding term root is 94fa10... and proposition id is f7c32a...
Proof:
Proof not loaded.
End of Section Conj_mul_SNo_prop_1__28__0
Beginning of Section Conj_mul_SNo_prop_1__28__13
L5286
Variable x : set
(*** Conj_mul_SNo_prop_1__28__13 TMR831BWAdvaEpS2616V4Gd9pzgtvoABtNH bounty of about 25 bars ***)
L5287
Variable y : set
L5288
Variable z : set
L5289
Variable w : set
L5290
Variable u : set
L5291
Variable v : set
L5292
Hypothesis H0 : SNo x
L5293
Hypothesis H1 : (βˆ€x2 : set, x2 ∈ SNoS_ (SNoLev x) β†’ (βˆ€y2 : set, SNo y2 β†’ (βˆ€P : prop, (SNo (x2 * y2) β†’ (βˆ€z2 : set, z2 ∈ SNoL x2 β†’ (βˆ€w2 : set, w2 ∈ SNoL y2 β†’ (z2 * y2 + x2 * w2) < x2 * y2 + z2 * w2)) β†’ (βˆ€z2 : set, z2 ∈ SNoR x2 β†’ (βˆ€w2 : set, w2 ∈ SNoR y2 β†’ (z2 * y2 + x2 * w2) < x2 * y2 + z2 * w2)) β†’ (βˆ€z2 : set, z2 ∈ SNoL x2 β†’ (βˆ€w2 : set, w2 ∈ SNoR y2 β†’ (x2 * y2 + z2 * w2) < z2 * y2 + x2 * w2)) β†’ (βˆ€z2 : set, z2 ∈ SNoR x2 β†’ (βˆ€w2 : set, w2 ∈ SNoL y2 β†’ (x2 * y2 + z2 * w2) < z2 * y2 + x2 * w2)) β†’ P) β†’ P)))
L5294
Hypothesis H2 : SNo y
L5295
Hypothesis H3 : (βˆ€x2 : set, x2 ∈ SNoS_ (SNoLev y) β†’ (βˆ€P : prop, (SNo (x * x2) β†’ (βˆ€y2 : set, y2 ∈ SNoL x β†’ (βˆ€z2 : set, z2 ∈ SNoL x2 β†’ (y2 * x2 + x * z2) < x * x2 + y2 * z2)) β†’ (βˆ€y2 : set, y2 ∈ SNoR x β†’ (βˆ€z2 : set, z2 ∈ SNoR x2 β†’ (y2 * x2 + x * z2) < x * x2 + y2 * z2)) β†’ (βˆ€y2 : set, y2 ∈ SNoL x β†’ (βˆ€z2 : set, z2 ∈ SNoR x2 β†’ (x * x2 + y2 * z2) < y2 * x2 + x * z2)) β†’ (βˆ€y2 : set, y2 ∈ SNoR x β†’ (βˆ€z2 : set, z2 ∈ SNoL x2 β†’ (x * x2 + y2 * z2) < y2 * x2 + x * z2)) β†’ P) β†’ P))
L5296
Hypothesis H4 : (βˆ€x2 : set, x2 ∈ SNoR x β†’ (βˆ€y2 : set, SNo y2 β†’ SNo (x2 * y2)))
L5297
Hypothesis H5 : (βˆ€x2 : set, x2 ∈ SNoR x β†’ SNo (x2 * y))
L5298
Hypothesis H6 : z ∈ SNoR x
L5299
Hypothesis H7 : w ∈ SNoR y
L5300
Hypothesis H8 : SNo z
L5301
Hypothesis H9 : SNoLev z ∈ SNoLev x
L5302
Hypothesis H10 : x < z
L5303
Hypothesis H11 : SNo w
L5304
Hypothesis H12 : SNoLev w ∈ SNoLev y
L5305
Hypothesis H14 : v ∈ SNoL y
L5306
Hypothesis H15 : SNo u
L5307
Hypothesis H16 : SNoLev u ∈ SNoLev x
L5308
Hypothesis H17 : x < u
L5309
Hypothesis H18 : SNo v
L5310
Hypothesis H19 : SNo (z * y)
L5311
Hypothesis H20 : SNo (x * w)
L5312
Hypothesis H21 : SNo (z * w)
L5313
Hypothesis H22 : SNo (u * y)
L5314
Hypothesis H23 : SNo (x * v)
L5315
Hypothesis H24 : SNo (u * v)
L5316
Hypothesis H25 : SNo (z * v)
L5317
Hypothesis H26 : SNo (u * w)
L5318
Hypothesis H27 : v < w
L5319
Theorem. (Conj_mul_SNo_prop_1__28__13)
(βˆ€x2 : set, x2 ∈ SNoR x β†’ (x * w + x2 * v) < x2 * w + x * v) β†’ (z * y + x * w + u * v) < u * y + x * v + z * w
In Proofgold the corresponding term root is 4bf70b... and proposition id is 9d004f...
Proof:
Proof not loaded.
End of Section Conj_mul_SNo_prop_1__28__13
Beginning of Section Conj_mul_SNo_prop_1__28__18
L5325
Variable x : set
(*** Conj_mul_SNo_prop_1__28__18 TMXnpbhJNJkTf1338bGH7xwHL48xrz8RNrK bounty of about 25 bars ***)
L5326
Variable y : set
L5327
Variable z : set
L5328
Variable w : set
L5329
Variable u : set
L5330
Variable v : set
L5331
Hypothesis H0 : SNo x
L5332
Hypothesis H1 : (βˆ€x2 : set, x2 ∈ SNoS_ (SNoLev x) β†’ (βˆ€y2 : set, SNo y2 β†’ (βˆ€P : prop, (SNo (x2 * y2) β†’ (βˆ€z2 : set, z2 ∈ SNoL x2 β†’ (βˆ€w2 : set, w2 ∈ SNoL y2 β†’ (z2 * y2 + x2 * w2) < x2 * y2 + z2 * w2)) β†’ (βˆ€z2 : set, z2 ∈ SNoR x2 β†’ (βˆ€w2 : set, w2 ∈ SNoR y2 β†’ (z2 * y2 + x2 * w2) < x2 * y2 + z2 * w2)) β†’ (βˆ€z2 : set, z2 ∈ SNoL x2 β†’ (βˆ€w2 : set, w2 ∈ SNoR y2 β†’ (x2 * y2 + z2 * w2) < z2 * y2 + x2 * w2)) β†’ (βˆ€z2 : set, z2 ∈ SNoR x2 β†’ (βˆ€w2 : set, w2 ∈ SNoL y2 β†’ (x2 * y2 + z2 * w2) < z2 * y2 + x2 * w2)) β†’ P) β†’ P)))
L5333
Hypothesis H2 : SNo y
L5334
Hypothesis H3 : (βˆ€x2 : set, x2 ∈ SNoS_ (SNoLev y) β†’ (βˆ€P : prop, (SNo (x * x2) β†’ (βˆ€y2 : set, y2 ∈ SNoL x β†’ (βˆ€z2 : set, z2 ∈ SNoL x2 β†’ (y2 * x2 + x * z2) < x * x2 + y2 * z2)) β†’ (βˆ€y2 : set, y2 ∈ SNoR x β†’ (βˆ€z2 : set, z2 ∈ SNoR x2 β†’ (y2 * x2 + x * z2) < x * x2 + y2 * z2)) β†’ (βˆ€y2 : set, y2 ∈ SNoL x β†’ (βˆ€z2 : set, z2 ∈ SNoR x2 β†’ (x * x2 + y2 * z2) < y2 * x2 + x * z2)) β†’ (βˆ€y2 : set, y2 ∈ SNoR x β†’ (βˆ€z2 : set, z2 ∈ SNoL x2 β†’ (x * x2 + y2 * z2) < y2 * x2 + x * z2)) β†’ P) β†’ P))
L5335
Hypothesis H4 : (βˆ€x2 : set, x2 ∈ SNoR x β†’ (βˆ€y2 : set, SNo y2 β†’ SNo (x2 * y2)))
L5336
Hypothesis H5 : (βˆ€x2 : set, x2 ∈ SNoR x β†’ SNo (x2 * y))
L5337
Hypothesis H6 : z ∈ SNoR x
L5338
Hypothesis H7 : w ∈ SNoR y
L5339
Hypothesis H8 : SNo z
L5340
Hypothesis H9 : SNoLev z ∈ SNoLev x
L5341
Hypothesis H10 : x < z
L5342
Hypothesis H11 : SNo w
L5343
Hypothesis H12 : SNoLev w ∈ SNoLev y
L5344
Hypothesis H13 : u ∈ SNoR x
L5345
Hypothesis H14 : v ∈ SNoL y
L5346
Hypothesis H15 : SNo u
L5347
Hypothesis H16 : SNoLev u ∈ SNoLev x
L5348
Hypothesis H17 : x < u
L5349
Hypothesis H19 : SNo (z * y)
L5350
Hypothesis H20 : SNo (x * w)
L5351
Hypothesis H21 : SNo (z * w)
L5352
Hypothesis H22 : SNo (u * y)
L5353
Hypothesis H23 : SNo (x * v)
L5354
Hypothesis H24 : SNo (u * v)
L5355
Hypothesis H25 : SNo (z * v)
L5356
Hypothesis H26 : SNo (u * w)
L5357
Hypothesis H27 : v < w
L5358
Theorem. (Conj_mul_SNo_prop_1__28__18)
(βˆ€x2 : set, x2 ∈ SNoR x β†’ (x * w + x2 * v) < x2 * w + x * v) β†’ (z * y + x * w + u * v) < u * y + x * v + z * w
In Proofgold the corresponding term root is 8e1737... and proposition id is 185602...
Proof:
Proof not loaded.
End of Section Conj_mul_SNo_prop_1__28__18
Beginning of Section Conj_mul_SNo_prop_1__29__15
L5364
Variable x : set
(*** Conj_mul_SNo_prop_1__29__15 TMP7jVhEQZ4nc3TLqJTrjcumTZvEYEZmrHR bounty of about 25 bars ***)
L5365
Variable y : set
L5366
Variable z : set
L5367
Variable w : set
L5368
Variable u : set
L5369
Variable v : set
L5370
Hypothesis H0 : SNo x
L5371
Hypothesis H1 : (βˆ€x2 : set, x2 ∈ SNoS_ (SNoLev x) β†’ (βˆ€y2 : set, SNo y2 β†’ (βˆ€P : prop, (SNo (x2 * y2) β†’ (βˆ€z2 : set, z2 ∈ SNoL x2 β†’ (βˆ€w2 : set, w2 ∈ SNoL y2 β†’ (z2 * y2 + x2 * w2) < x2 * y2 + z2 * w2)) β†’ (βˆ€z2 : set, z2 ∈ SNoR x2 β†’ (βˆ€w2 : set, w2 ∈ SNoR y2 β†’ (z2 * y2 + x2 * w2) < x2 * y2 + z2 * w2)) β†’ (βˆ€z2 : set, z2 ∈ SNoL x2 β†’ (βˆ€w2 : set, w2 ∈ SNoR y2 β†’ (x2 * y2 + z2 * w2) < z2 * y2 + x2 * w2)) β†’ (βˆ€z2 : set, z2 ∈ SNoR x2 β†’ (βˆ€w2 : set, w2 ∈ SNoL y2 β†’ (x2 * y2 + z2 * w2) < z2 * y2 + x2 * w2)) β†’ P) β†’ P)))
L5372
Hypothesis H2 : SNo y
L5373
Hypothesis H3 : (βˆ€x2 : set, x2 ∈ SNoS_ (SNoLev y) β†’ (βˆ€P : prop, (SNo (x * x2) β†’ (βˆ€y2 : set, y2 ∈ SNoL x β†’ (βˆ€z2 : set, z2 ∈ SNoL x2 β†’ (y2 * x2 + x * z2) < x * x2 + y2 * z2)) β†’ (βˆ€y2 : set, y2 ∈ SNoR x β†’ (βˆ€z2 : set, z2 ∈ SNoR x2 β†’ (y2 * x2 + x * z2) < x * x2 + y2 * z2)) β†’ (βˆ€y2 : set, y2 ∈ SNoL x β†’ (βˆ€z2 : set, z2 ∈ SNoR x2 β†’ (x * x2 + y2 * z2) < y2 * x2 + x * z2)) β†’ (βˆ€y2 : set, y2 ∈ SNoR x β†’ (βˆ€z2 : set, z2 ∈ SNoL x2 β†’ (x * x2 + y2 * z2) < y2 * x2 + x * z2)) β†’ P) β†’ P))
L5374
Hypothesis H4 : (βˆ€x2 : set, x2 ∈ SNoR x β†’ (βˆ€y2 : set, SNo y2 β†’ SNo (x2 * y2)))
L5375
Hypothesis H5 : (βˆ€x2 : set, x2 ∈ SNoR x β†’ SNo (x2 * y))
L5376
Hypothesis H6 : z ∈ SNoR x
L5377
Hypothesis H7 : w ∈ SNoR y
L5378
Hypothesis H8 : SNo z
L5379
Hypothesis H9 : SNoLev z ∈ SNoLev x
L5380
Hypothesis H10 : x < z
L5381
Hypothesis H11 : SNo w
L5382
Hypothesis H12 : SNoLev w ∈ SNoLev y
L5383
Hypothesis H13 : y < w
L5384
Hypothesis H14 : u ∈ SNoR x
L5385
Hypothesis H16 : SNo u
L5386
Hypothesis H17 : SNoLev u ∈ SNoLev x
L5387
Hypothesis H18 : x < u
L5388
Hypothesis H19 : SNo v
L5389
Hypothesis H20 : v < y
L5390
Hypothesis H21 : SNo (z * y)
L5391
Hypothesis H22 : SNo (x * w)
L5392
Hypothesis H23 : SNo (z * w)
L5393
Hypothesis H24 : SNo (u * y)
L5394
Hypothesis H25 : SNo (x * v)
L5395
Hypothesis H26 : SNo (u * v)
L5396
Hypothesis H27 : SNo (z * v)
L5397
Hypothesis H28 : SNo (u * w)
L5398
Theorem. (Conj_mul_SNo_prop_1__29__15)
v < w β†’ (z * y + x * w + u * v) < u * y + x * v + z * w
In Proofgold the corresponding term root is 658a3b... and proposition id is 090919...
Proof:
Proof not loaded.
End of Section Conj_mul_SNo_prop_1__29__15
Beginning of Section Conj_mul_SNo_prop_1__29__22
L5404
Variable x : set
(*** Conj_mul_SNo_prop_1__29__22 TMT7sZTYvkgjFvy8zYquTee2QQeeCXt12aQ bounty of about 25 bars ***)
L5405
Variable y : set
L5406
Variable z : set
L5407
Variable w : set
L5408
Variable u : set
L5409
Variable v : set
L5410
Hypothesis H0 : SNo x
L5411
Hypothesis H1 : (βˆ€x2 : set, x2 ∈ SNoS_ (SNoLev x) β†’ (βˆ€y2 : set, SNo y2 β†’ (βˆ€P : prop, (SNo (x2 * y2) β†’ (βˆ€z2 : set, z2 ∈ SNoL x2 β†’ (βˆ€w2 : set, w2 ∈ SNoL y2 β†’ (z2 * y2 + x2 * w2) < x2 * y2 + z2 * w2)) β†’ (βˆ€z2 : set, z2 ∈ SNoR x2 β†’ (βˆ€w2 : set, w2 ∈ SNoR y2 β†’ (z2 * y2 + x2 * w2) < x2 * y2 + z2 * w2)) β†’ (βˆ€z2 : set, z2 ∈ SNoL x2 β†’ (βˆ€w2 : set, w2 ∈ SNoR y2 β†’ (x2 * y2 + z2 * w2) < z2 * y2 + x2 * w2)) β†’ (βˆ€z2 : set, z2 ∈ SNoR x2 β†’ (βˆ€w2 : set, w2 ∈ SNoL y2 β†’ (x2 * y2 + z2 * w2) < z2 * y2 + x2 * w2)) β†’ P) β†’ P)))
L5412
Hypothesis H2 : SNo y
L5413
Hypothesis H3 : (βˆ€x2 : set, x2 ∈ SNoS_ (SNoLev y) β†’ (βˆ€P : prop, (SNo (x * x2) β†’ (βˆ€y2 : set, y2 ∈ SNoL x β†’ (βˆ€z2 : set, z2 ∈ SNoL x2 β†’ (y2 * x2 + x * z2) < x * x2 + y2 * z2)) β†’ (βˆ€y2 : set, y2 ∈ SNoR x β†’ (βˆ€z2 : set, z2 ∈ SNoR x2 β†’ (y2 * x2 + x * z2) < x * x2 + y2 * z2)) β†’ (βˆ€y2 : set, y2 ∈ SNoL x β†’ (βˆ€z2 : set, z2 ∈ SNoR x2 β†’ (x * x2 + y2 * z2) < y2 * x2 + x * z2)) β†’ (βˆ€y2 : set, y2 ∈ SNoR x β†’ (βˆ€z2 : set, z2 ∈ SNoL x2 β†’ (x * x2 + y2 * z2) < y2 * x2 + x * z2)) β†’ P) β†’ P))
L5414
Hypothesis H4 : (βˆ€x2 : set, x2 ∈ SNoR x β†’ (βˆ€y2 : set, SNo y2 β†’ SNo (x2 * y2)))
L5415
Hypothesis H5 : (βˆ€x2 : set, x2 ∈ SNoR x β†’ SNo (x2 * y))
L5416
Hypothesis H6 : z ∈ SNoR x
L5417
Hypothesis H7 : w ∈ SNoR y
L5418
Hypothesis H8 : SNo z
L5419
Hypothesis H9 : SNoLev z ∈ SNoLev x
L5420
Hypothesis H10 : x < z
L5421
Hypothesis H11 : SNo w
L5422
Hypothesis H12 : SNoLev w ∈ SNoLev y
L5423
Hypothesis H13 : y < w
L5424
Hypothesis H14 : u ∈ SNoR x
L5425
Hypothesis H15 : v ∈ SNoL y
L5426
Hypothesis H16 : SNo u
L5427
Hypothesis H17 : SNoLev u ∈ SNoLev x
L5428
Hypothesis H18 : x < u
L5429
Hypothesis H19 : SNo v
L5430
Hypothesis H20 : v < y
L5431
Hypothesis H21 : SNo (z * y)
L5432
Hypothesis H23 : SNo (z * w)
L5433
Hypothesis H24 : SNo (u * y)
L5434
Hypothesis H25 : SNo (x * v)
L5435
Hypothesis H26 : SNo (u * v)
L5436
Hypothesis H27 : SNo (z * v)
L5437
Hypothesis H28 : SNo (u * w)
L5438
Theorem. (Conj_mul_SNo_prop_1__29__22)
v < w β†’ (z * y + x * w + u * v) < u * y + x * v + z * w
In Proofgold the corresponding term root is 8a0723... and proposition id is 720650...
Proof:
Proof not loaded.
End of Section Conj_mul_SNo_prop_1__29__22
Beginning of Section Conj_mul_SNo_prop_1__29__23
L5444
Variable x : set
(*** Conj_mul_SNo_prop_1__29__23 TMLW5dUF6k5Jxm2JD1o9k6jLDsdAJEuJhYz bounty of about 25 bars ***)
L5445
Variable y : set
L5446
Variable z : set
L5447
Variable w : set
L5448
Variable u : set
L5449
Variable v : set
L5450
Hypothesis H0 : SNo x
L5451
Hypothesis H1 : (βˆ€x2 : set, x2 ∈ SNoS_ (SNoLev x) β†’ (βˆ€y2 : set, SNo y2 β†’ (βˆ€P : prop, (SNo (x2 * y2) β†’ (βˆ€z2 : set, z2 ∈ SNoL x2 β†’ (βˆ€w2 : set, w2 ∈ SNoL y2 β†’ (z2 * y2 + x2 * w2) < x2 * y2 + z2 * w2)) β†’ (βˆ€z2 : set, z2 ∈ SNoR x2 β†’ (βˆ€w2 : set, w2 ∈ SNoR y2 β†’ (z2 * y2 + x2 * w2) < x2 * y2 + z2 * w2)) β†’ (βˆ€z2 : set, z2 ∈ SNoL x2 β†’ (βˆ€w2 : set, w2 ∈ SNoR y2 β†’ (x2 * y2 + z2 * w2) < z2 * y2 + x2 * w2)) β†’ (βˆ€z2 : set, z2 ∈ SNoR x2 β†’ (βˆ€w2 : set, w2 ∈ SNoL y2 β†’ (x2 * y2 + z2 * w2) < z2 * y2 + x2 * w2)) β†’ P) β†’ P)))
L5452
Hypothesis H2 : SNo y
L5453
Hypothesis H3 : (βˆ€x2 : set, x2 ∈ SNoS_ (SNoLev y) β†’ (βˆ€P : prop, (SNo (x * x2) β†’ (βˆ€y2 : set, y2 ∈ SNoL x β†’ (βˆ€z2 : set, z2 ∈ SNoL x2 β†’ (y2 * x2 + x * z2) < x * x2 + y2 * z2)) β†’ (βˆ€y2 : set, y2 ∈ SNoR x β†’ (βˆ€z2 : set, z2 ∈ SNoR x2 β†’ (y2 * x2 + x * z2) < x * x2 + y2 * z2)) β†’ (βˆ€y2 : set, y2 ∈ SNoL x β†’ (βˆ€z2 : set, z2 ∈ SNoR x2 β†’ (x * x2 + y2 * z2) < y2 * x2 + x * z2)) β†’ (βˆ€y2 : set, y2 ∈ SNoR x β†’ (βˆ€z2 : set, z2 ∈ SNoL x2 β†’ (x * x2 + y2 * z2) < y2 * x2 + x * z2)) β†’ P) β†’ P))
L5454
Hypothesis H4 : (βˆ€x2 : set, x2 ∈ SNoR x β†’ (βˆ€y2 : set, SNo y2 β†’ SNo (x2 * y2)))
L5455
Hypothesis H5 : (βˆ€x2 : set, x2 ∈ SNoR x β†’ SNo (x2 * y))
L5456
Hypothesis H6 : z ∈ SNoR x
L5457
Hypothesis H7 : w ∈ SNoR y
L5458
Hypothesis H8 : SNo z
L5459
Hypothesis H9 : SNoLev z ∈ SNoLev x
L5460
Hypothesis H10 : x < z
L5461
Hypothesis H11 : SNo w
L5462
Hypothesis H12 : SNoLev w ∈ SNoLev y
L5463
Hypothesis H13 : y < w
L5464
Hypothesis H14 : u ∈ SNoR x
L5465
Hypothesis H15 : v ∈ SNoL y
L5466
Hypothesis H16 : SNo u
L5467
Hypothesis H17 : SNoLev u ∈ SNoLev x
L5468
Hypothesis H18 : x < u
L5469
Hypothesis H19 : SNo v
L5470
Hypothesis H20 : v < y
L5471
Hypothesis H21 : SNo (z * y)
L5472
Hypothesis H22 : SNo (x * w)
L5473
Hypothesis H24 : SNo (u * y)
L5474
Hypothesis H25 : SNo (x * v)
L5475
Hypothesis H26 : SNo (u * v)
L5476
Hypothesis H27 : SNo (z * v)
L5477
Hypothesis H28 : SNo (u * w)
L5478
Theorem. (Conj_mul_SNo_prop_1__29__23)
v < w β†’ (z * y + x * w + u * v) < u * y + x * v + z * w
In Proofgold the corresponding term root is 71cb11... and proposition id is b9a459...
Proof:
Proof not loaded.
End of Section Conj_mul_SNo_prop_1__29__23
Beginning of Section Conj_mul_SNo_prop_1__29__26
L5484
Variable x : set
(*** Conj_mul_SNo_prop_1__29__26 TMFwRocbYCoAMZt9dxHUntt2MNBM95icjAr bounty of about 25 bars ***)
L5485
Variable y : set
L5486
Variable z : set
L5487
Variable w : set
L5488
Variable u : set
L5489
Variable v : set
L5490
Hypothesis H0 : SNo x
L5491
Hypothesis H1 : (βˆ€x2 : set, x2 ∈ SNoS_ (SNoLev x) β†’ (βˆ€y2 : set, SNo y2 β†’ (βˆ€P : prop, (SNo (x2 * y2) β†’ (βˆ€z2 : set, z2 ∈ SNoL x2 β†’ (βˆ€w2 : set, w2 ∈ SNoL y2 β†’ (z2 * y2 + x2 * w2) < x2 * y2 + z2 * w2)) β†’ (βˆ€z2 : set, z2 ∈ SNoR x2 β†’ (βˆ€w2 : set, w2 ∈ SNoR y2 β†’ (z2 * y2 + x2 * w2) < x2 * y2 + z2 * w2)) β†’ (βˆ€z2 : set, z2 ∈ SNoL x2 β†’ (βˆ€w2 : set, w2 ∈ SNoR y2 β†’ (x2 * y2 + z2 * w2) < z2 * y2 + x2 * w2)) β†’ (βˆ€z2 : set, z2 ∈ SNoR x2 β†’ (βˆ€w2 : set, w2 ∈ SNoL y2 β†’ (x2 * y2 + z2 * w2) < z2 * y2 + x2 * w2)) β†’ P) β†’ P)))
L5492
Hypothesis H2 : SNo y
L5493
Hypothesis H3 : (βˆ€x2 : set, x2 ∈ SNoS_ (SNoLev y) β†’ (βˆ€P : prop, (SNo (x * x2) β†’ (βˆ€y2 : set, y2 ∈ SNoL x β†’ (βˆ€z2 : set, z2 ∈ SNoL x2 β†’ (y2 * x2 + x * z2) < x * x2 + y2 * z2)) β†’ (βˆ€y2 : set, y2 ∈ SNoR x β†’ (βˆ€z2 : set, z2 ∈ SNoR x2 β†’ (y2 * x2 + x * z2) < x * x2 + y2 * z2)) β†’ (βˆ€y2 : set, y2 ∈ SNoL x β†’ (βˆ€z2 : set, z2 ∈ SNoR x2 β†’ (x * x2 + y2 * z2) < y2 * x2 + x * z2)) β†’ (βˆ€y2 : set, y2 ∈ SNoR x β†’ (βˆ€z2 : set, z2 ∈ SNoL x2 β†’ (x * x2 + y2 * z2) < y2 * x2 + x * z2)) β†’ P) β†’ P))
L5494
Hypothesis H4 : (βˆ€x2 : set, x2 ∈ SNoR x β†’ (βˆ€y2 : set, SNo y2 β†’ SNo (x2 * y2)))
L5495
Hypothesis H5 : (βˆ€x2 : set, x2 ∈ SNoR x β†’ SNo (x2 * y))
L5496
Hypothesis H6 : z ∈ SNoR x
L5497
Hypothesis H7 : w ∈ SNoR y
L5498
Hypothesis H8 : SNo z
L5499
Hypothesis H9 : SNoLev z ∈ SNoLev x
L5500
Hypothesis H10 : x < z
L5501
Hypothesis H11 : SNo w
L5502
Hypothesis H12 : SNoLev w ∈ SNoLev y
L5503
Hypothesis H13 : y < w
L5504
Hypothesis H14 : u ∈ SNoR x
L5505
Hypothesis H15 : v ∈ SNoL y
L5506
Hypothesis H16 : SNo u
L5507
Hypothesis H17 : SNoLev u ∈ SNoLev x
L5508
Hypothesis H18 : x < u
L5509
Hypothesis H19 : SNo v
L5510
Hypothesis H20 : v < y
L5511
Hypothesis H21 : SNo (z * y)
L5512
Hypothesis H22 : SNo (x * w)
L5513
Hypothesis H23 : SNo (z * w)
L5514
Hypothesis H24 : SNo (u * y)
L5515
Hypothesis H25 : SNo (x * v)
L5516
Hypothesis H27 : SNo (z * v)
L5517
Hypothesis H28 : SNo (u * w)
L5518
Theorem. (Conj_mul_SNo_prop_1__29__26)
v < w β†’ (z * y + x * w + u * v) < u * y + x * v + z * w
In Proofgold the corresponding term root is 253de8... and proposition id is 2795be...
Proof:
Proof not loaded.
End of Section Conj_mul_SNo_prop_1__29__26
Beginning of Section Conj_mul_SNo_prop_1__30__3
L5524
Variable x : set
(*** Conj_mul_SNo_prop_1__30__3 TMJWwbsYzbBenTSXEqvuscAGr23EDPZAebU bounty of about 25 bars ***)
L5525
Variable y : set
L5526
Variable z : set
L5527
Variable w : set
L5528
Variable u : set
L5529
Variable v : set
L5530
Hypothesis H0 : SNo x
L5531
Hypothesis H1 : (βˆ€x2 : set, x2 ∈ SNoS_ (SNoLev x) β†’ (βˆ€y2 : set, SNo y2 β†’ (βˆ€P : prop, (SNo (x2 * y2) β†’ (βˆ€z2 : set, z2 ∈ SNoL x2 β†’ (βˆ€w2 : set, w2 ∈ SNoL y2 β†’ (z2 * y2 + x2 * w2) < x2 * y2 + z2 * w2)) β†’ (βˆ€z2 : set, z2 ∈ SNoR x2 β†’ (βˆ€w2 : set, w2 ∈ SNoR y2 β†’ (z2 * y2 + x2 * w2) < x2 * y2 + z2 * w2)) β†’ (βˆ€z2 : set, z2 ∈ SNoL x2 β†’ (βˆ€w2 : set, w2 ∈ SNoR y2 β†’ (x2 * y2 + z2 * w2) < z2 * y2 + x2 * w2)) β†’ (βˆ€z2 : set, z2 ∈ SNoR x2 β†’ (βˆ€w2 : set, w2 ∈ SNoL y2 β†’ (x2 * y2 + z2 * w2) < z2 * y2 + x2 * w2)) β†’ P) β†’ P)))
L5532
Hypothesis H2 : SNo y
L5533
Hypothesis H4 : SNo (z * y)
L5534
Hypothesis H5 : SNo (w * y)
L5535
Hypothesis H6 : u ∈ SNoR y
L5536
Hypothesis H7 : SNo (z * u)
L5537
Hypothesis H8 : SNo (w * u)
L5538
Hypothesis H9 : v ∈ SNoR w
L5539
Hypothesis H10 : SNo (v * u)
L5540
Hypothesis H11 : SNo (v * y)
L5541
Hypothesis H12 : (z * y + v * u) < z * u + v * y
L5542
Theorem. (Conj_mul_SNo_prop_1__30__3)
(w * u + v * y) < w * y + v * u β†’ (z * y + w * u) < w * y + z * u
In Proofgold the corresponding term root is ed96dd... and proposition id is 216ddf...
Proof:
Proof not loaded.
End of Section Conj_mul_SNo_prop_1__30__3
Beginning of Section Conj_mul_SNo_prop_1__31__4
L5548
Variable x : set
(*** Conj_mul_SNo_prop_1__31__4 TMHqjm8xstCfcJ1ximjvdU7CcjiMih21cm6 bounty of about 25 bars ***)
L5549
Variable y : set
L5550
Variable z : set
L5551
Variable w : set
L5552
Variable u : set
L5553
Variable v : set
L5554
Hypothesis H0 : SNo x
L5555
Hypothesis H1 : (βˆ€x2 : set, x2 ∈ SNoS_ (SNoLev x) β†’ (βˆ€y2 : set, SNo y2 β†’ (βˆ€P : prop, (SNo (x2 * y2) β†’ (βˆ€z2 : set, z2 ∈ SNoL x2 β†’ (βˆ€w2 : set, w2 ∈ SNoL y2 β†’ (z2 * y2 + x2 * w2) < x2 * y2 + z2 * w2)) β†’ (βˆ€z2 : set, z2 ∈ SNoR x2 β†’ (βˆ€w2 : set, w2 ∈ SNoR y2 β†’ (z2 * y2 + x2 * w2) < x2 * y2 + z2 * w2)) β†’ (βˆ€z2 : set, z2 ∈ SNoL x2 β†’ (βˆ€w2 : set, w2 ∈ SNoR y2 β†’ (x2 * y2 + z2 * w2) < z2 * y2 + x2 * w2)) β†’ (βˆ€z2 : set, z2 ∈ SNoR x2 β†’ (βˆ€w2 : set, w2 ∈ SNoL y2 β†’ (x2 * y2 + z2 * w2) < z2 * y2 + x2 * w2)) β†’ P) β†’ P)))
L5556
Hypothesis H2 : SNo y
L5557
Hypothesis H3 : z ∈ SNoR x
L5558
Hypothesis H5 : SNo (z * y)
L5559
Hypothesis H6 : SNo (w * y)
L5560
Hypothesis H7 : u ∈ SNoR y
L5561
Hypothesis H8 : SNo (z * u)
L5562
Hypothesis H9 : SNo (w * u)
L5563
Hypothesis H10 : v ∈ SNoL z
L5564
Hypothesis H11 : v ∈ SNoR w
L5565
Hypothesis H12 : SNo (v * u)
L5566
Hypothesis H13 : SNo (v * y)
L5567
Theorem. (Conj_mul_SNo_prop_1__31__4)
(z * y + v * u) < z * u + v * y β†’ (z * y + w * u) < w * y + z * u
In Proofgold the corresponding term root is 60546e... and proposition id is 45ce9d...
Proof:
Proof not loaded.
End of Section Conj_mul_SNo_prop_1__31__4
Beginning of Section Conj_mul_SNo_prop_1__32__6
L5573
Variable x : set
(*** Conj_mul_SNo_prop_1__32__6 TMddyo9bXU62qgCsjWv5BUADyLwjeaqdKLP bounty of about 25 bars ***)
L5574
Variable y : set
L5575
Variable z : set
L5576
Variable w : set
L5577
Variable u : set
L5578
Variable v : set
L5579
Hypothesis H0 : SNo x
L5580
Hypothesis H1 : (βˆ€x2 : set, x2 ∈ SNoS_ (SNoLev x) β†’ (βˆ€y2 : set, SNo y2 β†’ (βˆ€P : prop, (SNo (x2 * y2) β†’ (βˆ€z2 : set, z2 ∈ SNoL x2 β†’ (βˆ€w2 : set, w2 ∈ SNoL y2 β†’ (z2 * y2 + x2 * w2) < x2 * y2 + z2 * w2)) β†’ (βˆ€z2 : set, z2 ∈ SNoR x2 β†’ (βˆ€w2 : set, w2 ∈ SNoR y2 β†’ (z2 * y2 + x2 * w2) < x2 * y2 + z2 * w2)) β†’ (βˆ€z2 : set, z2 ∈ SNoL x2 β†’ (βˆ€w2 : set, w2 ∈ SNoR y2 β†’ (x2 * y2 + z2 * w2) < z2 * y2 + x2 * w2)) β†’ (βˆ€z2 : set, z2 ∈ SNoR x2 β†’ (βˆ€w2 : set, w2 ∈ SNoL y2 β†’ (x2 * y2 + z2 * w2) < z2 * y2 + x2 * w2)) β†’ P) β†’ P)))
L5581
Hypothesis H2 : SNo y
L5582
Hypothesis H3 : z ∈ SNoR x
L5583
Hypothesis H4 : w ∈ SNoL x
L5584
Hypothesis H5 : SNo (z * y)
L5585
Hypothesis H7 : u ∈ SNoR y
L5586
Hypothesis H8 : SNo (z * u)
L5587
Hypothesis H9 : SNo (w * u)
L5588
Hypothesis H10 : v ∈ SNoL z
L5589
Hypothesis H11 : v ∈ SNoR w
L5590
Hypothesis H12 : v ∈ SNoS_ (SNoLev x)
L5591
Hypothesis H13 : SNo (v * u)
L5592
Theorem. (Conj_mul_SNo_prop_1__32__6)
SNo (v * y) β†’ (z * y + w * u) < w * y + z * u
In Proofgold the corresponding term root is 400923... and proposition id is 79b1e4...
Proof:
Proof not loaded.
End of Section Conj_mul_SNo_prop_1__32__6
Beginning of Section Conj_mul_SNo_prop_1__32__7
L5598
Variable x : set
(*** Conj_mul_SNo_prop_1__32__7 TMQpdtERREELk5UuA8DtZjY1SoCHXbGkWE8 bounty of about 25 bars ***)
L5599
Variable y : set
L5600
Variable z : set
L5601
Variable w : set
L5602
Variable u : set
L5603
Variable v : set
L5604
Hypothesis H0 : SNo x
L5605
Hypothesis H1 : (βˆ€x2 : set, x2 ∈ SNoS_ (SNoLev x) β†’ (βˆ€y2 : set, SNo y2 β†’ (βˆ€P : prop, (SNo (x2 * y2) β†’ (βˆ€z2 : set, z2 ∈ SNoL x2 β†’ (βˆ€w2 : set, w2 ∈ SNoL y2 β†’ (z2 * y2 + x2 * w2) < x2 * y2 + z2 * w2)) β†’ (βˆ€z2 : set, z2 ∈ SNoR x2 β†’ (βˆ€w2 : set, w2 ∈ SNoR y2 β†’ (z2 * y2 + x2 * w2) < x2 * y2 + z2 * w2)) β†’ (βˆ€z2 : set, z2 ∈ SNoL x2 β†’ (βˆ€w2 : set, w2 ∈ SNoR y2 β†’ (x2 * y2 + z2 * w2) < z2 * y2 + x2 * w2)) β†’ (βˆ€z2 : set, z2 ∈ SNoR x2 β†’ (βˆ€w2 : set, w2 ∈ SNoL y2 β†’ (x2 * y2 + z2 * w2) < z2 * y2 + x2 * w2)) β†’ P) β†’ P)))
L5606
Hypothesis H2 : SNo y
L5607
Hypothesis H3 : z ∈ SNoR x
L5608
Hypothesis H4 : w ∈ SNoL x
L5609
Hypothesis H5 : SNo (z * y)
L5610
Hypothesis H6 : SNo (w * y)
L5611
Hypothesis H8 : SNo (z * u)
L5612
Hypothesis H9 : SNo (w * u)
L5613
Hypothesis H10 : v ∈ SNoL z
L5614
Hypothesis H11 : v ∈ SNoR w
L5615
Hypothesis H12 : v ∈ SNoS_ (SNoLev x)
L5616
Hypothesis H13 : SNo (v * u)
L5617
Theorem. (Conj_mul_SNo_prop_1__32__7)
SNo (v * y) β†’ (z * y + w * u) < w * y + z * u
In Proofgold the corresponding term root is 010947... and proposition id is 0c3fa1...
Proof:
Proof not loaded.
End of Section Conj_mul_SNo_prop_1__32__7
Beginning of Section Conj_mul_SNo_prop_1__33__2
L5623
Variable x : set
(*** Conj_mul_SNo_prop_1__33__2 TMPXPKmUfM2JAfJKHmbCkCsEUrdb69xjKe3 bounty of about 25 bars ***)
L5624
Variable y : set
L5625
Variable z : set
L5626
Variable w : set
L5627
Variable u : set
L5628
Variable v : set
L5629
Hypothesis H0 : SNo x
L5630
Hypothesis H1 : (βˆ€x2 : set, x2 ∈ SNoS_ (SNoLev x) β†’ (βˆ€y2 : set, SNo y2 β†’ (βˆ€P : prop, (SNo (x2 * y2) β†’ (βˆ€z2 : set, z2 ∈ SNoL x2 β†’ (βˆ€w2 : set, w2 ∈ SNoL y2 β†’ (z2 * y2 + x2 * w2) < x2 * y2 + z2 * w2)) β†’ (βˆ€z2 : set, z2 ∈ SNoR x2 β†’ (βˆ€w2 : set, w2 ∈ SNoR y2 β†’ (z2 * y2 + x2 * w2) < x2 * y2 + z2 * w2)) β†’ (βˆ€z2 : set, z2 ∈ SNoL x2 β†’ (βˆ€w2 : set, w2 ∈ SNoR y2 β†’ (x2 * y2 + z2 * w2) < z2 * y2 + x2 * w2)) β†’ (βˆ€z2 : set, z2 ∈ SNoR x2 β†’ (βˆ€w2 : set, w2 ∈ SNoL y2 β†’ (x2 * y2 + z2 * w2) < z2 * y2 + x2 * w2)) β†’ P) β†’ P)))
L5631
Hypothesis H3 : z ∈ SNoR x
L5632
Hypothesis H4 : w ∈ SNoL x
L5633
Hypothesis H5 : SNo (z * y)
L5634
Hypothesis H6 : SNo (w * y)
L5635
Hypothesis H7 : u ∈ SNoR y
L5636
Hypothesis H8 : SNo u
L5637
Hypothesis H9 : SNo (z * u)
L5638
Hypothesis H10 : SNo (w * u)
L5639
Hypothesis H11 : v ∈ SNoL z
L5640
Hypothesis H12 : v ∈ SNoR w
L5641
Hypothesis H13 : v ∈ SNoS_ (SNoLev x)
L5642
Theorem. (Conj_mul_SNo_prop_1__33__2)
SNo (v * u) β†’ (z * y + w * u) < w * y + z * u
In Proofgold the corresponding term root is d5ffd9... and proposition id is dc1b4b...
Proof:
Proof not loaded.
End of Section Conj_mul_SNo_prop_1__33__2
Beginning of Section Conj_mul_SNo_prop_1__33__6
L5648
Variable x : set
(*** Conj_mul_SNo_prop_1__33__6 TMKZMkNfJcpWu4cxgoRT4E3Goh6BQLaVDyn bounty of about 25 bars ***)
L5649
Variable y : set
L5650
Variable z : set
L5651
Variable w : set
L5652
Variable u : set
L5653
Variable v : set
L5654
Hypothesis H0 : SNo x
L5655
Hypothesis H1 : (βˆ€x2 : set, x2 ∈ SNoS_ (SNoLev x) β†’ (βˆ€y2 : set, SNo y2 β†’ (βˆ€P : prop, (SNo (x2 * y2) β†’ (βˆ€z2 : set, z2 ∈ SNoL x2 β†’ (βˆ€w2 : set, w2 ∈ SNoL y2 β†’ (z2 * y2 + x2 * w2) < x2 * y2 + z2 * w2)) β†’ (βˆ€z2 : set, z2 ∈ SNoR x2 β†’ (βˆ€w2 : set, w2 ∈ SNoR y2 β†’ (z2 * y2 + x2 * w2) < x2 * y2 + z2 * w2)) β†’ (βˆ€z2 : set, z2 ∈ SNoL x2 β†’ (βˆ€w2 : set, w2 ∈ SNoR y2 β†’ (x2 * y2 + z2 * w2) < z2 * y2 + x2 * w2)) β†’ (βˆ€z2 : set, z2 ∈ SNoR x2 β†’ (βˆ€w2 : set, w2 ∈ SNoL y2 β†’ (x2 * y2 + z2 * w2) < z2 * y2 + x2 * w2)) β†’ P) β†’ P)))
L5656
Hypothesis H2 : SNo y
L5657
Hypothesis H3 : z ∈ SNoR x
L5658
Hypothesis H4 : w ∈ SNoL x
L5659
Hypothesis H5 : SNo (z * y)
L5660
Hypothesis H7 : u ∈ SNoR y
L5661
Hypothesis H8 : SNo u
L5662
Hypothesis H9 : SNo (z * u)
L5663
Hypothesis H10 : SNo (w * u)
L5664
Hypothesis H11 : v ∈ SNoL z
L5665
Hypothesis H12 : v ∈ SNoR w
L5666
Hypothesis H13 : v ∈ SNoS_ (SNoLev x)
L5667
Theorem. (Conj_mul_SNo_prop_1__33__6)
SNo (v * u) β†’ (z * y + w * u) < w * y + z * u
In Proofgold the corresponding term root is 15cfeb... and proposition id is 276247...
Proof:
Proof not loaded.
End of Section Conj_mul_SNo_prop_1__33__6
Beginning of Section Conj_mul_SNo_prop_1__35__2
L5673
Variable x : set
(*** Conj_mul_SNo_prop_1__35__2 TMQLLtZctYKRgFR4zo262tWJaGEkWa3zUAE bounty of about 25 bars ***)
L5674
Variable y : set
L5675
Variable z : set
L5676
Variable w : set
L5677
Variable u : set
L5678
Variable v : set
L5679
Hypothesis H0 : SNo x
L5680
Hypothesis H1 : (βˆ€x2 : set, x2 ∈ SNoS_ (SNoLev x) β†’ (βˆ€y2 : set, SNo y2 β†’ (βˆ€P : prop, (SNo (x2 * y2) β†’ (βˆ€z2 : set, z2 ∈ SNoL x2 β†’ (βˆ€w2 : set, w2 ∈ SNoL y2 β†’ (z2 * y2 + x2 * w2) < x2 * y2 + z2 * w2)) β†’ (βˆ€z2 : set, z2 ∈ SNoR x2 β†’ (βˆ€w2 : set, w2 ∈ SNoR y2 β†’ (z2 * y2 + x2 * w2) < x2 * y2 + z2 * w2)) β†’ (βˆ€z2 : set, z2 ∈ SNoL x2 β†’ (βˆ€w2 : set, w2 ∈ SNoR y2 β†’ (x2 * y2 + z2 * w2) < z2 * y2 + x2 * w2)) β†’ (βˆ€z2 : set, z2 ∈ SNoR x2 β†’ (βˆ€w2 : set, w2 ∈ SNoL y2 β†’ (x2 * y2 + z2 * w2) < z2 * y2 + x2 * w2)) β†’ P) β†’ P)))
L5681
Hypothesis H3 : z ∈ SNoR x
L5682
Hypothesis H4 : SNoLev z ∈ SNoLev x
L5683
Hypothesis H5 : w ∈ SNoL x
L5684
Hypothesis H6 : SNo (z * y)
L5685
Hypothesis H7 : SNo (w * y)
L5686
Hypothesis H8 : u ∈ SNoR y
L5687
Hypothesis H9 : SNo u
L5688
Hypothesis H10 : SNo (z * u)
L5689
Hypothesis H11 : SNo (w * u)
L5690
Hypothesis H12 : v ∈ SNoL z
L5691
Hypothesis H13 : v ∈ SNoR w
L5692
Hypothesis H14 : SNo v
L5693
Hypothesis H15 : SNoLev v ∈ SNoLev z
L5694
Theorem. (Conj_mul_SNo_prop_1__35__2)
SNoLev v ∈ SNoLev x β†’ (z * y + w * u) < w * y + z * u
In Proofgold the corresponding term root is dde1d6... and proposition id is 626658...
Proof:
Proof not loaded.
End of Section Conj_mul_SNo_prop_1__35__2
Beginning of Section Conj_mul_SNo_prop_1__39__15
L5700
Variable x : set
(*** Conj_mul_SNo_prop_1__39__15 TMaE15KFJJNTrPoiDrJJ7AxFnuXy8vjtjNR bounty of about 25 bars ***)
L5701
Variable y : set
L5702
Variable z : set
L5703
Variable w : set
L5704
Variable u : set
L5705
Variable v : set
L5706
Variable x2 : set
L5707
Hypothesis H0 : SNo y
L5708
Hypothesis H1 : (βˆ€y2 : set, y2 ∈ SNoS_ (SNoLev y) β†’ (βˆ€P : prop, (SNo (x * y2) β†’ (βˆ€z2 : set, z2 ∈ SNoL x β†’ (βˆ€w2 : set, w2 ∈ SNoL y2 β†’ (z2 * y2 + x * w2) < x * y2 + z2 * w2)) β†’ (βˆ€z2 : set, z2 ∈ SNoR x β†’ (βˆ€w2 : set, w2 ∈ SNoR y2 β†’ (z2 * y2 + x * w2) < x * y2 + z2 * w2)) β†’ (βˆ€z2 : set, z2 ∈ SNoL x β†’ (βˆ€w2 : set, w2 ∈ SNoR y2 β†’ (x * y2 + z2 * w2) < z2 * y2 + x * w2)) β†’ (βˆ€z2 : set, z2 ∈ SNoR x β†’ (βˆ€w2 : set, w2 ∈ SNoL y2 β†’ (x * y2 + z2 * w2) < z2 * y2 + x * w2)) β†’ P) β†’ P))
L5709
Hypothesis H2 : (βˆ€y2 : set, y2 ∈ SNoL x β†’ (βˆ€z2 : set, SNo z2 β†’ SNo (y2 * z2)))
L5710
Hypothesis H3 : (βˆ€y2 : set, y2 ∈ SNoR y β†’ SNo (x * y2))
L5711
Hypothesis H4 : w ∈ SNoR y
L5712
Hypothesis H5 : u ∈ SNoL x
L5713
Hypothesis H6 : v ∈ SNoR y
L5714
Hypothesis H7 : SNo (z * y)
L5715
Hypothesis H8 : SNo (x * w)
L5716
Hypothesis H9 : SNo (z * w)
L5717
Hypothesis H10 : SNo (u * y)
L5718
Hypothesis H11 : SNo (x * v)
L5719
Hypothesis H12 : SNo (u * v)
L5720
Hypothesis H13 : SNo (u * w)
L5721
Hypothesis H14 : (z * y + u * w) < u * y + z * w
L5722
Hypothesis H16 : x2 ∈ SNoL v
L5723
Hypothesis H17 : SNo x2
L5724
Hypothesis H18 : x2 ∈ SNoR y
L5725
Theorem. (Conj_mul_SNo_prop_1__39__15)
(x * w + u * x2) < x * x2 + u * w β†’ (z * y + x * w + u * v) < u * y + x * v + z * w
In Proofgold the corresponding term root is 048517... and proposition id is 726fdb...
Proof:
Proof not loaded.
End of Section Conj_mul_SNo_prop_1__39__15
Beginning of Section Conj_mul_SNo_prop_1__42__16
L5731
Variable x : set
(*** Conj_mul_SNo_prop_1__42__16 TMRzchoronbkspw5vJW1hPicyeoehU3zFLn bounty of about 25 bars ***)
L5732
Variable y : set
L5733
Variable z : set
L5734
Variable w : set
L5735
Variable u : set
L5736
Variable v : set
L5737
Variable x2 : set
L5738
Hypothesis H0 : SNo y
L5739
Hypothesis H1 : (βˆ€y2 : set, y2 ∈ SNoS_ (SNoLev y) β†’ (βˆ€P : prop, (SNo (x * y2) β†’ (βˆ€z2 : set, z2 ∈ SNoL x β†’ (βˆ€w2 : set, w2 ∈ SNoL y2 β†’ (z2 * y2 + x * w2) < x * y2 + z2 * w2)) β†’ (βˆ€z2 : set, z2 ∈ SNoR x β†’ (βˆ€w2 : set, w2 ∈ SNoR y2 β†’ (z2 * y2 + x * w2) < x * y2 + z2 * w2)) β†’ (βˆ€z2 : set, z2 ∈ SNoL x β†’ (βˆ€w2 : set, w2 ∈ SNoR y2 β†’ (x * y2 + z2 * w2) < z2 * y2 + x * w2)) β†’ (βˆ€z2 : set, z2 ∈ SNoR x β†’ (βˆ€w2 : set, w2 ∈ SNoL y2 β†’ (x * y2 + z2 * w2) < z2 * y2 + x * w2)) β†’ P) β†’ P))
L5740
Hypothesis H2 : (βˆ€y2 : set, y2 ∈ SNoR x β†’ (βˆ€z2 : set, SNo z2 β†’ SNo (y2 * z2)))
L5741
Hypothesis H3 : (βˆ€y2 : set, y2 ∈ SNoR y β†’ SNo (x * y2))
L5742
Hypothesis H4 : z ∈ SNoR x
L5743
Hypothesis H5 : w ∈ SNoR y
L5744
Hypothesis H6 : v ∈ SNoR y
L5745
Hypothesis H7 : SNo (z * y)
L5746
Hypothesis H8 : SNo (x * w)
L5747
Hypothesis H9 : SNo (z * w)
L5748
Hypothesis H10 : SNo (u * y)
L5749
Hypothesis H11 : SNo (x * v)
L5750
Hypothesis H12 : SNo (u * v)
L5751
Hypothesis H13 : SNo (z * v)
L5752
Hypothesis H14 : (z * y + u * v) < u * y + z * v
L5753
Hypothesis H15 : x2 ∈ SNoL w
L5754
Hypothesis H17 : SNo x2
L5755
Hypothesis H18 : x2 ∈ SNoR y
L5756
Theorem. (Conj_mul_SNo_prop_1__42__16)
(x * w + z * x2) < x * x2 + z * w β†’ (z * y + x * w + u * v) < u * y + x * v + z * w
In Proofgold the corresponding term root is 016d14... and proposition id is ac5df9...
Proof:
Proof not loaded.
End of Section Conj_mul_SNo_prop_1__42__16
Beginning of Section Conj_mul_SNo_prop_1__43__9
L5762
Variable x : set
(*** Conj_mul_SNo_prop_1__43__9 TMb8wBe8siM1CeCwNPqbGioUQFDF3WBtcw7 bounty of about 25 bars ***)
L5763
Variable y : set
L5764
Variable z : set
L5765
Variable w : set
L5766
Variable u : set
L5767
Variable v : set
L5768
Variable x2 : set
L5769
Hypothesis H0 : SNo y
L5770
Hypothesis H1 : (βˆ€y2 : set, y2 ∈ SNoS_ (SNoLev y) β†’ (βˆ€P : prop, (SNo (x * y2) β†’ (βˆ€z2 : set, z2 ∈ SNoL x β†’ (βˆ€w2 : set, w2 ∈ SNoL y2 β†’ (z2 * y2 + x * w2) < x * y2 + z2 * w2)) β†’ (βˆ€z2 : set, z2 ∈ SNoR x β†’ (βˆ€w2 : set, w2 ∈ SNoR y2 β†’ (z2 * y2 + x * w2) < x * y2 + z2 * w2)) β†’ (βˆ€z2 : set, z2 ∈ SNoL x β†’ (βˆ€w2 : set, w2 ∈ SNoR y2 β†’ (x * y2 + z2 * w2) < z2 * y2 + x * w2)) β†’ (βˆ€z2 : set, z2 ∈ SNoR x β†’ (βˆ€w2 : set, w2 ∈ SNoL y2 β†’ (x * y2 + z2 * w2) < z2 * y2 + x * w2)) β†’ P) β†’ P))
L5771
Hypothesis H2 : (βˆ€y2 : set, y2 ∈ SNoR x β†’ (βˆ€z2 : set, SNo z2 β†’ SNo (y2 * z2)))
L5772
Hypothesis H3 : (βˆ€y2 : set, y2 ∈ SNoR y β†’ SNo (x * y2))
L5773
Hypothesis H4 : z ∈ SNoR x
L5774
Hypothesis H5 : w ∈ SNoR y
L5775
Hypothesis H6 : v ∈ SNoR y
L5776
Hypothesis H7 : SNo v
L5777
Hypothesis H8 : SNoLev v ∈ SNoLev y
L5778
Hypothesis H10 : SNo (z * y)
L5779
Hypothesis H11 : SNo (x * w)
L5780
Hypothesis H12 : SNo (z * w)
L5781
Hypothesis H13 : SNo (u * y)
L5782
Hypothesis H14 : SNo (x * v)
L5783
Hypothesis H15 : SNo (u * v)
L5784
Hypothesis H16 : SNo (z * v)
L5785
Hypothesis H17 : (z * y + u * v) < u * y + z * v
L5786
Hypothesis H18 : x2 ∈ SNoL w
L5787
Hypothesis H19 : x2 ∈ SNoR v
L5788
Hypothesis H20 : SNo x2
L5789
Hypothesis H21 : SNoLev x2 ∈ SNoLev v
L5790
Hypothesis H22 : v < x2
L5791
Theorem. (Conj_mul_SNo_prop_1__43__9)
x2 ∈ SNoR y β†’ (z * y + x * w + u * v) < u * y + x * v + z * w
In Proofgold the corresponding term root is 3ce62e... and proposition id is 81348d...
Proof:
Proof not loaded.
End of Section Conj_mul_SNo_prop_1__43__9
Beginning of Section Conj_mul_SNo_prop_1__43__17
L5797
Variable x : set
(*** Conj_mul_SNo_prop_1__43__17 TMcfHhnu5Ycxw1wxrN2JjrGVaMXdMw4VsM1 bounty of about 25 bars ***)
L5798
Variable y : set
L5799
Variable z : set
L5800
Variable w : set
L5801
Variable u : set
L5802
Variable v : set
L5803
Variable x2 : set
L5804
Hypothesis H0 : SNo y
L5805
Hypothesis H1 : (βˆ€y2 : set, y2 ∈ SNoS_ (SNoLev y) β†’ (βˆ€P : prop, (SNo (x * y2) β†’ (βˆ€z2 : set, z2 ∈ SNoL x β†’ (βˆ€w2 : set, w2 ∈ SNoL y2 β†’ (z2 * y2 + x * w2) < x * y2 + z2 * w2)) β†’ (βˆ€z2 : set, z2 ∈ SNoR x β†’ (βˆ€w2 : set, w2 ∈ SNoR y2 β†’ (z2 * y2 + x * w2) < x * y2 + z2 * w2)) β†’ (βˆ€z2 : set, z2 ∈ SNoL x β†’ (βˆ€w2 : set, w2 ∈ SNoR y2 β†’ (x * y2 + z2 * w2) < z2 * y2 + x * w2)) β†’ (βˆ€z2 : set, z2 ∈ SNoR x β†’ (βˆ€w2 : set, w2 ∈ SNoL y2 β†’ (x * y2 + z2 * w2) < z2 * y2 + x * w2)) β†’ P) β†’ P))
L5806
Hypothesis H2 : (βˆ€y2 : set, y2 ∈ SNoR x β†’ (βˆ€z2 : set, SNo z2 β†’ SNo (y2 * z2)))
L5807
Hypothesis H3 : (βˆ€y2 : set, y2 ∈ SNoR y β†’ SNo (x * y2))
L5808
Hypothesis H4 : z ∈ SNoR x
L5809
Hypothesis H5 : w ∈ SNoR y
L5810
Hypothesis H6 : v ∈ SNoR y
L5811
Hypothesis H7 : SNo v
L5812
Hypothesis H8 : SNoLev v ∈ SNoLev y
L5813
Hypothesis H9 : y < v
L5814
Hypothesis H10 : SNo (z * y)
L5815
Hypothesis H11 : SNo (x * w)
L5816
Hypothesis H12 : SNo (z * w)
L5817
Hypothesis H13 : SNo (u * y)
L5818
Hypothesis H14 : SNo (x * v)
L5819
Hypothesis H15 : SNo (u * v)
L5820
Hypothesis H16 : SNo (z * v)
L5821
Hypothesis H18 : x2 ∈ SNoL w
L5822
Hypothesis H19 : x2 ∈ SNoR v
L5823
Hypothesis H20 : SNo x2
L5824
Hypothesis H21 : SNoLev x2 ∈ SNoLev v
L5825
Hypothesis H22 : v < x2
L5826
Theorem. (Conj_mul_SNo_prop_1__43__17)
x2 ∈ SNoR y β†’ (z * y + x * w + u * v) < u * y + x * v + z * w
In Proofgold the corresponding term root is ec83ad... and proposition id is 62517c...
Proof:
Proof not loaded.
End of Section Conj_mul_SNo_prop_1__43__17
Beginning of Section Conj_mul_SNo_prop_1__44__12
L5832
Variable x : set
(*** Conj_mul_SNo_prop_1__44__12 TMZ2aAX36kuGCc5aYPtLGQNoWFQqGTJr4yF bounty of about 25 bars ***)
L5833
Variable y : set
L5834
Variable z : set
L5835
Variable w : set
L5836
Variable u : set
L5837
Variable v : set
L5838
Hypothesis H0 : SNo y
L5839
Hypothesis H1 : (βˆ€x2 : set, x2 ∈ SNoS_ (SNoLev y) β†’ (βˆ€P : prop, (SNo (x * x2) β†’ (βˆ€y2 : set, y2 ∈ SNoL x β†’ (βˆ€z2 : set, z2 ∈ SNoL x2 β†’ (y2 * x2 + x * z2) < x * x2 + y2 * z2)) β†’ (βˆ€y2 : set, y2 ∈ SNoR x β†’ (βˆ€z2 : set, z2 ∈ SNoR x2 β†’ (y2 * x2 + x * z2) < x * x2 + y2 * z2)) β†’ (βˆ€y2 : set, y2 ∈ SNoL x β†’ (βˆ€z2 : set, z2 ∈ SNoR x2 β†’ (x * x2 + y2 * z2) < y2 * x2 + x * z2)) β†’ (βˆ€y2 : set, y2 ∈ SNoR x β†’ (βˆ€z2 : set, z2 ∈ SNoL x2 β†’ (x * x2 + y2 * z2) < y2 * x2 + x * z2)) β†’ P) β†’ P))
L5840
Hypothesis H2 : (βˆ€x2 : set, x2 ∈ SNoL x β†’ (βˆ€y2 : set, SNo y2 β†’ SNo (x2 * y2)))
L5841
Hypothesis H3 : (βˆ€x2 : set, x2 ∈ SNoR x β†’ (βˆ€y2 : set, SNo y2 β†’ SNo (x2 * y2)))
L5842
Hypothesis H4 : (βˆ€x2 : set, x2 ∈ SNoR y β†’ SNo (x * x2))
L5843
Hypothesis H5 : z ∈ SNoR x
L5844
Hypothesis H6 : w ∈ SNoR y
L5845
Hypothesis H7 : SNo w
L5846
Hypothesis H8 : SNoLev w ∈ SNoLev y
L5847
Hypothesis H9 : y < w
L5848
Hypothesis H10 : u ∈ SNoL x
L5849
Hypothesis H11 : v ∈ SNoR y
L5850
Hypothesis H13 : SNoLev v ∈ SNoLev y
L5851
Hypothesis H14 : y < v
L5852
Hypothesis H15 : SNo (z * y)
L5853
Hypothesis H16 : SNo (x * w)
L5854
Hypothesis H17 : SNo (z * w)
L5855
Hypothesis H18 : SNo (u * y)
L5856
Hypothesis H19 : SNo (x * v)
L5857
Hypothesis H20 : SNo (u * v)
L5858
Hypothesis H21 : SNo (z * v)
L5859
Hypothesis H22 : SNo (u * w)
L5860
Hypothesis H23 : (z * y + u * w) < u * y + z * w
L5861
Hypothesis H24 : (z * y + u * v) < u * y + z * v
L5862
Hypothesis H25 : (x * w + u * v) < u * w + x * v β†’ (z * y + x * w + u * v) < u * y + x * v + z * w
L5863
Theorem. (Conj_mul_SNo_prop_1__44__12)
((x * w + z * v) < x * v + z * w β†’ (z * y + x * w + u * v) < u * y + x * v + z * w) β†’ (z * y + x * w + u * v) < u * y + x * v + z * w
In Proofgold the corresponding term root is 1b3a92... and proposition id is 282e43...
Proof:
Proof not loaded.
End of Section Conj_mul_SNo_prop_1__44__12
Beginning of Section Conj_mul_SNo_prop_1__44__19
L5869
Variable x : set
(*** Conj_mul_SNo_prop_1__44__19 TMRfA5BuCZuaDYGw2YFPQJAxGxsUgA3LHaX bounty of about 25 bars ***)
L5870
Variable y : set
L5871
Variable z : set
L5872
Variable w : set
L5873
Variable u : set
L5874
Variable v : set
L5875
Hypothesis H0 : SNo y
L5876
Hypothesis H1 : (βˆ€x2 : set, x2 ∈ SNoS_ (SNoLev y) β†’ (βˆ€P : prop, (SNo (x * x2) β†’ (βˆ€y2 : set, y2 ∈ SNoL x β†’ (βˆ€z2 : set, z2 ∈ SNoL x2 β†’ (y2 * x2 + x * z2) < x * x2 + y2 * z2)) β†’ (βˆ€y2 : set, y2 ∈ SNoR x β†’ (βˆ€z2 : set, z2 ∈ SNoR x2 β†’ (y2 * x2 + x * z2) < x * x2 + y2 * z2)) β†’ (βˆ€y2 : set, y2 ∈ SNoL x β†’ (βˆ€z2 : set, z2 ∈ SNoR x2 β†’ (x * x2 + y2 * z2) < y2 * x2 + x * z2)) β†’ (βˆ€y2 : set, y2 ∈ SNoR x β†’ (βˆ€z2 : set, z2 ∈ SNoL x2 β†’ (x * x2 + y2 * z2) < y2 * x2 + x * z2)) β†’ P) β†’ P))
L5877
Hypothesis H2 : (βˆ€x2 : set, x2 ∈ SNoL x β†’ (βˆ€y2 : set, SNo y2 β†’ SNo (x2 * y2)))
L5878
Hypothesis H3 : (βˆ€x2 : set, x2 ∈ SNoR x β†’ (βˆ€y2 : set, SNo y2 β†’ SNo (x2 * y2)))
L5879
Hypothesis H4 : (βˆ€x2 : set, x2 ∈ SNoR y β†’ SNo (x * x2))
L5880
Hypothesis H5 : z ∈ SNoR x
L5881
Hypothesis H6 : w ∈ SNoR y
L5882
Hypothesis H7 : SNo w
L5883
Hypothesis H8 : SNoLev w ∈ SNoLev y
L5884
Hypothesis H9 : y < w
L5885
Hypothesis H10 : u ∈ SNoL x
L5886
Hypothesis H11 : v ∈ SNoR y
L5887
Hypothesis H12 : SNo v
L5888
Hypothesis H13 : SNoLev v ∈ SNoLev y
L5889
Hypothesis H14 : y < v
L5890
Hypothesis H15 : SNo (z * y)
L5891
Hypothesis H16 : SNo (x * w)
L5892
Hypothesis H17 : SNo (z * w)
L5893
Hypothesis H18 : SNo (u * y)
L5894
Hypothesis H20 : SNo (u * v)
L5895
Hypothesis H21 : SNo (z * v)
L5896
Hypothesis H22 : SNo (u * w)
L5897
Hypothesis H23 : (z * y + u * w) < u * y + z * w
L5898
Hypothesis H24 : (z * y + u * v) < u * y + z * v
L5899
Hypothesis H25 : (x * w + u * v) < u * w + x * v β†’ (z * y + x * w + u * v) < u * y + x * v + z * w
L5900
Theorem. (Conj_mul_SNo_prop_1__44__19)
((x * w + z * v) < x * v + z * w β†’ (z * y + x * w + u * v) < u * y + x * v + z * w) β†’ (z * y + x * w + u * v) < u * y + x * v + z * w
In Proofgold the corresponding term root is a1c036... and proposition id is cc0732...
Proof:
Proof not loaded.
End of Section Conj_mul_SNo_prop_1__44__19
Beginning of Section Conj_mul_SNo_prop_1__44__25
L5906
Variable x : set
(*** Conj_mul_SNo_prop_1__44__25 TMVmXL12KbM5FiTcQEbr6PVhvPTALkK5tGo bounty of about 25 bars ***)
L5907
Variable y : set
L5908
Variable z : set
L5909
Variable w : set
L5910
Variable u : set
L5911
Variable v : set
L5912
Hypothesis H0 : SNo y
L5913
Hypothesis H1 : (βˆ€x2 : set, x2 ∈ SNoS_ (SNoLev y) β†’ (βˆ€P : prop, (SNo (x * x2) β†’ (βˆ€y2 : set, y2 ∈ SNoL x β†’ (βˆ€z2 : set, z2 ∈ SNoL x2 β†’ (y2 * x2 + x * z2) < x * x2 + y2 * z2)) β†’ (βˆ€y2 : set, y2 ∈ SNoR x β†’ (βˆ€z2 : set, z2 ∈ SNoR x2 β†’ (y2 * x2 + x * z2) < x * x2 + y2 * z2)) β†’ (βˆ€y2 : set, y2 ∈ SNoL x β†’ (βˆ€z2 : set, z2 ∈ SNoR x2 β†’ (x * x2 + y2 * z2) < y2 * x2 + x * z2)) β†’ (βˆ€y2 : set, y2 ∈ SNoR x β†’ (βˆ€z2 : set, z2 ∈ SNoL x2 β†’ (x * x2 + y2 * z2) < y2 * x2 + x * z2)) β†’ P) β†’ P))
L5914
Hypothesis H2 : (βˆ€x2 : set, x2 ∈ SNoL x β†’ (βˆ€y2 : set, SNo y2 β†’ SNo (x2 * y2)))
L5915
Hypothesis H3 : (βˆ€x2 : set, x2 ∈ SNoR x β†’ (βˆ€y2 : set, SNo y2 β†’ SNo (x2 * y2)))
L5916
Hypothesis H4 : (βˆ€x2 : set, x2 ∈ SNoR y β†’ SNo (x * x2))
L5917
Hypothesis H5 : z ∈ SNoR x
L5918
Hypothesis H6 : w ∈ SNoR y
L5919
Hypothesis H7 : SNo w
L5920
Hypothesis H8 : SNoLev w ∈ SNoLev y
L5921
Hypothesis H9 : y < w
L5922
Hypothesis H10 : u ∈ SNoL x
L5923
Hypothesis H11 : v ∈ SNoR y
L5924
Hypothesis H12 : SNo v
L5925
Hypothesis H13 : SNoLev v ∈ SNoLev y
L5926
Hypothesis H14 : y < v
L5927
Hypothesis H15 : SNo (z * y)
L5928
Hypothesis H16 : SNo (x * w)
L5929
Hypothesis H17 : SNo (z * w)
L5930
Hypothesis H18 : SNo (u * y)
L5931
Hypothesis H19 : SNo (x * v)
L5932
Hypothesis H20 : SNo (u * v)
L5933
Hypothesis H21 : SNo (z * v)
L5934
Hypothesis H22 : SNo (u * w)
L5935
Hypothesis H23 : (z * y + u * w) < u * y + z * w
L5936
Hypothesis H24 : (z * y + u * v) < u * y + z * v
L5937
Theorem. (Conj_mul_SNo_prop_1__44__25)
((x * w + z * v) < x * v + z * w β†’ (z * y + x * w + u * v) < u * y + x * v + z * w) β†’ (z * y + x * w + u * v) < u * y + x * v + z * w
In Proofgold the corresponding term root is a3dda4... and proposition id is 36cbb1...
Proof:
Proof not loaded.
End of Section Conj_mul_SNo_prop_1__44__25
Beginning of Section Conj_mul_SNo_prop_1__45__10
L5943
Variable x : set
(*** Conj_mul_SNo_prop_1__45__10 TMbhVdqtAMQiUPuMPRFFYdyctmibMMvGdL4 bounty of about 25 bars ***)
L5944
Variable y : set
L5945
Variable z : set
L5946
Variable w : set
L5947
Variable u : set
L5948
Variable v : set
L5949
Hypothesis H0 : SNo y
L5950
Hypothesis H1 : (βˆ€x2 : set, x2 ∈ SNoS_ (SNoLev y) β†’ (βˆ€P : prop, (SNo (x * x2) β†’ (βˆ€y2 : set, y2 ∈ SNoL x β†’ (βˆ€z2 : set, z2 ∈ SNoL x2 β†’ (y2 * x2 + x * z2) < x * x2 + y2 * z2)) β†’ (βˆ€y2 : set, y2 ∈ SNoR x β†’ (βˆ€z2 : set, z2 ∈ SNoR x2 β†’ (y2 * x2 + x * z2) < x * x2 + y2 * z2)) β†’ (βˆ€y2 : set, y2 ∈ SNoL x β†’ (βˆ€z2 : set, z2 ∈ SNoR x2 β†’ (x * x2 + y2 * z2) < y2 * x2 + x * z2)) β†’ (βˆ€y2 : set, y2 ∈ SNoR x β†’ (βˆ€z2 : set, z2 ∈ SNoL x2 β†’ (x * x2 + y2 * z2) < y2 * x2 + x * z2)) β†’ P) β†’ P))
L5951
Hypothesis H2 : (βˆ€x2 : set, x2 ∈ SNoL x β†’ (βˆ€y2 : set, SNo y2 β†’ SNo (x2 * y2)))
L5952
Hypothesis H3 : (βˆ€x2 : set, x2 ∈ SNoR x β†’ (βˆ€y2 : set, SNo y2 β†’ SNo (x2 * y2)))
L5953
Hypothesis H4 : (βˆ€x2 : set, x2 ∈ SNoR y β†’ SNo (x * x2))
L5954
Hypothesis H5 : z ∈ SNoR x
L5955
Hypothesis H6 : w ∈ SNoR y
L5956
Hypothesis H7 : SNo w
L5957
Hypothesis H8 : SNoLev w ∈ SNoLev y
L5958
Hypothesis H9 : y < w
L5959
Hypothesis H11 : v ∈ SNoR y
L5960
Hypothesis H12 : SNo v
L5961
Hypothesis H13 : SNoLev v ∈ SNoLev y
L5962
Hypothesis H14 : y < v
L5963
Hypothesis H15 : SNo (z * y)
L5964
Hypothesis H16 : SNo (x * w)
L5965
Hypothesis H17 : SNo (z * w)
L5966
Hypothesis H18 : SNo (u * y)
L5967
Hypothesis H19 : SNo (x * v)
L5968
Hypothesis H20 : SNo (u * v)
L5969
Hypothesis H21 : SNo (z * v)
L5970
Hypothesis H22 : SNo (u * w)
L5971
Hypothesis H23 : (z * y + u * w) < u * y + z * w
L5972
Hypothesis H24 : (z * y + u * v) < u * y + z * v
L5973
Theorem. (Conj_mul_SNo_prop_1__45__10)
((x * w + u * v) < u * w + x * v β†’ (z * y + x * w + u * v) < u * y + x * v + z * w) β†’ (z * y + x * w + u * v) < u * y + x * v + z * w
In Proofgold the corresponding term root is b15387... and proposition id is 068d73...
Proof:
Proof not loaded.
End of Section Conj_mul_SNo_prop_1__45__10
Beginning of Section Conj_mul_SNo_prop_1__46__15
L5979
Variable x : set
(*** Conj_mul_SNo_prop_1__46__15 TMKDHokpSB1qGgVrt2R5c1JczgKm3r8yFih bounty of about 25 bars ***)
L5980
Variable y : set
L5981
Variable z : set
L5982
Variable w : set
L5983
Variable u : set
L5984
Variable v : set
L5985
Hypothesis H0 : SNo y
L5986
Hypothesis H1 : (βˆ€x2 : set, x2 ∈ SNoS_ (SNoLev y) β†’ (βˆ€P : prop, (SNo (x * x2) β†’ (βˆ€y2 : set, y2 ∈ SNoL x β†’ (βˆ€z2 : set, z2 ∈ SNoL x2 β†’ (y2 * x2 + x * z2) < x * x2 + y2 * z2)) β†’ (βˆ€y2 : set, y2 ∈ SNoR x β†’ (βˆ€z2 : set, z2 ∈ SNoR x2 β†’ (y2 * x2 + x * z2) < x * x2 + y2 * z2)) β†’ (βˆ€y2 : set, y2 ∈ SNoL x β†’ (βˆ€z2 : set, z2 ∈ SNoR x2 β†’ (x * x2 + y2 * z2) < y2 * x2 + x * z2)) β†’ (βˆ€y2 : set, y2 ∈ SNoR x β†’ (βˆ€z2 : set, z2 ∈ SNoL x2 β†’ (x * x2 + y2 * z2) < y2 * x2 + x * z2)) β†’ P) β†’ P))
L5987
Hypothesis H2 : (βˆ€x2 : set, x2 ∈ SNoL x β†’ (βˆ€y2 : set, SNo y2 β†’ SNo (x2 * y2)))
L5988
Hypothesis H3 : (βˆ€x2 : set, x2 ∈ SNoR x β†’ (βˆ€y2 : set, SNo y2 β†’ SNo (x2 * y2)))
L5989
Hypothesis H4 : (βˆ€x2 : set, x2 ∈ SNoR y β†’ SNo (x * x2))
L5990
Hypothesis H5 : z ∈ SNoR x
L5991
Hypothesis H6 : w ∈ SNoR y
L5992
Hypothesis H7 : SNo w
L5993
Hypothesis H8 : SNoLev w ∈ SNoLev y
L5994
Hypothesis H9 : y < w
L5995
Hypothesis H10 : u ∈ SNoL x
L5996
Hypothesis H11 : v ∈ SNoR y
L5997
Hypothesis H12 : SNo v
L5998
Hypothesis H13 : SNoLev v ∈ SNoLev y
L5999
Hypothesis H14 : y < v
L6000
Hypothesis H16 : SNo (x * w)
L6001
Hypothesis H17 : SNo (z * w)
L6002
Hypothesis H18 : SNo (u * y)
L6003
Hypothesis H19 : SNo (x * v)
L6004
Hypothesis H20 : SNo (u * v)
L6005
Hypothesis H21 : SNo (z * v)
L6006
Hypothesis H22 : SNo (u * w)
L6007
Hypothesis H23 : (βˆ€x2 : set, x2 ∈ SNoR y β†’ (z * y + u * x2) < u * y + z * x2)
L6008
Hypothesis H24 : (z * y + u * w) < u * y + z * w
L6009
Theorem. (Conj_mul_SNo_prop_1__46__15)
(z * y + u * v) < u * y + z * v β†’ (z * y + x * w + u * v) < u * y + x * v + z * w
In Proofgold the corresponding term root is e60d1b... and proposition id is 990ab2...
Proof:
Proof not loaded.
End of Section Conj_mul_SNo_prop_1__46__15
Beginning of Section Conj_mul_SNo_prop_1__48__5
L6015
Variable x : set
(*** Conj_mul_SNo_prop_1__48__5 TMTQdHn8p7KcNs7EcTWn8mvm9b3aukMakyA bounty of about 25 bars ***)
L6016
Variable y : set
L6017
Variable z : set
L6018
Variable w : set
L6019
Variable u : set
L6020
Variable v : set
L6021
Hypothesis H0 : SNo x
L6022
Hypothesis H1 : (βˆ€x2 : set, x2 ∈ SNoS_ (SNoLev x) β†’ (βˆ€y2 : set, SNo y2 β†’ (βˆ€P : prop, (SNo (x2 * y2) β†’ (βˆ€z2 : set, z2 ∈ SNoL x2 β†’ (βˆ€w2 : set, w2 ∈ SNoL y2 β†’ (z2 * y2 + x2 * w2) < x2 * y2 + z2 * w2)) β†’ (βˆ€z2 : set, z2 ∈ SNoR x2 β†’ (βˆ€w2 : set, w2 ∈ SNoR y2 β†’ (z2 * y2 + x2 * w2) < x2 * y2 + z2 * w2)) β†’ (βˆ€z2 : set, z2 ∈ SNoL x2 β†’ (βˆ€w2 : set, w2 ∈ SNoR y2 β†’ (x2 * y2 + z2 * w2) < z2 * y2 + x2 * w2)) β†’ (βˆ€z2 : set, z2 ∈ SNoR x2 β†’ (βˆ€w2 : set, w2 ∈ SNoL y2 β†’ (x2 * y2 + z2 * w2) < z2 * y2 + x2 * w2)) β†’ P) β†’ P)))
L6023
Hypothesis H2 : SNo y
L6024
Hypothesis H3 : (βˆ€x2 : set, x2 ∈ SNoS_ (SNoLev y) β†’ (βˆ€P : prop, (SNo (x * x2) β†’ (βˆ€y2 : set, y2 ∈ SNoL x β†’ (βˆ€z2 : set, z2 ∈ SNoL x2 β†’ (y2 * x2 + x * z2) < x * x2 + y2 * z2)) β†’ (βˆ€y2 : set, y2 ∈ SNoR x β†’ (βˆ€z2 : set, z2 ∈ SNoR x2 β†’ (y2 * x2 + x * z2) < x * x2 + y2 * z2)) β†’ (βˆ€y2 : set, y2 ∈ SNoL x β†’ (βˆ€z2 : set, z2 ∈ SNoR x2 β†’ (x * x2 + y2 * z2) < y2 * x2 + x * z2)) β†’ (βˆ€y2 : set, y2 ∈ SNoR x β†’ (βˆ€z2 : set, z2 ∈ SNoL x2 β†’ (x * x2 + y2 * z2) < y2 * x2 + x * z2)) β†’ P) β†’ P))
L6025
Hypothesis H4 : (βˆ€x2 : set, x2 ∈ SNoL x β†’ (βˆ€y2 : set, SNo y2 β†’ SNo (x2 * y2)))
L6026
Hypothesis H6 : (βˆ€x2 : set, x2 ∈ SNoR y β†’ SNo (x * x2))
L6027
Hypothesis H7 : z ∈ SNoR x
L6028
Hypothesis H8 : w ∈ SNoR y
L6029
Hypothesis H9 : SNo z
L6030
Hypothesis H10 : SNoLev z ∈ SNoLev x
L6031
Hypothesis H11 : SNo w
L6032
Hypothesis H12 : SNoLev w ∈ SNoLev y
L6033
Hypothesis H13 : y < w
L6034
Hypothesis H14 : u ∈ SNoL x
L6035
Hypothesis H15 : v ∈ SNoR y
L6036
Hypothesis H16 : SNo u
L6037
Hypothesis H17 : SNo v
L6038
Hypothesis H18 : SNoLev v ∈ SNoLev y
L6039
Hypothesis H19 : y < v
L6040
Hypothesis H20 : SNo (z * y)
L6041
Hypothesis H21 : SNo (x * w)
L6042
Hypothesis H22 : SNo (z * w)
L6043
Hypothesis H23 : SNo (u * y)
L6044
Hypothesis H24 : SNo (x * v)
L6045
Hypothesis H25 : SNo (u * v)
L6046
Hypothesis H26 : SNo (z * v)
L6047
Hypothesis H27 : SNo (u * w)
L6048
Hypothesis H28 : u < z
L6049
Theorem. (Conj_mul_SNo_prop_1__48__5)
(βˆ€x2 : set, x2 ∈ SNoR y β†’ (z * y + u * x2) < u * y + z * x2) β†’ (z * y + x * w + u * v) < u * y + x * v + z * w
In Proofgold the corresponding term root is 592712... and proposition id is 2439c5...
Proof:
Proof not loaded.
End of Section Conj_mul_SNo_prop_1__48__5
Beginning of Section Conj_mul_SNo_prop_1__49__9
L6055
Variable x : set
(*** Conj_mul_SNo_prop_1__49__9 TMGzkC27BsSYzFDQBPWS269mwz4ydMRj7wd bounty of about 25 bars ***)
L6056
Variable y : set
L6057
Variable z : set
L6058
Variable w : set
L6059
Variable u : set
L6060
Variable v : set
L6061
Hypothesis H0 : SNo x
L6062
Hypothesis H1 : (βˆ€x2 : set, x2 ∈ SNoS_ (SNoLev x) β†’ (βˆ€y2 : set, SNo y2 β†’ (βˆ€P : prop, (SNo (x2 * y2) β†’ (βˆ€z2 : set, z2 ∈ SNoL x2 β†’ (βˆ€w2 : set, w2 ∈ SNoL y2 β†’ (z2 * y2 + x2 * w2) < x2 * y2 + z2 * w2)) β†’ (βˆ€z2 : set, z2 ∈ SNoR x2 β†’ (βˆ€w2 : set, w2 ∈ SNoR y2 β†’ (z2 * y2 + x2 * w2) < x2 * y2 + z2 * w2)) β†’ (βˆ€z2 : set, z2 ∈ SNoL x2 β†’ (βˆ€w2 : set, w2 ∈ SNoR y2 β†’ (x2 * y2 + z2 * w2) < z2 * y2 + x2 * w2)) β†’ (βˆ€z2 : set, z2 ∈ SNoR x2 β†’ (βˆ€w2 : set, w2 ∈ SNoL y2 β†’ (x2 * y2 + z2 * w2) < z2 * y2 + x2 * w2)) β†’ P) β†’ P)))
L6063
Hypothesis H2 : SNo y
L6064
Hypothesis H3 : (βˆ€x2 : set, x2 ∈ SNoS_ (SNoLev y) β†’ (βˆ€P : prop, (SNo (x * x2) β†’ (βˆ€y2 : set, y2 ∈ SNoL x β†’ (βˆ€z2 : set, z2 ∈ SNoL x2 β†’ (y2 * x2 + x * z2) < x * x2 + y2 * z2)) β†’ (βˆ€y2 : set, y2 ∈ SNoR x β†’ (βˆ€z2 : set, z2 ∈ SNoR x2 β†’ (y2 * x2 + x * z2) < x * x2 + y2 * z2)) β†’ (βˆ€y2 : set, y2 ∈ SNoL x β†’ (βˆ€z2 : set, z2 ∈ SNoR x2 β†’ (x * x2 + y2 * z2) < y2 * x2 + x * z2)) β†’ (βˆ€y2 : set, y2 ∈ SNoR x β†’ (βˆ€z2 : set, z2 ∈ SNoL x2 β†’ (x * x2 + y2 * z2) < y2 * x2 + x * z2)) β†’ P) β†’ P))
L6065
Hypothesis H4 : (βˆ€x2 : set, x2 ∈ SNoL x β†’ (βˆ€y2 : set, SNo y2 β†’ SNo (x2 * y2)))
L6066
Hypothesis H5 : (βˆ€x2 : set, x2 ∈ SNoR x β†’ (βˆ€y2 : set, SNo y2 β†’ SNo (x2 * y2)))
L6067
Hypothesis H6 : (βˆ€x2 : set, x2 ∈ SNoR y β†’ SNo (x * x2))
L6068
Hypothesis H7 : z ∈ SNoR x
L6069
Hypothesis H8 : w ∈ SNoR y
L6070
Hypothesis H10 : SNoLev z ∈ SNoLev x
L6071
Hypothesis H11 : x < z
L6072
Hypothesis H12 : SNo w
L6073
Hypothesis H13 : SNoLev w ∈ SNoLev y
L6074
Hypothesis H14 : y < w
L6075
Hypothesis H15 : u ∈ SNoL x
L6076
Hypothesis H16 : v ∈ SNoR y
L6077
Hypothesis H17 : SNo u
L6078
Hypothesis H18 : u < x
L6079
Hypothesis H19 : SNo v
L6080
Hypothesis H20 : SNoLev v ∈ SNoLev y
L6081
Hypothesis H21 : y < v
L6082
Hypothesis H22 : SNo (z * y)
L6083
Hypothesis H23 : SNo (x * w)
L6084
Hypothesis H24 : SNo (z * w)
L6085
Hypothesis H25 : SNo (u * y)
L6086
Hypothesis H26 : SNo (x * v)
L6087
Hypothesis H27 : SNo (u * v)
L6088
Hypothesis H28 : SNo (z * v)
L6089
Hypothesis H29 : SNo (u * w)
L6090
Theorem. (Conj_mul_SNo_prop_1__49__9)
u < z β†’ (z * y + x * w + u * v) < u * y + x * v + z * w
In Proofgold the corresponding term root is 73e3e6... and proposition id is 12f17e...
Proof:
Proof not loaded.
End of Section Conj_mul_SNo_prop_1__49__9
Beginning of Section Conj_mul_SNo_prop_1__49__15
L6096
Variable x : set
(*** Conj_mul_SNo_prop_1__49__15 TMWcHKwu2R8RcEwvkTcaFNaPBT8mKLsLSnX bounty of about 25 bars ***)
L6097
Variable y : set
L6098
Variable z : set
L6099
Variable w : set
L6100
Variable u : set
L6101
Variable v : set
L6102
Hypothesis H0 : SNo x
L6103
Hypothesis H1 : (βˆ€x2 : set, x2 ∈ SNoS_ (SNoLev x) β†’ (βˆ€y2 : set, SNo y2 β†’ (βˆ€P : prop, (SNo (x2 * y2) β†’ (βˆ€z2 : set, z2 ∈ SNoL x2 β†’ (βˆ€w2 : set, w2 ∈ SNoL y2 β†’ (z2 * y2 + x2 * w2) < x2 * y2 + z2 * w2)) β†’ (βˆ€z2 : set, z2 ∈ SNoR x2 β†’ (βˆ€w2 : set, w2 ∈ SNoR y2 β†’ (z2 * y2 + x2 * w2) < x2 * y2 + z2 * w2)) β†’ (βˆ€z2 : set, z2 ∈ SNoL x2 β†’ (βˆ€w2 : set, w2 ∈ SNoR y2 β†’ (x2 * y2 + z2 * w2) < z2 * y2 + x2 * w2)) β†’ (βˆ€z2 : set, z2 ∈ SNoR x2 β†’ (βˆ€w2 : set, w2 ∈ SNoL y2 β†’ (x2 * y2 + z2 * w2) < z2 * y2 + x2 * w2)) β†’ P) β†’ P)))
L6104
Hypothesis H2 : SNo y
L6105
Hypothesis H3 : (βˆ€x2 : set, x2 ∈ SNoS_ (SNoLev y) β†’ (βˆ€P : prop, (SNo (x * x2) β†’ (βˆ€y2 : set, y2 ∈ SNoL x β†’ (βˆ€z2 : set, z2 ∈ SNoL x2 β†’ (y2 * x2 + x * z2) < x * x2 + y2 * z2)) β†’ (βˆ€y2 : set, y2 ∈ SNoR x β†’ (βˆ€z2 : set, z2 ∈ SNoR x2 β†’ (y2 * x2 + x * z2) < x * x2 + y2 * z2)) β†’ (βˆ€y2 : set, y2 ∈ SNoL x β†’ (βˆ€z2 : set, z2 ∈ SNoR x2 β†’ (x * x2 + y2 * z2) < y2 * x2 + x * z2)) β†’ (βˆ€y2 : set, y2 ∈ SNoR x β†’ (βˆ€z2 : set, z2 ∈ SNoL x2 β†’ (x * x2 + y2 * z2) < y2 * x2 + x * z2)) β†’ P) β†’ P))
L6106
Hypothesis H4 : (βˆ€x2 : set, x2 ∈ SNoL x β†’ (βˆ€y2 : set, SNo y2 β†’ SNo (x2 * y2)))
L6107
Hypothesis H5 : (βˆ€x2 : set, x2 ∈ SNoR x β†’ (βˆ€y2 : set, SNo y2 β†’ SNo (x2 * y2)))
L6108
Hypothesis H6 : (βˆ€x2 : set, x2 ∈ SNoR y β†’ SNo (x * x2))
L6109
Hypothesis H7 : z ∈ SNoR x
L6110
Hypothesis H8 : w ∈ SNoR y
L6111
Hypothesis H9 : SNo z
L6112
Hypothesis H10 : SNoLev z ∈ SNoLev x
L6113
Hypothesis H11 : x < z
L6114
Hypothesis H12 : SNo w
L6115
Hypothesis H13 : SNoLev w ∈ SNoLev y
L6116
Hypothesis H14 : y < w
L6117
Hypothesis H16 : v ∈ SNoR y
L6118
Hypothesis H17 : SNo u
L6119
Hypothesis H18 : u < x
L6120
Hypothesis H19 : SNo v
L6121
Hypothesis H20 : SNoLev v ∈ SNoLev y
L6122
Hypothesis H21 : y < v
L6123
Hypothesis H22 : SNo (z * y)
L6124
Hypothesis H23 : SNo (x * w)
L6125
Hypothesis H24 : SNo (z * w)
L6126
Hypothesis H25 : SNo (u * y)
L6127
Hypothesis H26 : SNo (x * v)
L6128
Hypothesis H27 : SNo (u * v)
L6129
Hypothesis H28 : SNo (z * v)
L6130
Hypothesis H29 : SNo (u * w)
L6131
Theorem. (Conj_mul_SNo_prop_1__49__15)
u < z β†’ (z * y + x * w + u * v) < u * y + x * v + z * w
In Proofgold the corresponding term root is 21fe15... and proposition id is b129f8...
Proof:
Proof not loaded.
End of Section Conj_mul_SNo_prop_1__49__15
Beginning of Section Conj_mul_SNo_prop_1__51__0
L6137
Variable x : set
(*** Conj_mul_SNo_prop_1__51__0 TMHMzpNex99xmrp3V29PLXXCFhce1x9TPXG bounty of about 25 bars ***)
L6138
Variable y : set
L6139
Variable z : set
L6140
Variable w : set
L6141
Variable u : set
L6142
Variable v : set
L6143
Hypothesis H1 : (βˆ€x2 : set, x2 ∈ SNoS_ (SNoLev x) β†’ (βˆ€y2 : set, SNo y2 β†’ (βˆ€P : prop, (SNo (x2 * y2) β†’ (βˆ€z2 : set, z2 ∈ SNoL x2 β†’ (βˆ€w2 : set, w2 ∈ SNoL y2 β†’ (z2 * y2 + x2 * w2) < x2 * y2 + z2 * w2)) β†’ (βˆ€z2 : set, z2 ∈ SNoR x2 β†’ (βˆ€w2 : set, w2 ∈ SNoR y2 β†’ (z2 * y2 + x2 * w2) < x2 * y2 + z2 * w2)) β†’ (βˆ€z2 : set, z2 ∈ SNoL x2 β†’ (βˆ€w2 : set, w2 ∈ SNoR y2 β†’ (x2 * y2 + z2 * w2) < z2 * y2 + x2 * w2)) β†’ (βˆ€z2 : set, z2 ∈ SNoR x2 β†’ (βˆ€w2 : set, w2 ∈ SNoL y2 β†’ (x2 * y2 + z2 * w2) < z2 * y2 + x2 * w2)) β†’ P) β†’ P)))
L6144
Hypothesis H2 : SNo y
L6145
Hypothesis H3 : z ∈ SNoL x
L6146
Hypothesis H4 : w ∈ SNoR x
L6147
Hypothesis H5 : SNo (z * y)
L6148
Hypothesis H6 : SNo (w * y)
L6149
Hypothesis H7 : u ∈ SNoL y
L6150
Hypothesis H8 : SNo (z * u)
L6151
Hypothesis H9 : SNo (w * u)
L6152
Hypothesis H10 : v ∈ SNoL w
L6153
Hypothesis H11 : v ∈ SNoR z
L6154
Hypothesis H12 : SNo (v * u)
L6155
Hypothesis H13 : SNo (v * y)
L6156
Theorem. (Conj_mul_SNo_prop_1__51__0)
(z * y + v * u) < z * u + v * y β†’ (z * y + w * u) < w * y + z * u
In Proofgold the corresponding term root is 7d5e5b... and proposition id is ccbf77...
Proof:
Proof not loaded.
End of Section Conj_mul_SNo_prop_1__51__0
Beginning of Section Conj_mul_SNo_prop_1__51__2
L6162
Variable x : set
(*** Conj_mul_SNo_prop_1__51__2 TMTFZyTEfkCkEXiPpP6WQQ4Eqote9LNmJh7 bounty of about 25 bars ***)
L6163
Variable y : set
L6164
Variable z : set
L6165
Variable w : set
L6166
Variable u : set
L6167
Variable v : set
L6168
Hypothesis H0 : SNo x
L6169
Hypothesis H1 : (βˆ€x2 : set, x2 ∈ SNoS_ (SNoLev x) β†’ (βˆ€y2 : set, SNo y2 β†’ (βˆ€P : prop, (SNo (x2 * y2) β†’ (βˆ€z2 : set, z2 ∈ SNoL x2 β†’ (βˆ€w2 : set, w2 ∈ SNoL y2 β†’ (z2 * y2 + x2 * w2) < x2 * y2 + z2 * w2)) β†’ (βˆ€z2 : set, z2 ∈ SNoR x2 β†’ (βˆ€w2 : set, w2 ∈ SNoR y2 β†’ (z2 * y2 + x2 * w2) < x2 * y2 + z2 * w2)) β†’ (βˆ€z2 : set, z2 ∈ SNoL x2 β†’ (βˆ€w2 : set, w2 ∈ SNoR y2 β†’ (x2 * y2 + z2 * w2) < z2 * y2 + x2 * w2)) β†’ (βˆ€z2 : set, z2 ∈ SNoR x2 β†’ (βˆ€w2 : set, w2 ∈ SNoL y2 β†’ (x2 * y2 + z2 * w2) < z2 * y2 + x2 * w2)) β†’ P) β†’ P)))
L6170
Hypothesis H3 : z ∈ SNoL x
L6171
Hypothesis H4 : w ∈ SNoR x
L6172
Hypothesis H5 : SNo (z * y)
L6173
Hypothesis H6 : SNo (w * y)
L6174
Hypothesis H7 : u ∈ SNoL y
L6175
Hypothesis H8 : SNo (z * u)
L6176
Hypothesis H9 : SNo (w * u)
L6177
Hypothesis H10 : v ∈ SNoL w
L6178
Hypothesis H11 : v ∈ SNoR z
L6179
Hypothesis H12 : SNo (v * u)
L6180
Hypothesis H13 : SNo (v * y)
L6181
Theorem. (Conj_mul_SNo_prop_1__51__2)
(z * y + v * u) < z * u + v * y β†’ (z * y + w * u) < w * y + z * u
In Proofgold the corresponding term root is 0fe284... and proposition id is aea9a7...
Proof:
Proof not loaded.
End of Section Conj_mul_SNo_prop_1__51__2
Beginning of Section Conj_mul_SNo_prop_1__51__12
L6187
Variable x : set
(*** Conj_mul_SNo_prop_1__51__12 TMaHVkV1AdjrPSGRJKH2vMU2LTyam4mSF4E bounty of about 25 bars ***)
L6188
Variable y : set
L6189
Variable z : set
L6190
Variable w : set
L6191
Variable u : set
L6192
Variable v : set
L6193
Hypothesis H0 : SNo x
L6194
Hypothesis H1 : (βˆ€x2 : set, x2 ∈ SNoS_ (SNoLev x) β†’ (βˆ€y2 : set, SNo y2 β†’ (βˆ€P : prop, (SNo (x2 * y2) β†’ (βˆ€z2 : set, z2 ∈ SNoL x2 β†’ (βˆ€w2 : set, w2 ∈ SNoL y2 β†’ (z2 * y2 + x2 * w2) < x2 * y2 + z2 * w2)) β†’ (βˆ€z2 : set, z2 ∈ SNoR x2 β†’ (βˆ€w2 : set, w2 ∈ SNoR y2 β†’ (z2 * y2 + x2 * w2) < x2 * y2 + z2 * w2)) β†’ (βˆ€z2 : set, z2 ∈ SNoL x2 β†’ (βˆ€w2 : set, w2 ∈ SNoR y2 β†’ (x2 * y2 + z2 * w2) < z2 * y2 + x2 * w2)) β†’ (βˆ€z2 : set, z2 ∈ SNoR x2 β†’ (βˆ€w2 : set, w2 ∈ SNoL y2 β†’ (x2 * y2 + z2 * w2) < z2 * y2 + x2 * w2)) β†’ P) β†’ P)))
L6195
Hypothesis H2 : SNo y
L6196
Hypothesis H3 : z ∈ SNoL x
L6197
Hypothesis H4 : w ∈ SNoR x
L6198
Hypothesis H5 : SNo (z * y)
L6199
Hypothesis H6 : SNo (w * y)
L6200
Hypothesis H7 : u ∈ SNoL y
L6201
Hypothesis H8 : SNo (z * u)
L6202
Hypothesis H9 : SNo (w * u)
L6203
Hypothesis H10 : v ∈ SNoL w
L6204
Hypothesis H11 : v ∈ SNoR z
L6205
Hypothesis H13 : SNo (v * y)
L6206
Theorem. (Conj_mul_SNo_prop_1__51__12)
(z * y + v * u) < z * u + v * y β†’ (z * y + w * u) < w * y + z * u
In Proofgold the corresponding term root is 2b24e9... and proposition id is 990d2e...
Proof:
Proof not loaded.
End of Section Conj_mul_SNo_prop_1__51__12
Beginning of Section Conj_mul_SNo_prop_1__52__12
L6212
Variable x : set
(*** Conj_mul_SNo_prop_1__52__12 TMZpNvsMkcESbzvC1UrsPQKFgGYekkiwmbL bounty of about 25 bars ***)
L6213
Variable y : set
L6214
Variable z : set
L6215
Variable w : set
L6216
Variable u : set
L6217
Variable v : set
L6218
Hypothesis H0 : SNo x
L6219
Hypothesis H1 : (βˆ€x2 : set, x2 ∈ SNoS_ (SNoLev x) β†’ (βˆ€y2 : set, SNo y2 β†’ (βˆ€P : prop, (SNo (x2 * y2) β†’ (βˆ€z2 : set, z2 ∈ SNoL x2 β†’ (βˆ€w2 : set, w2 ∈ SNoL y2 β†’ (z2 * y2 + x2 * w2) < x2 * y2 + z2 * w2)) β†’ (βˆ€z2 : set, z2 ∈ SNoR x2 β†’ (βˆ€w2 : set, w2 ∈ SNoR y2 β†’ (z2 * y2 + x2 * w2) < x2 * y2 + z2 * w2)) β†’ (βˆ€z2 : set, z2 ∈ SNoL x2 β†’ (βˆ€w2 : set, w2 ∈ SNoR y2 β†’ (x2 * y2 + z2 * w2) < z2 * y2 + x2 * w2)) β†’ (βˆ€z2 : set, z2 ∈ SNoR x2 β†’ (βˆ€w2 : set, w2 ∈ SNoL y2 β†’ (x2 * y2 + z2 * w2) < z2 * y2 + x2 * w2)) β†’ P) β†’ P)))
L6220
Hypothesis H2 : SNo y
L6221
Hypothesis H3 : z ∈ SNoL x
L6222
Hypothesis H4 : w ∈ SNoR x
L6223
Hypothesis H5 : SNo (z * y)
L6224
Hypothesis H6 : SNo (w * y)
L6225
Hypothesis H7 : u ∈ SNoL y
L6226
Hypothesis H8 : SNo (z * u)
L6227
Hypothesis H9 : SNo (w * u)
L6228
Hypothesis H10 : v ∈ SNoL w
L6229
Hypothesis H11 : v ∈ SNoR z
L6230
Hypothesis H13 : SNo (v * u)
L6231
Theorem. (Conj_mul_SNo_prop_1__52__12)
SNo (v * y) β†’ (z * y + w * u) < w * y + z * u
In Proofgold the corresponding term root is 222543... and proposition id is f2492e...
Proof:
Proof not loaded.
End of Section Conj_mul_SNo_prop_1__52__12
Beginning of Section Conj_mul_SNo_prop_1__54__4
L6237
Variable x : set
(*** Conj_mul_SNo_prop_1__54__4 TMK18JiQ6Qf9cM3ycUAigsbnuQLn39wiQPs bounty of about 25 bars ***)
L6238
Variable y : set
L6239
Variable z : set
L6240
Variable w : set
L6241
Variable u : set
L6242
Variable v : set
L6243
Hypothesis H0 : SNo x
L6244
Hypothesis H1 : (βˆ€x2 : set, x2 ∈ SNoS_ (SNoLev x) β†’ (βˆ€y2 : set, SNo y2 β†’ (βˆ€P : prop, (SNo (x2 * y2) β†’ (βˆ€z2 : set, z2 ∈ SNoL x2 β†’ (βˆ€w2 : set, w2 ∈ SNoL y2 β†’ (z2 * y2 + x2 * w2) < x2 * y2 + z2 * w2)) β†’ (βˆ€z2 : set, z2 ∈ SNoR x2 β†’ (βˆ€w2 : set, w2 ∈ SNoR y2 β†’ (z2 * y2 + x2 * w2) < x2 * y2 + z2 * w2)) β†’ (βˆ€z2 : set, z2 ∈ SNoL x2 β†’ (βˆ€w2 : set, w2 ∈ SNoR y2 β†’ (x2 * y2 + z2 * w2) < z2 * y2 + x2 * w2)) β†’ (βˆ€z2 : set, z2 ∈ SNoR x2 β†’ (βˆ€w2 : set, w2 ∈ SNoL y2 β†’ (x2 * y2 + z2 * w2) < z2 * y2 + x2 * w2)) β†’ P) β†’ P)))
L6245
Hypothesis H2 : SNo y
L6246
Hypothesis H3 : z ∈ SNoL x
L6247
Hypothesis H5 : SNo (z * y)
L6248
Hypothesis H6 : SNo (w * y)
L6249
Hypothesis H7 : u ∈ SNoL y
L6250
Hypothesis H8 : SNo u
L6251
Hypothesis H9 : SNo (z * u)
L6252
Hypothesis H10 : SNo (w * u)
L6253
Hypothesis H11 : v ∈ SNoL w
L6254
Hypothesis H12 : v ∈ SNoR z
L6255
Hypothesis H13 : SNo v
L6256
Hypothesis H14 : SNoLev v ∈ SNoLev x
L6257
Theorem. (Conj_mul_SNo_prop_1__54__4)
v ∈ SNoS_ (SNoLev x) β†’ (z * y + w * u) < w * y + z * u
In Proofgold the corresponding term root is dbc342... and proposition id is f9e5b8...
Proof:
Proof not loaded.
End of Section Conj_mul_SNo_prop_1__54__4
Beginning of Section Conj_mul_SNo_prop_1__55__1
L6263
Variable x : set
(*** Conj_mul_SNo_prop_1__55__1 TMG2rXoggoXeMrFpkPrsYH4gMMYYmyuGdzo bounty of about 25 bars ***)
L6264
Variable y : set
L6265
Variable z : set
L6266
Variable w : set
L6267
Variable u : set
L6268
Variable v : set
L6269
Hypothesis H0 : SNo x
L6270
Hypothesis H2 : SNo y
L6271
Hypothesis H3 : z ∈ SNoL x
L6272
Hypothesis H4 : SNoLev z ∈ SNoLev x
L6273
Hypothesis H5 : w ∈ SNoR x
L6274
Hypothesis H6 : SNo (z * y)
L6275
Hypothesis H7 : SNo (w * y)
L6276
Hypothesis H8 : u ∈ SNoL y
L6277
Hypothesis H9 : SNo u
L6278
Hypothesis H10 : SNo (z * u)
L6279
Hypothesis H11 : SNo (w * u)
L6280
Hypothesis H12 : v ∈ SNoL w
L6281
Hypothesis H13 : v ∈ SNoR z
L6282
Hypothesis H14 : SNo v
L6283
Hypothesis H15 : SNoLev v ∈ SNoLev z
L6284
Theorem. (Conj_mul_SNo_prop_1__55__1)
SNoLev v ∈ SNoLev x β†’ (z * y + w * u) < w * y + z * u
In Proofgold the corresponding term root is 8a88e9... and proposition id is c00a9d...
Proof:
Proof not loaded.
End of Section Conj_mul_SNo_prop_1__55__1
Beginning of Section Conj_mul_SNo_prop_1__58__11
L6290
Variable x : set
(*** Conj_mul_SNo_prop_1__58__11 TMFHGR3w1KWUEB8TttZsGmGYdgVujMhLqDM bounty of about 25 bars ***)
L6291
Variable y : set
L6292
Variable z : set
L6293
Variable w : set
L6294
Variable u : set
L6295
Variable v : set
L6296
Variable x2 : set
L6297
Hypothesis H0 : SNo y
L6298
Hypothesis H1 : (βˆ€y2 : set, y2 ∈ SNoS_ (SNoLev y) β†’ (βˆ€P : prop, (SNo (x * y2) β†’ (βˆ€z2 : set, z2 ∈ SNoL x β†’ (βˆ€w2 : set, w2 ∈ SNoL y2 β†’ (z2 * y2 + x * w2) < x * y2 + z2 * w2)) β†’ (βˆ€z2 : set, z2 ∈ SNoR x β†’ (βˆ€w2 : set, w2 ∈ SNoR y2 β†’ (z2 * y2 + x * w2) < x * y2 + z2 * w2)) β†’ (βˆ€z2 : set, z2 ∈ SNoL x β†’ (βˆ€w2 : set, w2 ∈ SNoR y2 β†’ (x * y2 + z2 * w2) < z2 * y2 + x * w2)) β†’ (βˆ€z2 : set, z2 ∈ SNoR x β†’ (βˆ€w2 : set, w2 ∈ SNoL y2 β†’ (x * y2 + z2 * w2) < z2 * y2 + x * w2)) β†’ P) β†’ P))
L6299
Hypothesis H2 : (βˆ€y2 : set, y2 ∈ SNoL x β†’ (βˆ€z2 : set, SNo z2 β†’ SNo (y2 * z2)))
L6300
Hypothesis H3 : (βˆ€y2 : set, y2 ∈ SNoL y β†’ SNo (x * y2))
L6301
Hypothesis H4 : z ∈ SNoL x
L6302
Hypothesis H5 : v ∈ SNoL y
L6303
Hypothesis H6 : SNo (z * y)
L6304
Hypothesis H7 : SNo (x * w)
L6305
Hypothesis H8 : SNo (z * w)
L6306
Hypothesis H9 : SNo (u * y)
L6307
Hypothesis H10 : SNo (x * v)
L6308
Hypothesis H12 : SNo (z * v)
L6309
Hypothesis H13 : (z * y + u * v) < u * y + z * v
L6310
Hypothesis H14 : x2 ∈ SNoL v
L6311
Hypothesis H15 : SNo x2
L6312
Hypothesis H16 : x2 ∈ SNoL y
L6313
Hypothesis H17 : (x * w + z * x2) < x * x2 + z * w
L6314
Theorem. (Conj_mul_SNo_prop_1__58__11)
(x * x2 + z * v) < x * v + z * x2 β†’ (z * y + x * w + u * v) < u * y + x * v + z * w
In Proofgold the corresponding term root is 2ad8e7... and proposition id is 154588...
Proof:
Proof not loaded.
End of Section Conj_mul_SNo_prop_1__58__11
Beginning of Section Conj_mul_SNo_prop_1__58__17
L6320
Variable x : set
(*** Conj_mul_SNo_prop_1__58__17 TMadhBP2tARcVgvEQjb92i4PKxmwfrH1fxb bounty of about 25 bars ***)
L6321
Variable y : set
L6322
Variable z : set
L6323
Variable w : set
L6324
Variable u : set
L6325
Variable v : set
L6326
Variable x2 : set
L6327
Hypothesis H0 : SNo y
L6328
Hypothesis H1 : (βˆ€y2 : set, y2 ∈ SNoS_ (SNoLev y) β†’ (βˆ€P : prop, (SNo (x * y2) β†’ (βˆ€z2 : set, z2 ∈ SNoL x β†’ (βˆ€w2 : set, w2 ∈ SNoL y2 β†’ (z2 * y2 + x * w2) < x * y2 + z2 * w2)) β†’ (βˆ€z2 : set, z2 ∈ SNoR x β†’ (βˆ€w2 : set, w2 ∈ SNoR y2 β†’ (z2 * y2 + x * w2) < x * y2 + z2 * w2)) β†’ (βˆ€z2 : set, z2 ∈ SNoL x β†’ (βˆ€w2 : set, w2 ∈ SNoR y2 β†’ (x * y2 + z2 * w2) < z2 * y2 + x * w2)) β†’ (βˆ€z2 : set, z2 ∈ SNoR x β†’ (βˆ€w2 : set, w2 ∈ SNoL y2 β†’ (x * y2 + z2 * w2) < z2 * y2 + x * w2)) β†’ P) β†’ P))
L6329
Hypothesis H2 : (βˆ€y2 : set, y2 ∈ SNoL x β†’ (βˆ€z2 : set, SNo z2 β†’ SNo (y2 * z2)))
L6330
Hypothesis H3 : (βˆ€y2 : set, y2 ∈ SNoL y β†’ SNo (x * y2))
L6331
Hypothesis H4 : z ∈ SNoL x
L6332
Hypothesis H5 : v ∈ SNoL y
L6333
Hypothesis H6 : SNo (z * y)
L6334
Hypothesis H7 : SNo (x * w)
L6335
Hypothesis H8 : SNo (z * w)
L6336
Hypothesis H9 : SNo (u * y)
L6337
Hypothesis H10 : SNo (x * v)
L6338
Hypothesis H11 : SNo (u * v)
L6339
Hypothesis H12 : SNo (z * v)
L6340
Hypothesis H13 : (z * y + u * v) < u * y + z * v
L6341
Hypothesis H14 : x2 ∈ SNoL v
L6342
Hypothesis H15 : SNo x2
L6343
Hypothesis H16 : x2 ∈ SNoL y
L6344
Theorem. (Conj_mul_SNo_prop_1__58__17)
(x * x2 + z * v) < x * v + z * x2 β†’ (z * y + x * w + u * v) < u * y + x * v + z * w
In Proofgold the corresponding term root is b8b017... and proposition id is e27ed7...
Proof:
Proof not loaded.
End of Section Conj_mul_SNo_prop_1__58__17
Beginning of Section Conj_mul_SNo_prop_1__62__10
L6350
Variable x : set
(*** Conj_mul_SNo_prop_1__62__10 TMJWxfjahZWQnm2Z8ag5DgsnwdCmBjt8qeM bounty of about 25 bars ***)
L6351
Variable y : set
L6352
Variable z : set
L6353
Variable w : set
L6354
Variable u : set
L6355
Variable v : set
L6356
Variable x2 : set
L6357
Hypothesis H0 : SNo y
L6358
Hypothesis H1 : (βˆ€y2 : set, y2 ∈ SNoS_ (SNoLev y) β†’ (βˆ€P : prop, (SNo (x * y2) β†’ (βˆ€z2 : set, z2 ∈ SNoL x β†’ (βˆ€w2 : set, w2 ∈ SNoL y2 β†’ (z2 * y2 + x * w2) < x * y2 + z2 * w2)) β†’ (βˆ€z2 : set, z2 ∈ SNoR x β†’ (βˆ€w2 : set, w2 ∈ SNoR y2 β†’ (z2 * y2 + x * w2) < x * y2 + z2 * w2)) β†’ (βˆ€z2 : set, z2 ∈ SNoL x β†’ (βˆ€w2 : set, w2 ∈ SNoR y2 β†’ (x * y2 + z2 * w2) < z2 * y2 + x * w2)) β†’ (βˆ€z2 : set, z2 ∈ SNoR x β†’ (βˆ€w2 : set, w2 ∈ SNoL y2 β†’ (x * y2 + z2 * w2) < z2 * y2 + x * w2)) β†’ P) β†’ P))
L6359
Hypothesis H2 : (βˆ€y2 : set, y2 ∈ SNoR x β†’ (βˆ€z2 : set, SNo z2 β†’ SNo (y2 * z2)))
L6360
Hypothesis H3 : (βˆ€y2 : set, y2 ∈ SNoL y β†’ SNo (x * y2))
L6361
Hypothesis H4 : w ∈ SNoL y
L6362
Hypothesis H5 : u ∈ SNoR x
L6363
Hypothesis H6 : v ∈ SNoL y
L6364
Hypothesis H7 : SNo (z * y)
L6365
Hypothesis H8 : SNo (x * w)
L6366
Hypothesis H9 : SNo (z * w)
L6367
Hypothesis H11 : SNo (x * v)
L6368
Hypothesis H12 : SNo (u * v)
L6369
Hypothesis H13 : SNo (u * w)
L6370
Hypothesis H14 : (z * y + u * w) < u * y + z * w
L6371
Hypothesis H15 : x2 ∈ SNoL w
L6372
Hypothesis H16 : x2 ∈ SNoR v
L6373
Hypothesis H17 : SNo x2
L6374
Hypothesis H18 : x2 ∈ SNoL y
L6375
Theorem. (Conj_mul_SNo_prop_1__62__10)
(x * w + u * x2) < x * x2 + u * w β†’ (z * y + x * w + u * v) < u * y + x * v + z * w
In Proofgold the corresponding term root is c90d2b... and proposition id is 702543...
Proof:
Proof not loaded.
End of Section Conj_mul_SNo_prop_1__62__10
Beginning of Section Conj_mul_SNo_prop_1__62__17
L6381
Variable x : set
(*** Conj_mul_SNo_prop_1__62__17 TMWmUqtMvcvpsSKQZSNcSRXBVPAfAXzanvu bounty of about 25 bars ***)
L6382
Variable y : set
L6383
Variable z : set
L6384
Variable w : set
L6385
Variable u : set
L6386
Variable v : set
L6387
Variable x2 : set
L6388
Hypothesis H0 : SNo y
L6389
Hypothesis H1 : (βˆ€y2 : set, y2 ∈ SNoS_ (SNoLev y) β†’ (βˆ€P : prop, (SNo (x * y2) β†’ (βˆ€z2 : set, z2 ∈ SNoL x β†’ (βˆ€w2 : set, w2 ∈ SNoL y2 β†’ (z2 * y2 + x * w2) < x * y2 + z2 * w2)) β†’ (βˆ€z2 : set, z2 ∈ SNoR x β†’ (βˆ€w2 : set, w2 ∈ SNoR y2 β†’ (z2 * y2 + x * w2) < x * y2 + z2 * w2)) β†’ (βˆ€z2 : set, z2 ∈ SNoL x β†’ (βˆ€w2 : set, w2 ∈ SNoR y2 β†’ (x * y2 + z2 * w2) < z2 * y2 + x * w2)) β†’ (βˆ€z2 : set, z2 ∈ SNoR x β†’ (βˆ€w2 : set, w2 ∈ SNoL y2 β†’ (x * y2 + z2 * w2) < z2 * y2 + x * w2)) β†’ P) β†’ P))
L6390
Hypothesis H2 : (βˆ€y2 : set, y2 ∈ SNoR x β†’ (βˆ€z2 : set, SNo z2 β†’ SNo (y2 * z2)))
L6391
Hypothesis H3 : (βˆ€y2 : set, y2 ∈ SNoL y β†’ SNo (x * y2))
L6392
Hypothesis H4 : w ∈ SNoL y
L6393
Hypothesis H5 : u ∈ SNoR x
L6394
Hypothesis H6 : v ∈ SNoL y
L6395
Hypothesis H7 : SNo (z * y)
L6396
Hypothesis H8 : SNo (x * w)
L6397
Hypothesis H9 : SNo (z * w)
L6398
Hypothesis H10 : SNo (u * y)
L6399
Hypothesis H11 : SNo (x * v)
L6400
Hypothesis H12 : SNo (u * v)
L6401
Hypothesis H13 : SNo (u * w)
L6402
Hypothesis H14 : (z * y + u * w) < u * y + z * w
L6403
Hypothesis H15 : x2 ∈ SNoL w
L6404
Hypothesis H16 : x2 ∈ SNoR v
L6405
Hypothesis H18 : x2 ∈ SNoL y
L6406
Theorem. (Conj_mul_SNo_prop_1__62__17)
(x * w + u * x2) < x * x2 + u * w β†’ (z * y + x * w + u * v) < u * y + x * v + z * w
In Proofgold the corresponding term root is 494941... and proposition id is ebde20...
Proof:
Proof not loaded.
End of Section Conj_mul_SNo_prop_1__62__17
Beginning of Section Conj_mul_SNo_prop_1__65__5
L6412
Variable x : set
(*** Conj_mul_SNo_prop_1__65__5 TMK53EJuJx72mEV2YTRSPeg6j1bf2Jwe6jY bounty of about 25 bars ***)
L6413
Variable y : set
L6414
Variable z : set
L6415
Variable w : set
L6416
Variable u : set
L6417
Variable v : set
L6418
Hypothesis H0 : SNo y
L6419
Hypothesis H1 : (βˆ€x2 : set, x2 ∈ SNoS_ (SNoLev y) β†’ (βˆ€P : prop, (SNo (x * x2) β†’ (βˆ€y2 : set, y2 ∈ SNoL x β†’ (βˆ€z2 : set, z2 ∈ SNoL x2 β†’ (y2 * x2 + x * z2) < x * x2 + y2 * z2)) β†’ (βˆ€y2 : set, y2 ∈ SNoR x β†’ (βˆ€z2 : set, z2 ∈ SNoR x2 β†’ (y2 * x2 + x * z2) < x * x2 + y2 * z2)) β†’ (βˆ€y2 : set, y2 ∈ SNoL x β†’ (βˆ€z2 : set, z2 ∈ SNoR x2 β†’ (x * x2 + y2 * z2) < y2 * x2 + x * z2)) β†’ (βˆ€y2 : set, y2 ∈ SNoR x β†’ (βˆ€z2 : set, z2 ∈ SNoL x2 β†’ (x * x2 + y2 * z2) < y2 * x2 + x * z2)) β†’ P) β†’ P))
L6420
Hypothesis H2 : (βˆ€x2 : set, x2 ∈ SNoL x β†’ (βˆ€y2 : set, SNo y2 β†’ SNo (x2 * y2)))
L6421
Hypothesis H3 : (βˆ€x2 : set, x2 ∈ SNoR x β†’ (βˆ€y2 : set, SNo y2 β†’ SNo (x2 * y2)))
L6422
Hypothesis H4 : (βˆ€x2 : set, x2 ∈ SNoL y β†’ SNo (x * x2))
L6423
Hypothesis H6 : w ∈ SNoL y
L6424
Hypothesis H7 : SNo w
L6425
Hypothesis H8 : SNoLev w ∈ SNoLev y
L6426
Hypothesis H9 : w < y
L6427
Hypothesis H10 : u ∈ SNoR x
L6428
Hypothesis H11 : v ∈ SNoL y
L6429
Hypothesis H12 : SNo v
L6430
Hypothesis H13 : SNoLev v ∈ SNoLev y
L6431
Hypothesis H14 : v < y
L6432
Hypothesis H15 : SNo (z * y)
L6433
Hypothesis H16 : SNo (x * w)
L6434
Hypothesis H17 : SNo (z * w)
L6435
Hypothesis H18 : SNo (u * y)
L6436
Hypothesis H19 : SNo (x * v)
L6437
Hypothesis H20 : SNo (u * v)
L6438
Hypothesis H21 : SNo (z * v)
L6439
Hypothesis H22 : SNo (u * w)
L6440
Hypothesis H23 : (z * y + u * w) < u * y + z * w
L6441
Hypothesis H24 : (z * y + u * v) < u * y + z * v
L6442
Theorem. (Conj_mul_SNo_prop_1__65__5)
((x * w + u * v) < u * w + x * v β†’ (z * y + x * w + u * v) < u * y + x * v + z * w) β†’ (z * y + x * w + u * v) < u * y + x * v + z * w
In Proofgold the corresponding term root is 96b45a... and proposition id is 3073c1...
Proof:
Proof not loaded.
End of Section Conj_mul_SNo_prop_1__65__5
Beginning of Section Conj_mul_SNo_prop_1__65__15
L6448
Variable x : set
(*** Conj_mul_SNo_prop_1__65__15 TMQHvDANbkyAzj79EqP3qYr6o2SUdSdWb2g bounty of about 25 bars ***)
L6449
Variable y : set
L6450
Variable z : set
L6451
Variable w : set
L6452
Variable u : set
L6453
Variable v : set
L6454
Hypothesis H0 : SNo y
L6455
Hypothesis H1 : (βˆ€x2 : set, x2 ∈ SNoS_ (SNoLev y) β†’ (βˆ€P : prop, (SNo (x * x2) β†’ (βˆ€y2 : set, y2 ∈ SNoL x β†’ (βˆ€z2 : set, z2 ∈ SNoL x2 β†’ (y2 * x2 + x * z2) < x * x2 + y2 * z2)) β†’ (βˆ€y2 : set, y2 ∈ SNoR x β†’ (βˆ€z2 : set, z2 ∈ SNoR x2 β†’ (y2 * x2 + x * z2) < x * x2 + y2 * z2)) β†’ (βˆ€y2 : set, y2 ∈ SNoL x β†’ (βˆ€z2 : set, z2 ∈ SNoR x2 β†’ (x * x2 + y2 * z2) < y2 * x2 + x * z2)) β†’ (βˆ€y2 : set, y2 ∈ SNoR x β†’ (βˆ€z2 : set, z2 ∈ SNoL x2 β†’ (x * x2 + y2 * z2) < y2 * x2 + x * z2)) β†’ P) β†’ P))
L6456
Hypothesis H2 : (βˆ€x2 : set, x2 ∈ SNoL x β†’ (βˆ€y2 : set, SNo y2 β†’ SNo (x2 * y2)))
L6457
Hypothesis H3 : (βˆ€x2 : set, x2 ∈ SNoR x β†’ (βˆ€y2 : set, SNo y2 β†’ SNo (x2 * y2)))
L6458
Hypothesis H4 : (βˆ€x2 : set, x2 ∈ SNoL y β†’ SNo (x * x2))
L6459
Hypothesis H5 : z ∈ SNoL x
L6460
Hypothesis H6 : w ∈ SNoL y
L6461
Hypothesis H7 : SNo w
L6462
Hypothesis H8 : SNoLev w ∈ SNoLev y
L6463
Hypothesis H9 : w < y
L6464
Hypothesis H10 : u ∈ SNoR x
L6465
Hypothesis H11 : v ∈ SNoL y
L6466
Hypothesis H12 : SNo v
L6467
Hypothesis H13 : SNoLev v ∈ SNoLev y
L6468
Hypothesis H14 : v < y
L6469
Hypothesis H16 : SNo (x * w)
L6470
Hypothesis H17 : SNo (z * w)
L6471
Hypothesis H18 : SNo (u * y)
L6472
Hypothesis H19 : SNo (x * v)
L6473
Hypothesis H20 : SNo (u * v)
L6474
Hypothesis H21 : SNo (z * v)
L6475
Hypothesis H22 : SNo (u * w)
L6476
Hypothesis H23 : (z * y + u * w) < u * y + z * w
L6477
Hypothesis H24 : (z * y + u * v) < u * y + z * v
L6478
Theorem. (Conj_mul_SNo_prop_1__65__15)
((x * w + u * v) < u * w + x * v β†’ (z * y + x * w + u * v) < u * y + x * v + z * w) β†’ (z * y + x * w + u * v) < u * y + x * v + z * w
In Proofgold the corresponding term root is 232f85... and proposition id is f69d6f...
Proof:
Proof not loaded.
End of Section Conj_mul_SNo_prop_1__65__15
Beginning of Section Conj_mul_SNo_prop_1__66__7
L6484
Variable x : set
(*** Conj_mul_SNo_prop_1__66__7 TMZNugXfyHChGQpdxDctKgrQ9PrHUjsFcwD bounty of about 25 bars ***)
L6485
Variable y : set
L6486
Variable z : set
L6487
Variable w : set
L6488
Variable u : set
L6489
Variable v : set
L6490
Hypothesis H0 : SNo y
L6491
Hypothesis H1 : (βˆ€x2 : set, x2 ∈ SNoS_ (SNoLev y) β†’ (βˆ€P : prop, (SNo (x * x2) β†’ (βˆ€y2 : set, y2 ∈ SNoL x β†’ (βˆ€z2 : set, z2 ∈ SNoL x2 β†’ (y2 * x2 + x * z2) < x * x2 + y2 * z2)) β†’ (βˆ€y2 : set, y2 ∈ SNoR x β†’ (βˆ€z2 : set, z2 ∈ SNoR x2 β†’ (y2 * x2 + x * z2) < x * x2 + y2 * z2)) β†’ (βˆ€y2 : set, y2 ∈ SNoL x β†’ (βˆ€z2 : set, z2 ∈ SNoR x2 β†’ (x * x2 + y2 * z2) < y2 * x2 + x * z2)) β†’ (βˆ€y2 : set, y2 ∈ SNoR x β†’ (βˆ€z2 : set, z2 ∈ SNoL x2 β†’ (x * x2 + y2 * z2) < y2 * x2 + x * z2)) β†’ P) β†’ P))
L6492
Hypothesis H2 : (βˆ€x2 : set, x2 ∈ SNoL x β†’ (βˆ€y2 : set, SNo y2 β†’ SNo (x2 * y2)))
L6493
Hypothesis H3 : (βˆ€x2 : set, x2 ∈ SNoR x β†’ (βˆ€y2 : set, SNo y2 β†’ SNo (x2 * y2)))
L6494
Hypothesis H4 : (βˆ€x2 : set, x2 ∈ SNoL y β†’ SNo (x * x2))
L6495
Hypothesis H5 : z ∈ SNoL x
L6496
Hypothesis H6 : w ∈ SNoL y
L6497
Hypothesis H8 : SNoLev w ∈ SNoLev y
L6498
Hypothesis H9 : w < y
L6499
Hypothesis H10 : u ∈ SNoR x
L6500
Hypothesis H11 : v ∈ SNoL y
L6501
Hypothesis H12 : SNo v
L6502
Hypothesis H13 : SNoLev v ∈ SNoLev y
L6503
Hypothesis H14 : v < y
L6504
Hypothesis H15 : SNo (z * y)
L6505
Hypothesis H16 : SNo (x * w)
L6506
Hypothesis H17 : SNo (z * w)
L6507
Hypothesis H18 : SNo (u * y)
L6508
Hypothesis H19 : SNo (x * v)
L6509
Hypothesis H20 : SNo (u * v)
L6510
Hypothesis H21 : SNo (z * v)
L6511
Hypothesis H22 : SNo (u * w)
L6512
Hypothesis H23 : (βˆ€x2 : set, x2 ∈ SNoL y β†’ (z * y + u * x2) < u * y + z * x2)
L6513
Hypothesis H24 : (z * y + u * w) < u * y + z * w
L6514
Theorem. (Conj_mul_SNo_prop_1__66__7)
(z * y + u * v) < u * y + z * v β†’ (z * y + x * w + u * v) < u * y + x * v + z * w
In Proofgold the corresponding term root is b9cd84... and proposition id is e5789f...
Proof:
Proof not loaded.
End of Section Conj_mul_SNo_prop_1__66__7
Beginning of Section Conj_mul_SNo_prop_1__67__5
L6520
Variable x : set
(*** Conj_mul_SNo_prop_1__67__5 TMN87rRJmRWisZVw9bktEpDqpRRpDchYKTb bounty of about 25 bars ***)
L6521
Variable y : set
L6522
Variable z : set
L6523
Variable w : set
L6524
Variable u : set
L6525
Variable v : set
L6526
Hypothesis H0 : SNo y
L6527
Hypothesis H1 : (βˆ€x2 : set, x2 ∈ SNoS_ (SNoLev y) β†’ (βˆ€P : prop, (SNo (x * x2) β†’ (βˆ€y2 : set, y2 ∈ SNoL x β†’ (βˆ€z2 : set, z2 ∈ SNoL x2 β†’ (y2 * x2 + x * z2) < x * x2 + y2 * z2)) β†’ (βˆ€y2 : set, y2 ∈ SNoR x β†’ (βˆ€z2 : set, z2 ∈ SNoR x2 β†’ (y2 * x2 + x * z2) < x * x2 + y2 * z2)) β†’ (βˆ€y2 : set, y2 ∈ SNoL x β†’ (βˆ€z2 : set, z2 ∈ SNoR x2 β†’ (x * x2 + y2 * z2) < y2 * x2 + x * z2)) β†’ (βˆ€y2 : set, y2 ∈ SNoR x β†’ (βˆ€z2 : set, z2 ∈ SNoL x2 β†’ (x * x2 + y2 * z2) < y2 * x2 + x * z2)) β†’ P) β†’ P))
L6528
Hypothesis H2 : (βˆ€x2 : set, x2 ∈ SNoL x β†’ (βˆ€y2 : set, SNo y2 β†’ SNo (x2 * y2)))
L6529
Hypothesis H3 : (βˆ€x2 : set, x2 ∈ SNoR x β†’ (βˆ€y2 : set, SNo y2 β†’ SNo (x2 * y2)))
L6530
Hypothesis H4 : (βˆ€x2 : set, x2 ∈ SNoL y β†’ SNo (x * x2))
L6531
Hypothesis H6 : w ∈ SNoL y
L6532
Hypothesis H7 : SNo w
L6533
Hypothesis H8 : SNoLev w ∈ SNoLev y
L6534
Hypothesis H9 : w < y
L6535
Hypothesis H10 : u ∈ SNoR x
L6536
Hypothesis H11 : v ∈ SNoL y
L6537
Hypothesis H12 : SNo v
L6538
Hypothesis H13 : SNoLev v ∈ SNoLev y
L6539
Hypothesis H14 : v < y
L6540
Hypothesis H15 : SNo (z * y)
L6541
Hypothesis H16 : SNo (x * w)
L6542
Hypothesis H17 : SNo (z * w)
L6543
Hypothesis H18 : SNo (u * y)
L6544
Hypothesis H19 : SNo (x * v)
L6545
Hypothesis H20 : SNo (u * v)
L6546
Hypothesis H21 : SNo (z * v)
L6547
Hypothesis H22 : SNo (u * w)
L6548
Hypothesis H23 : (βˆ€x2 : set, x2 ∈ SNoL y β†’ (z * y + u * x2) < u * y + z * x2)
L6549
Theorem. (Conj_mul_SNo_prop_1__67__5)
(z * y + u * w) < u * y + z * w β†’ (z * y + x * w + u * v) < u * y + x * v + z * w
In Proofgold the corresponding term root is ed71d4... and proposition id is 907c60...
Proof:
Proof not loaded.
End of Section Conj_mul_SNo_prop_1__67__5
Beginning of Section Conj_mul_SNo_prop_1__68__13
L6555
Variable x : set
(*** Conj_mul_SNo_prop_1__68__13 TMLZku422zzc6voMUCnt6roL8PiuofkncqC bounty of about 25 bars ***)
L6556
Variable y : set
L6557
Variable z : set
L6558
Variable w : set
L6559
Variable u : set
L6560
Variable v : set
L6561
Hypothesis H0 : SNo x
L6562
Hypothesis H1 : (βˆ€x2 : set, x2 ∈ SNoS_ (SNoLev x) β†’ (βˆ€y2 : set, SNo y2 β†’ (βˆ€P : prop, (SNo (x2 * y2) β†’ (βˆ€z2 : set, z2 ∈ SNoL x2 β†’ (βˆ€w2 : set, w2 ∈ SNoL y2 β†’ (z2 * y2 + x2 * w2) < x2 * y2 + z2 * w2)) β†’ (βˆ€z2 : set, z2 ∈ SNoR x2 β†’ (βˆ€w2 : set, w2 ∈ SNoR y2 β†’ (z2 * y2 + x2 * w2) < x2 * y2 + z2 * w2)) β†’ (βˆ€z2 : set, z2 ∈ SNoL x2 β†’ (βˆ€w2 : set, w2 ∈ SNoR y2 β†’ (x2 * y2 + z2 * w2) < z2 * y2 + x2 * w2)) β†’ (βˆ€z2 : set, z2 ∈ SNoR x2 β†’ (βˆ€w2 : set, w2 ∈ SNoL y2 β†’ (x2 * y2 + z2 * w2) < z2 * y2 + x2 * w2)) β†’ P) β†’ P)))
L6563
Hypothesis H2 : SNo y
L6564
Hypothesis H3 : (βˆ€x2 : set, x2 ∈ SNoS_ (SNoLev y) β†’ (βˆ€P : prop, (SNo (x * x2) β†’ (βˆ€y2 : set, y2 ∈ SNoL x β†’ (βˆ€z2 : set, z2 ∈ SNoL x2 β†’ (y2 * x2 + x * z2) < x * x2 + y2 * z2)) β†’ (βˆ€y2 : set, y2 ∈ SNoR x β†’ (βˆ€z2 : set, z2 ∈ SNoR x2 β†’ (y2 * x2 + x * z2) < x * x2 + y2 * z2)) β†’ (βˆ€y2 : set, y2 ∈ SNoL x β†’ (βˆ€z2 : set, z2 ∈ SNoR x2 β†’ (x * x2 + y2 * z2) < y2 * x2 + x * z2)) β†’ (βˆ€y2 : set, y2 ∈ SNoR x β†’ (βˆ€z2 : set, z2 ∈ SNoL x2 β†’ (x * x2 + y2 * z2) < y2 * x2 + x * z2)) β†’ P) β†’ P))
L6565
Hypothesis H4 : (βˆ€x2 : set, x2 ∈ SNoL x β†’ (βˆ€y2 : set, SNo y2 β†’ SNo (x2 * y2)))
L6566
Hypothesis H5 : (βˆ€x2 : set, x2 ∈ SNoR x β†’ (βˆ€y2 : set, SNo y2 β†’ SNo (x2 * y2)))
L6567
Hypothesis H6 : (βˆ€x2 : set, x2 ∈ SNoL y β†’ SNo (x * x2))
L6568
Hypothesis H7 : z ∈ SNoL x
L6569
Hypothesis H8 : w ∈ SNoL y
L6570
Hypothesis H9 : SNo z
L6571
Hypothesis H10 : SNoLev z ∈ SNoLev x
L6572
Hypothesis H11 : SNo w
L6573
Hypothesis H12 : SNoLev w ∈ SNoLev y
L6574
Hypothesis H14 : u ∈ SNoR x
L6575
Hypothesis H15 : v ∈ SNoL y
L6576
Hypothesis H16 : SNo u
L6577
Hypothesis H17 : SNo v
L6578
Hypothesis H18 : SNoLev v ∈ SNoLev y
L6579
Hypothesis H19 : v < y
L6580
Hypothesis H20 : SNo (z * y)
L6581
Hypothesis H21 : SNo (x * w)
L6582
Hypothesis H22 : SNo (z * w)
L6583
Hypothesis H23 : SNo (u * y)
L6584
Hypothesis H24 : SNo (x * v)
L6585
Hypothesis H25 : SNo (u * v)
L6586
Hypothesis H26 : SNo (z * v)
L6587
Hypothesis H27 : SNo (u * w)
L6588
Hypothesis H28 : z < u
L6589
Theorem. (Conj_mul_SNo_prop_1__68__13)
(βˆ€x2 : set, x2 ∈ SNoL y β†’ (z * y + u * x2) < u * y + z * x2) β†’ (z * y + x * w + u * v) < u * y + x * v + z * w
In Proofgold the corresponding term root is 7ad490... and proposition id is 45a668...
Proof:
Proof not loaded.
End of Section Conj_mul_SNo_prop_1__68__13
Beginning of Section Conj_mul_SNo_prop_1__69__0
L6595
Variable x : set
(*** Conj_mul_SNo_prop_1__69__0 TMQqZyabXML6eBsMAdGKPVNQytW287r9FvS bounty of about 25 bars ***)
L6596
Variable y : set
L6597
Variable z : set
L6598
Variable w : set
L6599
Variable u : set
L6600
Variable v : set
L6601
Hypothesis H1 : (βˆ€x2 : set, x2 ∈ SNoS_ (SNoLev x) β†’ (βˆ€y2 : set, SNo y2 β†’ (βˆ€P : prop, (SNo (x2 * y2) β†’ (βˆ€z2 : set, z2 ∈ SNoL x2 β†’ (βˆ€w2 : set, w2 ∈ SNoL y2 β†’ (z2 * y2 + x2 * w2) < x2 * y2 + z2 * w2)) β†’ (βˆ€z2 : set, z2 ∈ SNoR x2 β†’ (βˆ€w2 : set, w2 ∈ SNoR y2 β†’ (z2 * y2 + x2 * w2) < x2 * y2 + z2 * w2)) β†’ (βˆ€z2 : set, z2 ∈ SNoL x2 β†’ (βˆ€w2 : set, w2 ∈ SNoR y2 β†’ (x2 * y2 + z2 * w2) < z2 * y2 + x2 * w2)) β†’ (βˆ€z2 : set, z2 ∈ SNoR x2 β†’ (βˆ€w2 : set, w2 ∈ SNoL y2 β†’ (x2 * y2 + z2 * w2) < z2 * y2 + x2 * w2)) β†’ P) β†’ P)))
L6602
Hypothesis H2 : SNo y
L6603
Hypothesis H3 : (βˆ€x2 : set, x2 ∈ SNoS_ (SNoLev y) β†’ (βˆ€P : prop, (SNo (x * x2) β†’ (βˆ€y2 : set, y2 ∈ SNoL x β†’ (βˆ€z2 : set, z2 ∈ SNoL x2 β†’ (y2 * x2 + x * z2) < x * x2 + y2 * z2)) β†’ (βˆ€y2 : set, y2 ∈ SNoR x β†’ (βˆ€z2 : set, z2 ∈ SNoR x2 β†’ (y2 * x2 + x * z2) < x * x2 + y2 * z2)) β†’ (βˆ€y2 : set, y2 ∈ SNoL x β†’ (βˆ€z2 : set, z2 ∈ SNoR x2 β†’ (x * x2 + y2 * z2) < y2 * x2 + x * z2)) β†’ (βˆ€y2 : set, y2 ∈ SNoR x β†’ (βˆ€z2 : set, z2 ∈ SNoL x2 β†’ (x * x2 + y2 * z2) < y2 * x2 + x * z2)) β†’ P) β†’ P))
L6604
Hypothesis H4 : (βˆ€x2 : set, x2 ∈ SNoL x β†’ (βˆ€y2 : set, SNo y2 β†’ SNo (x2 * y2)))
L6605
Hypothesis H5 : (βˆ€x2 : set, x2 ∈ SNoR x β†’ (βˆ€y2 : set, SNo y2 β†’ SNo (x2 * y2)))
L6606
Hypothesis H6 : (βˆ€x2 : set, x2 ∈ SNoL y β†’ SNo (x * x2))
L6607
Hypothesis H7 : z ∈ SNoL x
L6608
Hypothesis H8 : w ∈ SNoL y
L6609
Hypothesis H9 : SNo z
L6610
Hypothesis H10 : SNoLev z ∈ SNoLev x
L6611
Hypothesis H11 : z < x
L6612
Hypothesis H12 : SNo w
L6613
Hypothesis H13 : SNoLev w ∈ SNoLev y
L6614
Hypothesis H14 : w < y
L6615
Hypothesis H15 : u ∈ SNoR x
L6616
Hypothesis H16 : v ∈ SNoL y
L6617
Hypothesis H17 : SNo u
L6618
Hypothesis H18 : x < u
L6619
Hypothesis H19 : SNo v
L6620
Hypothesis H20 : SNoLev v ∈ SNoLev y
L6621
Hypothesis H21 : v < y
L6622
Hypothesis H22 : SNo (z * y)
L6623
Hypothesis H23 : SNo (x * w)
L6624
Hypothesis H24 : SNo (z * w)
L6625
Hypothesis H25 : SNo (u * y)
L6626
Hypothesis H26 : SNo (x * v)
L6627
Hypothesis H27 : SNo (u * v)
L6628
Hypothesis H28 : SNo (z * v)
L6629
Hypothesis H29 : SNo (u * w)
L6630
Theorem. (Conj_mul_SNo_prop_1__69__0)
z < u β†’ (z * y + x * w + u * v) < u * y + x * v + z * w
In Proofgold the corresponding term root is 8bc761... and proposition id is d8ff17...
Proof:
Proof not loaded.
End of Section Conj_mul_SNo_prop_1__69__0
Beginning of Section Conj_mul_SNo_prop_1__71__10
L6636
Variable x : set
(*** Conj_mul_SNo_prop_1__71__10 TMXTiSf4Xp2S8PCJpUC6Y6fYHKrGvPJiJLb bounty of about 25 bars ***)
L6637
Variable y : set
L6638
Variable z : set
L6639
Variable w : set
L6640
Variable u : set
L6641
Variable v : set
L6642
Hypothesis H0 : SNo y
L6643
Hypothesis H1 : (βˆ€x2 : set, x2 ∈ SNoS_ (SNoLev y) β†’ (βˆ€P : prop, (SNo (x * x2) β†’ (βˆ€y2 : set, y2 ∈ SNoL x β†’ (βˆ€z2 : set, z2 ∈ SNoL x2 β†’ (y2 * x2 + x * z2) < x * x2 + y2 * z2)) β†’ (βˆ€y2 : set, y2 ∈ SNoR x β†’ (βˆ€z2 : set, z2 ∈ SNoR x2 β†’ (y2 * x2 + x * z2) < x * x2 + y2 * z2)) β†’ (βˆ€y2 : set, y2 ∈ SNoL x β†’ (βˆ€z2 : set, z2 ∈ SNoR x2 β†’ (x * x2 + y2 * z2) < y2 * x2 + x * z2)) β†’ (βˆ€y2 : set, y2 ∈ SNoR x β†’ (βˆ€z2 : set, z2 ∈ SNoL x2 β†’ (x * x2 + y2 * z2) < y2 * x2 + x * z2)) β†’ P) β†’ P))
L6644
Hypothesis H2 : z ∈ SNoL y
L6645
Hypothesis H3 : w ∈ SNoR y
L6646
Hypothesis H4 : SNo (x * z)
L6647
Hypothesis H5 : SNo (x * w)
L6648
Hypothesis H6 : u ∈ SNoL x
L6649
Hypothesis H7 : SNo (u * z)
L6650
Hypothesis H8 : SNo (u * w)
L6651
Hypothesis H9 : v ∈ SNoL w
L6652
Hypothesis H11 : SNo (u * v)
L6653
Hypothesis H12 : SNo (x * v)
L6654
Theorem. (Conj_mul_SNo_prop_1__71__10)
(u * w + x * v) < x * w + u * v β†’ (x * z + u * w) < u * z + x * w
In Proofgold the corresponding term root is 5f8639... and proposition id is 5892cc...
Proof:
Proof not loaded.
End of Section Conj_mul_SNo_prop_1__71__10
Beginning of Section Conj_mul_SNo_prop_1__73__4
L6660
Variable x : set
(*** Conj_mul_SNo_prop_1__73__4 TMdakstiETEwm1MNzP56MyibSAGLpXVieRH bounty of about 25 bars ***)
L6661
Variable y : set
L6662
Variable z : set
L6663
Variable w : set
L6664
Variable u : set
L6665
Variable v : set
L6666
Hypothesis H0 : SNo y
L6667
Hypothesis H1 : (βˆ€x2 : set, x2 ∈ SNoS_ (SNoLev y) β†’ (βˆ€P : prop, (SNo (x * x2) β†’ (βˆ€y2 : set, y2 ∈ SNoL x β†’ (βˆ€z2 : set, z2 ∈ SNoL x2 β†’ (y2 * x2 + x * z2) < x * x2 + y2 * z2)) β†’ (βˆ€y2 : set, y2 ∈ SNoR x β†’ (βˆ€z2 : set, z2 ∈ SNoR x2 β†’ (y2 * x2 + x * z2) < x * x2 + y2 * z2)) β†’ (βˆ€y2 : set, y2 ∈ SNoL x β†’ (βˆ€z2 : set, z2 ∈ SNoR x2 β†’ (x * x2 + y2 * z2) < y2 * x2 + x * z2)) β†’ (βˆ€y2 : set, y2 ∈ SNoR x β†’ (βˆ€z2 : set, z2 ∈ SNoL x2 β†’ (x * x2 + y2 * z2) < y2 * x2 + x * z2)) β†’ P) β†’ P))
L6668
Hypothesis H2 : (βˆ€x2 : set, x2 ∈ SNoL x β†’ (βˆ€y2 : set, SNo y2 β†’ SNo (x2 * y2)))
L6669
Hypothesis H3 : z ∈ SNoL y
L6670
Hypothesis H5 : SNo (x * z)
L6671
Hypothesis H6 : SNo (x * w)
L6672
Hypothesis H7 : u ∈ SNoL x
L6673
Hypothesis H8 : SNo (u * z)
L6674
Hypothesis H9 : SNo (u * w)
L6675
Hypothesis H10 : v ∈ SNoL w
L6676
Hypothesis H11 : v ∈ SNoR z
L6677
Hypothesis H12 : SNo v
L6678
Hypothesis H13 : v ∈ SNoS_ (SNoLev y)
L6679
Theorem. (Conj_mul_SNo_prop_1__73__4)
SNo (u * v) β†’ (x * z + u * w) < u * z + x * w
In Proofgold the corresponding term root is 633d35... and proposition id is df61eb...
Proof:
Proof not loaded.
End of Section Conj_mul_SNo_prop_1__73__4
Beginning of Section Conj_mul_SNo_prop_1__76__0
L6685
Variable x : set
(*** Conj_mul_SNo_prop_1__76__0 TMTydfwLB2avggdVUwgi3qZhQwwSPzogzHg bounty of about 25 bars ***)
L6686
Variable y : set
L6687
Variable z : set
L6688
Variable w : set
L6689
Variable u : set
L6690
Hypothesis H1 : (βˆ€v : set, v ∈ SNoS_ (SNoLev y) β†’ (βˆ€P : prop, (SNo (x * v) β†’ (βˆ€x2 : set, x2 ∈ SNoL x β†’ (βˆ€y2 : set, y2 ∈ SNoL v β†’ (x2 * v + x * y2) < x * v + x2 * y2)) β†’ (βˆ€x2 : set, x2 ∈ SNoR x β†’ (βˆ€y2 : set, y2 ∈ SNoR v β†’ (x2 * v + x * y2) < x * v + x2 * y2)) β†’ (βˆ€x2 : set, x2 ∈ SNoL x β†’ (βˆ€y2 : set, y2 ∈ SNoR v β†’ (x * v + x2 * y2) < x2 * v + x * y2)) β†’ (βˆ€x2 : set, x2 ∈ SNoR x β†’ (βˆ€y2 : set, y2 ∈ SNoL v β†’ (x * v + x2 * y2) < x2 * v + x * y2)) β†’ P) β†’ P))
L6691
Hypothesis H2 : (βˆ€v : set, v ∈ SNoL x β†’ (βˆ€x2 : set, SNo x2 β†’ SNo (v * x2)))
L6692
Hypothesis H3 : z ∈ SNoL y
L6693
Hypothesis H4 : SNo z
L6694
Hypothesis H5 : SNoLev z ∈ SNoLev y
L6695
Hypothesis H6 : w ∈ SNoR y
L6696
Hypothesis H7 : SNo w
L6697
Hypothesis H8 : SNo (x * z)
L6698
Hypothesis H9 : SNo (x * w)
L6699
Hypothesis H10 : z < w
L6700
Hypothesis H11 : u ∈ SNoL x
L6701
Theorem. (Conj_mul_SNo_prop_1__76__0)
SNo (u * z) β†’ (x * z + u * w) < u * z + x * w
In Proofgold the corresponding term root is bf5383... and proposition id is 420ee4...
Proof:
Proof not loaded.
End of Section Conj_mul_SNo_prop_1__76__0
Beginning of Section Conj_mul_SNo_prop_1__78__0
L6707
Variable x : set
(*** Conj_mul_SNo_prop_1__78__0 TMP2eeazDWma3JpH27FpCHEBvmzEc9tbMno bounty of about 25 bars ***)
L6708
Variable y : set
L6709
Variable z : set
L6710
Variable w : set
L6711
Variable u : set
L6712
Variable v : set
L6713
Variable x2 : set
L6714
Hypothesis H1 : (βˆ€y2 : set, y2 ∈ SNoS_ (SNoLev x) β†’ (βˆ€z2 : set, SNo z2 β†’ (βˆ€P : prop, (SNo (y2 * z2) β†’ (βˆ€w2 : set, w2 ∈ SNoL y2 β†’ (βˆ€u2 : set, u2 ∈ SNoL z2 β†’ (w2 * z2 + y2 * u2) < y2 * z2 + w2 * u2)) β†’ (βˆ€w2 : set, w2 ∈ SNoR y2 β†’ (βˆ€u2 : set, u2 ∈ SNoR z2 β†’ (w2 * z2 + y2 * u2) < y2 * z2 + w2 * u2)) β†’ (βˆ€w2 : set, w2 ∈ SNoL y2 β†’ (βˆ€u2 : set, u2 ∈ SNoR z2 β†’ (y2 * z2 + w2 * u2) < w2 * z2 + y2 * u2)) β†’ (βˆ€w2 : set, w2 ∈ SNoR y2 β†’ (βˆ€u2 : set, u2 ∈ SNoL z2 β†’ (y2 * z2 + w2 * u2) < w2 * z2 + y2 * u2)) β†’ P) β†’ P)))
L6715
Hypothesis H2 : SNo y
L6716
Hypothesis H3 : (βˆ€y2 : set, y2 ∈ SNoL x β†’ (βˆ€z2 : set, SNo z2 β†’ SNo (y2 * z2)))
L6717
Hypothesis H4 : (βˆ€y2 : set, y2 ∈ SNoL x β†’ SNo (y2 * y))
L6718
Hypothesis H5 : z ∈ SNoL x
L6719
Hypothesis H6 : w ∈ SNoL y
L6720
Hypothesis H7 : SNo w
L6721
Hypothesis H8 : SNo u
L6722
Hypothesis H9 : SNoLev u ∈ SNoLev x
L6723
Hypothesis H10 : u < x
L6724
Hypothesis H11 : SNo (z * y)
L6725
Hypothesis H12 : SNo (x * w)
L6726
Hypothesis H13 : SNo (z * w)
L6727
Hypothesis H14 : SNo (u * y)
L6728
Hypothesis H15 : SNo (x * v)
L6729
Hypothesis H16 : SNo (u * v)
L6730
Hypothesis H17 : SNo (u * w)
L6731
Hypothesis H18 : (x * w + u * v) < u * w + x * v
L6732
Hypothesis H19 : x2 ∈ SNoR z
L6733
Hypothesis H20 : x2 ∈ SNoL u
L6734
Hypothesis H21 : (x2 * y + u * w) < u * y + x2 * w
L6735
Theorem. (Conj_mul_SNo_prop_1__78__0)
(z * y + x2 * w) < x2 * y + z * w β†’ (z * y + x * w + u * v) < u * y + x * v + z * w
In Proofgold the corresponding term root is 4c5b91... and proposition id is 259e85...
Proof:
Proof not loaded.
End of Section Conj_mul_SNo_prop_1__78__0
Beginning of Section Conj_mul_SNo_prop_1__81__2
L6741
Variable x : set
(*** Conj_mul_SNo_prop_1__81__2 TMQwvgY6pLBzKZb4QrM29iTGXMF2dBvzELW bounty of about 25 bars ***)
L6742
Variable y : set
L6743
Variable z : set
L6744
Variable w : set
L6745
Variable u : set
L6746
Variable v : set
L6747
Variable x2 : set
L6748
Hypothesis H0 : SNo x
L6749
Hypothesis H1 : (βˆ€y2 : set, y2 ∈ SNoS_ (SNoLev x) β†’ (βˆ€z2 : set, SNo z2 β†’ (βˆ€P : prop, (SNo (y2 * z2) β†’ (βˆ€w2 : set, w2 ∈ SNoL y2 β†’ (βˆ€u2 : set, u2 ∈ SNoL z2 β†’ (w2 * z2 + y2 * u2) < y2 * z2 + w2 * u2)) β†’ (βˆ€w2 : set, w2 ∈ SNoR y2 β†’ (βˆ€u2 : set, u2 ∈ SNoR z2 β†’ (w2 * z2 + y2 * u2) < y2 * z2 + w2 * u2)) β†’ (βˆ€w2 : set, w2 ∈ SNoL y2 β†’ (βˆ€u2 : set, u2 ∈ SNoR z2 β†’ (y2 * z2 + w2 * u2) < w2 * z2 + y2 * u2)) β†’ (βˆ€w2 : set, w2 ∈ SNoR y2 β†’ (βˆ€u2 : set, u2 ∈ SNoL z2 β†’ (y2 * z2 + w2 * u2) < w2 * z2 + y2 * u2)) β†’ P) β†’ P)))
L6750
Hypothesis H3 : (βˆ€y2 : set, y2 ∈ SNoL x β†’ (βˆ€z2 : set, SNo z2 β†’ SNo (y2 * z2)))
L6751
Hypothesis H4 : (βˆ€y2 : set, y2 ∈ SNoL x β†’ SNo (y2 * y))
L6752
Hypothesis H5 : SNo z
L6753
Hypothesis H6 : SNoLev z ∈ SNoLev x
L6754
Hypothesis H7 : z < x
L6755
Hypothesis H8 : u ∈ SNoL x
L6756
Hypothesis H9 : v ∈ SNoR y
L6757
Hypothesis H10 : SNo v
L6758
Hypothesis H11 : SNo (z * y)
L6759
Hypothesis H12 : SNo (x * w)
L6760
Hypothesis H13 : SNo (z * w)
L6761
Hypothesis H14 : SNo (u * y)
L6762
Hypothesis H15 : SNo (x * v)
L6763
Hypothesis H16 : SNo (u * v)
L6764
Hypothesis H17 : SNo (z * v)
L6765
Hypothesis H18 : (x * w + z * v) < z * w + x * v
L6766
Hypothesis H19 : x2 ∈ SNoL z
L6767
Hypothesis H20 : x2 ∈ SNoR u
L6768
Hypothesis H21 : (z * y + x2 * v) < x2 * y + z * v
L6769
Theorem. (Conj_mul_SNo_prop_1__81__2)
(x2 * y + u * v) < u * y + x2 * v β†’ (z * y + x * w + u * v) < u * y + x * v + z * w
In Proofgold the corresponding term root is 6e288a... and proposition id is ed5547...
Proof:
Proof not loaded.
End of Section Conj_mul_SNo_prop_1__81__2
Beginning of Section Conj_mul_SNo_prop_1__81__3
L6775
Variable x : set
(*** Conj_mul_SNo_prop_1__81__3 TMc6hoJ3TapeX6iyJamS8JUpRjTYpMqzVf8 bounty of about 25 bars ***)
L6776
Variable y : set
L6777
Variable z : set
L6778
Variable w : set
L6779
Variable u : set
L6780
Variable v : set
L6781
Variable x2 : set
L6782
Hypothesis H0 : SNo x
L6783
Hypothesis H1 : (βˆ€y2 : set, y2 ∈ SNoS_ (SNoLev x) β†’ (βˆ€z2 : set, SNo z2 β†’ (βˆ€P : prop, (SNo (y2 * z2) β†’ (βˆ€w2 : set, w2 ∈ SNoL y2 β†’ (βˆ€u2 : set, u2 ∈ SNoL z2 β†’ (w2 * z2 + y2 * u2) < y2 * z2 + w2 * u2)) β†’ (βˆ€w2 : set, w2 ∈ SNoR y2 β†’ (βˆ€u2 : set, u2 ∈ SNoR z2 β†’ (w2 * z2 + y2 * u2) < y2 * z2 + w2 * u2)) β†’ (βˆ€w2 : set, w2 ∈ SNoL y2 β†’ (βˆ€u2 : set, u2 ∈ SNoR z2 β†’ (y2 * z2 + w2 * u2) < w2 * z2 + y2 * u2)) β†’ (βˆ€w2 : set, w2 ∈ SNoR y2 β†’ (βˆ€u2 : set, u2 ∈ SNoL z2 β†’ (y2 * z2 + w2 * u2) < w2 * z2 + y2 * u2)) β†’ P) β†’ P)))
L6784
Hypothesis H2 : SNo y
L6785
Hypothesis H4 : (βˆ€y2 : set, y2 ∈ SNoL x β†’ SNo (y2 * y))
L6786
Hypothesis H5 : SNo z
L6787
Hypothesis H6 : SNoLev z ∈ SNoLev x
L6788
Hypothesis H7 : z < x
L6789
Hypothesis H8 : u ∈ SNoL x
L6790
Hypothesis H9 : v ∈ SNoR y
L6791
Hypothesis H10 : SNo v
L6792
Hypothesis H11 : SNo (z * y)
L6793
Hypothesis H12 : SNo (x * w)
L6794
Hypothesis H13 : SNo (z * w)
L6795
Hypothesis H14 : SNo (u * y)
L6796
Hypothesis H15 : SNo (x * v)
L6797
Hypothesis H16 : SNo (u * v)
L6798
Hypothesis H17 : SNo (z * v)
L6799
Hypothesis H18 : (x * w + z * v) < z * w + x * v
L6800
Hypothesis H19 : x2 ∈ SNoL z
L6801
Hypothesis H20 : x2 ∈ SNoR u
L6802
Hypothesis H21 : (z * y + x2 * v) < x2 * y + z * v
L6803
Theorem. (Conj_mul_SNo_prop_1__81__3)
(x2 * y + u * v) < u * y + x2 * v β†’ (z * y + x * w + u * v) < u * y + x * v + z * w
In Proofgold the corresponding term root is c15d19... and proposition id is 1b2103...
Proof:
Proof not loaded.
End of Section Conj_mul_SNo_prop_1__81__3
Beginning of Section Conj_mul_SNo_prop_1__83__6
L6809
Variable x : set
(*** Conj_mul_SNo_prop_1__83__6 TMHEucZqS6Kp9uKVoztJr2GeybxKyRxYP5J bounty of about 25 bars ***)
L6810
Variable y : set
L6811
Variable z : set
L6812
Variable w : set
L6813
Variable u : set
L6814
Variable v : set
L6815
Hypothesis H0 : SNo x
L6816
Hypothesis H1 : (βˆ€x2 : set, x2 ∈ SNoS_ (SNoLev x) β†’ (βˆ€y2 : set, SNo y2 β†’ (βˆ€P : prop, (SNo (x2 * y2) β†’ (βˆ€z2 : set, z2 ∈ SNoL x2 β†’ (βˆ€w2 : set, w2 ∈ SNoL y2 β†’ (z2 * y2 + x2 * w2) < x2 * y2 + z2 * w2)) β†’ (βˆ€z2 : set, z2 ∈ SNoR x2 β†’ (βˆ€w2 : set, w2 ∈ SNoR y2 β†’ (z2 * y2 + x2 * w2) < x2 * y2 + z2 * w2)) β†’ (βˆ€z2 : set, z2 ∈ SNoL x2 β†’ (βˆ€w2 : set, w2 ∈ SNoR y2 β†’ (x2 * y2 + z2 * w2) < z2 * y2 + x2 * w2)) β†’ (βˆ€z2 : set, z2 ∈ SNoR x2 β†’ (βˆ€w2 : set, w2 ∈ SNoL y2 β†’ (x2 * y2 + z2 * w2) < z2 * y2 + x2 * w2)) β†’ P) β†’ P)))
L6817
Hypothesis H2 : SNo y
L6818
Hypothesis H3 : (βˆ€x2 : set, x2 ∈ SNoL x β†’ (βˆ€y2 : set, SNo y2 β†’ SNo (x2 * y2)))
L6819
Hypothesis H4 : (βˆ€x2 : set, x2 ∈ SNoL x β†’ SNo (x2 * y))
L6820
Hypothesis H5 : z ∈ SNoL x
L6821
Hypothesis H7 : SNo z
L6822
Hypothesis H8 : SNoLev z ∈ SNoLev x
L6823
Hypothesis H9 : z < x
L6824
Hypothesis H10 : SNo w
L6825
Hypothesis H11 : u ∈ SNoL x
L6826
Hypothesis H12 : v ∈ SNoR y
L6827
Hypothesis H13 : SNo u
L6828
Hypothesis H14 : SNoLev u ∈ SNoLev x
L6829
Hypothesis H15 : u < x
L6830
Hypothesis H16 : SNo v
L6831
Hypothesis H17 : SNo (z * y)
L6832
Hypothesis H18 : SNo (x * w)
L6833
Hypothesis H19 : SNo (z * w)
L6834
Hypothesis H20 : SNo (u * y)
L6835
Hypothesis H21 : SNo (x * v)
L6836
Hypothesis H22 : SNo (u * v)
L6837
Hypothesis H23 : SNo (z * v)
L6838
Hypothesis H24 : SNo (u * w)
L6839
Hypothesis H25 : (x * w + z * v) < z * w + x * v
L6840
Hypothesis H26 : (x * w + u * v) < u * w + x * v
L6841
Hypothesis H27 : (z * y + u * v) < u * y + z * v β†’ (z * y + x * w + u * v) < u * y + x * v + z * w
L6842
Theorem. (Conj_mul_SNo_prop_1__83__6)
((z * y + u * w) < u * y + z * w β†’ (z * y + x * w + u * v) < u * y + x * v + z * w) β†’ (z * y + x * w + u * v) < u * y + x * v + z * w
In Proofgold the corresponding term root is f5a72e... and proposition id is a652c9...
Proof:
Proof not loaded.
End of Section Conj_mul_SNo_prop_1__83__6
Beginning of Section Conj_mul_SNo_prop_1__83__21
L6848
Variable x : set
(*** Conj_mul_SNo_prop_1__83__21 TMXakydoGNtjsaTyzSSMC7XDECZZpkRLbCm bounty of about 25 bars ***)
L6849
Variable y : set
L6850
Variable z : set
L6851
Variable w : set
L6852
Variable u : set
L6853
Variable v : set
L6854
Hypothesis H0 : SNo x
L6855
Hypothesis H1 : (βˆ€x2 : set, x2 ∈ SNoS_ (SNoLev x) β†’ (βˆ€y2 : set, SNo y2 β†’ (βˆ€P : prop, (SNo (x2 * y2) β†’ (βˆ€z2 : set, z2 ∈ SNoL x2 β†’ (βˆ€w2 : set, w2 ∈ SNoL y2 β†’ (z2 * y2 + x2 * w2) < x2 * y2 + z2 * w2)) β†’ (βˆ€z2 : set, z2 ∈ SNoR x2 β†’ (βˆ€w2 : set, w2 ∈ SNoR y2 β†’ (z2 * y2 + x2 * w2) < x2 * y2 + z2 * w2)) β†’ (βˆ€z2 : set, z2 ∈ SNoL x2 β†’ (βˆ€w2 : set, w2 ∈ SNoR y2 β†’ (x2 * y2 + z2 * w2) < z2 * y2 + x2 * w2)) β†’ (βˆ€z2 : set, z2 ∈ SNoR x2 β†’ (βˆ€w2 : set, w2 ∈ SNoL y2 β†’ (x2 * y2 + z2 * w2) < z2 * y2 + x2 * w2)) β†’ P) β†’ P)))
L6856
Hypothesis H2 : SNo y
L6857
Hypothesis H3 : (βˆ€x2 : set, x2 ∈ SNoL x β†’ (βˆ€y2 : set, SNo y2 β†’ SNo (x2 * y2)))
L6858
Hypothesis H4 : (βˆ€x2 : set, x2 ∈ SNoL x β†’ SNo (x2 * y))
L6859
Hypothesis H5 : z ∈ SNoL x
L6860
Hypothesis H6 : w ∈ SNoL y
L6861
Hypothesis H7 : SNo z
L6862
Hypothesis H8 : SNoLev z ∈ SNoLev x
L6863
Hypothesis H9 : z < x
L6864
Hypothesis H10 : SNo w
L6865
Hypothesis H11 : u ∈ SNoL x
L6866
Hypothesis H12 : v ∈ SNoR y
L6867
Hypothesis H13 : SNo u
L6868
Hypothesis H14 : SNoLev u ∈ SNoLev x
L6869
Hypothesis H15 : u < x
L6870
Hypothesis H16 : SNo v
L6871
Hypothesis H17 : SNo (z * y)
L6872
Hypothesis H18 : SNo (x * w)
L6873
Hypothesis H19 : SNo (z * w)
L6874
Hypothesis H20 : SNo (u * y)
L6875
Hypothesis H22 : SNo (u * v)
L6876
Hypothesis H23 : SNo (z * v)
L6877
Hypothesis H24 : SNo (u * w)
L6878
Hypothesis H25 : (x * w + z * v) < z * w + x * v
L6879
Hypothesis H26 : (x * w + u * v) < u * w + x * v
L6880
Hypothesis H27 : (z * y + u * v) < u * y + z * v β†’ (z * y + x * w + u * v) < u * y + x * v + z * w
L6881
Theorem. (Conj_mul_SNo_prop_1__83__21)
((z * y + u * w) < u * y + z * w β†’ (z * y + x * w + u * v) < u * y + x * v + z * w) β†’ (z * y + x * w + u * v) < u * y + x * v + z * w
In Proofgold the corresponding term root is 3051ca... and proposition id is d353dc...
Proof:
Proof not loaded.
End of Section Conj_mul_SNo_prop_1__83__21
Beginning of Section Conj_mul_SNo_prop_1__84__18
L6887
Variable x : set
(*** Conj_mul_SNo_prop_1__84__18 TMYeTkCobZWuwiYNubz566BQ9RpbC6ajKjs bounty of about 25 bars ***)
L6888
Variable y : set
L6889
Variable z : set
L6890
Variable w : set
L6891
Variable u : set
L6892
Variable v : set
L6893
Hypothesis H0 : SNo x
L6894
Hypothesis H1 : (βˆ€x2 : set, x2 ∈ SNoS_ (SNoLev x) β†’ (βˆ€y2 : set, SNo y2 β†’ (βˆ€P : prop, (SNo (x2 * y2) β†’ (βˆ€z2 : set, z2 ∈ SNoL x2 β†’ (βˆ€w2 : set, w2 ∈ SNoL y2 β†’ (z2 * y2 + x2 * w2) < x2 * y2 + z2 * w2)) β†’ (βˆ€z2 : set, z2 ∈ SNoR x2 β†’ (βˆ€w2 : set, w2 ∈ SNoR y2 β†’ (z2 * y2 + x2 * w2) < x2 * y2 + z2 * w2)) β†’ (βˆ€z2 : set, z2 ∈ SNoL x2 β†’ (βˆ€w2 : set, w2 ∈ SNoR y2 β†’ (x2 * y2 + z2 * w2) < z2 * y2 + x2 * w2)) β†’ (βˆ€z2 : set, z2 ∈ SNoR x2 β†’ (βˆ€w2 : set, w2 ∈ SNoL y2 β†’ (x2 * y2 + z2 * w2) < z2 * y2 + x2 * w2)) β†’ P) β†’ P)))
L6895
Hypothesis H2 : SNo y
L6896
Hypothesis H3 : (βˆ€x2 : set, x2 ∈ SNoL x β†’ (βˆ€y2 : set, SNo y2 β†’ SNo (x2 * y2)))
L6897
Hypothesis H4 : (βˆ€x2 : set, x2 ∈ SNoL x β†’ SNo (x2 * y))
L6898
Hypothesis H5 : z ∈ SNoL x
L6899
Hypothesis H6 : w ∈ SNoL y
L6900
Hypothesis H7 : SNo z
L6901
Hypothesis H8 : SNoLev z ∈ SNoLev x
L6902
Hypothesis H9 : z < x
L6903
Hypothesis H10 : SNo w
L6904
Hypothesis H11 : u ∈ SNoL x
L6905
Hypothesis H12 : v ∈ SNoR y
L6906
Hypothesis H13 : SNo u
L6907
Hypothesis H14 : SNoLev u ∈ SNoLev x
L6908
Hypothesis H15 : u < x
L6909
Hypothesis H16 : SNo v
L6910
Hypothesis H17 : SNo (z * y)
L6911
Hypothesis H19 : SNo (z * w)
L6912
Hypothesis H20 : SNo (u * y)
L6913
Hypothesis H21 : SNo (x * v)
L6914
Hypothesis H22 : SNo (u * v)
L6915
Hypothesis H23 : SNo (z * v)
L6916
Hypothesis H24 : SNo (u * w)
L6917
Hypothesis H25 : (x * w + z * v) < z * w + x * v
L6918
Hypothesis H26 : (x * w + u * v) < u * w + x * v
L6919
Theorem. (Conj_mul_SNo_prop_1__84__18)
((z * y + u * v) < u * y + z * v β†’ (z * y + x * w + u * v) < u * y + x * v + z * w) β†’ (z * y + x * w + u * v) < u * y + x * v + z * w
In Proofgold the corresponding term root is 9b1257... and proposition id is d7aaf4...
Proof:
Proof not loaded.
End of Section Conj_mul_SNo_prop_1__84__18
Beginning of Section Conj_mul_SNo_prop_1__85__16
L6925
Variable x : set
(*** Conj_mul_SNo_prop_1__85__16 TMX6VLTGgXnQe3uADa4PtjmzbxLN2E6XTq7 bounty of about 25 bars ***)
L6926
Variable y : set
L6927
Variable z : set
L6928
Variable w : set
L6929
Variable u : set
L6930
Variable v : set
L6931
Hypothesis H0 : SNo x
L6932
Hypothesis H1 : (βˆ€x2 : set, x2 ∈ SNoS_ (SNoLev x) β†’ (βˆ€y2 : set, SNo y2 β†’ (βˆ€P : prop, (SNo (x2 * y2) β†’ (βˆ€z2 : set, z2 ∈ SNoL x2 β†’ (βˆ€w2 : set, w2 ∈ SNoL y2 β†’ (z2 * y2 + x2 * w2) < x2 * y2 + z2 * w2)) β†’ (βˆ€z2 : set, z2 ∈ SNoR x2 β†’ (βˆ€w2 : set, w2 ∈ SNoR y2 β†’ (z2 * y2 + x2 * w2) < x2 * y2 + z2 * w2)) β†’ (βˆ€z2 : set, z2 ∈ SNoL x2 β†’ (βˆ€w2 : set, w2 ∈ SNoR y2 β†’ (x2 * y2 + z2 * w2) < z2 * y2 + x2 * w2)) β†’ (βˆ€z2 : set, z2 ∈ SNoR x2 β†’ (βˆ€w2 : set, w2 ∈ SNoL y2 β†’ (x2 * y2 + z2 * w2) < z2 * y2 + x2 * w2)) β†’ P) β†’ P)))
L6933
Hypothesis H2 : SNo y
L6934
Hypothesis H3 : (βˆ€x2 : set, x2 ∈ SNoL x β†’ (βˆ€y2 : set, SNo y2 β†’ SNo (x2 * y2)))
L6935
Hypothesis H4 : (βˆ€x2 : set, x2 ∈ SNoL x β†’ SNo (x2 * y))
L6936
Hypothesis H5 : z ∈ SNoL x
L6937
Hypothesis H6 : w ∈ SNoL y
L6938
Hypothesis H7 : SNo z
L6939
Hypothesis H8 : SNoLev z ∈ SNoLev x
L6940
Hypothesis H9 : z < x
L6941
Hypothesis H10 : SNo w
L6942
Hypothesis H11 : u ∈ SNoL x
L6943
Hypothesis H12 : v ∈ SNoR y
L6944
Hypothesis H13 : SNo u
L6945
Hypothesis H14 : SNoLev u ∈ SNoLev x
L6946
Hypothesis H15 : u < x
L6947
Hypothesis H17 : SNo (z * y)
L6948
Hypothesis H18 : SNo (x * w)
L6949
Hypothesis H19 : SNo (z * w)
L6950
Hypothesis H20 : SNo (u * y)
L6951
Hypothesis H21 : SNo (x * v)
L6952
Hypothesis H22 : SNo (u * v)
L6953
Hypothesis H23 : SNo (z * v)
L6954
Hypothesis H24 : SNo (u * w)
L6955
Hypothesis H25 : (βˆ€x2 : set, x2 ∈ SNoL x β†’ (x * w + x2 * v) < x2 * w + x * v)
L6956
Hypothesis H26 : (x * w + z * v) < z * w + x * v
L6957
Theorem. (Conj_mul_SNo_prop_1__85__16)
(x * w + u * v) < u * w + x * v β†’ (z * y + x * w + u * v) < u * y + x * v + z * w
In Proofgold the corresponding term root is 61b513... and proposition id is ed3e39...
Proof:
Proof not loaded.
End of Section Conj_mul_SNo_prop_1__85__16
Beginning of Section Conj_mul_SNo_prop_1__85__24
L6963
Variable x : set
(*** Conj_mul_SNo_prop_1__85__24 TMK68LwebqizwKxuB3iSwDa7fYr62ANByt3 bounty of about 25 bars ***)
L6964
Variable y : set
L6965
Variable z : set
L6966
Variable w : set
L6967
Variable u : set
L6968
Variable v : set
L6969
Hypothesis H0 : SNo x
L6970
Hypothesis H1 : (βˆ€x2 : set, x2 ∈ SNoS_ (SNoLev x) β†’ (βˆ€y2 : set, SNo y2 β†’ (βˆ€P : prop, (SNo (x2 * y2) β†’ (βˆ€z2 : set, z2 ∈ SNoL x2 β†’ (βˆ€w2 : set, w2 ∈ SNoL y2 β†’ (z2 * y2 + x2 * w2) < x2 * y2 + z2 * w2)) β†’ (βˆ€z2 : set, z2 ∈ SNoR x2 β†’ (βˆ€w2 : set, w2 ∈ SNoR y2 β†’ (z2 * y2 + x2 * w2) < x2 * y2 + z2 * w2)) β†’ (βˆ€z2 : set, z2 ∈ SNoL x2 β†’ (βˆ€w2 : set, w2 ∈ SNoR y2 β†’ (x2 * y2 + z2 * w2) < z2 * y2 + x2 * w2)) β†’ (βˆ€z2 : set, z2 ∈ SNoR x2 β†’ (βˆ€w2 : set, w2 ∈ SNoL y2 β†’ (x2 * y2 + z2 * w2) < z2 * y2 + x2 * w2)) β†’ P) β†’ P)))
L6971
Hypothesis H2 : SNo y
L6972
Hypothesis H3 : (βˆ€x2 : set, x2 ∈ SNoL x β†’ (βˆ€y2 : set, SNo y2 β†’ SNo (x2 * y2)))
L6973
Hypothesis H4 : (βˆ€x2 : set, x2 ∈ SNoL x β†’ SNo (x2 * y))
L6974
Hypothesis H5 : z ∈ SNoL x
L6975
Hypothesis H6 : w ∈ SNoL y
L6976
Hypothesis H7 : SNo z
L6977
Hypothesis H8 : SNoLev z ∈ SNoLev x
L6978
Hypothesis H9 : z < x
L6979
Hypothesis H10 : SNo w
L6980
Hypothesis H11 : u ∈ SNoL x
L6981
Hypothesis H12 : v ∈ SNoR y
L6982
Hypothesis H13 : SNo u
L6983
Hypothesis H14 : SNoLev u ∈ SNoLev x
L6984
Hypothesis H15 : u < x
L6985
Hypothesis H16 : SNo v
L6986
Hypothesis H17 : SNo (z * y)
L6987
Hypothesis H18 : SNo (x * w)
L6988
Hypothesis H19 : SNo (z * w)
L6989
Hypothesis H20 : SNo (u * y)
L6990
Hypothesis H21 : SNo (x * v)
L6991
Hypothesis H22 : SNo (u * v)
L6992
Hypothesis H23 : SNo (z * v)
L6993
Hypothesis H25 : (βˆ€x2 : set, x2 ∈ SNoL x β†’ (x * w + x2 * v) < x2 * w + x * v)
L6994
Hypothesis H26 : (x * w + z * v) < z * w + x * v
L6995
Theorem. (Conj_mul_SNo_prop_1__85__24)
(x * w + u * v) < u * w + x * v β†’ (z * y + x * w + u * v) < u * y + x * v + z * w
In Proofgold the corresponding term root is cabddb... and proposition id is 06bf90...
Proof:
Proof not loaded.
End of Section Conj_mul_SNo_prop_1__85__24
Beginning of Section Conj_mul_SNo_prop_1__86__11
L7001
Variable x : set
(*** Conj_mul_SNo_prop_1__86__11 TMFaK6wjQZsMkHciUswxtYR3aQXzMiTR7a6 bounty of about 25 bars ***)
L7002
Variable y : set
L7003
Variable z : set
L7004
Variable w : set
L7005
Variable u : set
L7006
Variable v : set
L7007
Hypothesis H0 : SNo x
L7008
Hypothesis H1 : (βˆ€x2 : set, x2 ∈ SNoS_ (SNoLev x) β†’ (βˆ€y2 : set, SNo y2 β†’ (βˆ€P : prop, (SNo (x2 * y2) β†’ (βˆ€z2 : set, z2 ∈ SNoL x2 β†’ (βˆ€w2 : set, w2 ∈ SNoL y2 β†’ (z2 * y2 + x2 * w2) < x2 * y2 + z2 * w2)) β†’ (βˆ€z2 : set, z2 ∈ SNoR x2 β†’ (βˆ€w2 : set, w2 ∈ SNoR y2 β†’ (z2 * y2 + x2 * w2) < x2 * y2 + z2 * w2)) β†’ (βˆ€z2 : set, z2 ∈ SNoL x2 β†’ (βˆ€w2 : set, w2 ∈ SNoR y2 β†’ (x2 * y2 + z2 * w2) < z2 * y2 + x2 * w2)) β†’ (βˆ€z2 : set, z2 ∈ SNoR x2 β†’ (βˆ€w2 : set, w2 ∈ SNoL y2 β†’ (x2 * y2 + z2 * w2) < z2 * y2 + x2 * w2)) β†’ P) β†’ P)))
L7009
Hypothesis H2 : SNo y
L7010
Hypothesis H3 : (βˆ€x2 : set, x2 ∈ SNoL x β†’ (βˆ€y2 : set, SNo y2 β†’ SNo (x2 * y2)))
L7011
Hypothesis H4 : (βˆ€x2 : set, x2 ∈ SNoL x β†’ SNo (x2 * y))
L7012
Hypothesis H5 : z ∈ SNoL x
L7013
Hypothesis H6 : w ∈ SNoL y
L7014
Hypothesis H7 : SNo z
L7015
Hypothesis H8 : SNoLev z ∈ SNoLev x
L7016
Hypothesis H9 : z < x
L7017
Hypothesis H10 : SNo w
L7018
Hypothesis H12 : v ∈ SNoR y
L7019
Hypothesis H13 : SNo u
L7020
Hypothesis H14 : SNoLev u ∈ SNoLev x
L7021
Hypothesis H15 : u < x
L7022
Hypothesis H16 : SNo v
L7023
Hypothesis H17 : SNo (z * y)
L7024
Hypothesis H18 : SNo (x * w)
L7025
Hypothesis H19 : SNo (z * w)
L7026
Hypothesis H20 : SNo (u * y)
L7027
Hypothesis H21 : SNo (x * v)
L7028
Hypothesis H22 : SNo (u * v)
L7029
Hypothesis H23 : SNo (z * v)
L7030
Hypothesis H24 : SNo (u * w)
L7031
Hypothesis H25 : (βˆ€x2 : set, x2 ∈ SNoL x β†’ (x * w + x2 * v) < x2 * w + x * v)
L7032
Theorem. (Conj_mul_SNo_prop_1__86__11)
(x * w + z * v) < z * w + x * v β†’ (z * y + x * w + u * v) < u * y + x * v + z * w
In Proofgold the corresponding term root is e7d783... and proposition id is 438d59...
Proof:
Proof not loaded.
End of Section Conj_mul_SNo_prop_1__86__11
Beginning of Section Conj_mul_SNo_prop_1__86__25
L7038
Variable x : set
(*** Conj_mul_SNo_prop_1__86__25 TMd6S7dtSnz7wu5UARtZjZsPhDeaH3wHVgB bounty of about 25 bars ***)
L7039
Variable y : set
L7040
Variable z : set
L7041
Variable w : set
L7042
Variable u : set
L7043
Variable v : set
L7044
Hypothesis H0 : SNo x
L7045
Hypothesis H1 : (βˆ€x2 : set, x2 ∈ SNoS_ (SNoLev x) β†’ (βˆ€y2 : set, SNo y2 β†’ (βˆ€P : prop, (SNo (x2 * y2) β†’ (βˆ€z2 : set, z2 ∈ SNoL x2 β†’ (βˆ€w2 : set, w2 ∈ SNoL y2 β†’ (z2 * y2 + x2 * w2) < x2 * y2 + z2 * w2)) β†’ (βˆ€z2 : set, z2 ∈ SNoR x2 β†’ (βˆ€w2 : set, w2 ∈ SNoR y2 β†’ (z2 * y2 + x2 * w2) < x2 * y2 + z2 * w2)) β†’ (βˆ€z2 : set, z2 ∈ SNoL x2 β†’ (βˆ€w2 : set, w2 ∈ SNoR y2 β†’ (x2 * y2 + z2 * w2) < z2 * y2 + x2 * w2)) β†’ (βˆ€z2 : set, z2 ∈ SNoR x2 β†’ (βˆ€w2 : set, w2 ∈ SNoL y2 β†’ (x2 * y2 + z2 * w2) < z2 * y2 + x2 * w2)) β†’ P) β†’ P)))
L7046
Hypothesis H2 : SNo y
L7047
Hypothesis H3 : (βˆ€x2 : set, x2 ∈ SNoL x β†’ (βˆ€y2 : set, SNo y2 β†’ SNo (x2 * y2)))
L7048
Hypothesis H4 : (βˆ€x2 : set, x2 ∈ SNoL x β†’ SNo (x2 * y))
L7049
Hypothesis H5 : z ∈ SNoL x
L7050
Hypothesis H6 : w ∈ SNoL y
L7051
Hypothesis H7 : SNo z
L7052
Hypothesis H8 : SNoLev z ∈ SNoLev x
L7053
Hypothesis H9 : z < x
L7054
Hypothesis H10 : SNo w
L7055
Hypothesis H11 : u ∈ SNoL x
L7056
Hypothesis H12 : v ∈ SNoR y
L7057
Hypothesis H13 : SNo u
L7058
Hypothesis H14 : SNoLev u ∈ SNoLev x
L7059
Hypothesis H15 : u < x
L7060
Hypothesis H16 : SNo v
L7061
Hypothesis H17 : SNo (z * y)
L7062
Hypothesis H18 : SNo (x * w)
L7063
Hypothesis H19 : SNo (z * w)
L7064
Hypothesis H20 : SNo (u * y)
L7065
Hypothesis H21 : SNo (x * v)
L7066
Hypothesis H22 : SNo (u * v)
L7067
Hypothesis H23 : SNo (z * v)
L7068
Hypothesis H24 : SNo (u * w)
L7069
Theorem. (Conj_mul_SNo_prop_1__86__25)
(x * w + z * v) < z * w + x * v β†’ (z * y + x * w + u * v) < u * y + x * v + z * w
In Proofgold the corresponding term root is f70943... and proposition id is 712de8...
Proof:
Proof not loaded.
End of Section Conj_mul_SNo_prop_1__86__25
Beginning of Section Conj_mul_SNo_prop_1__87__23
L7075
Variable x : set
(*** Conj_mul_SNo_prop_1__87__23 TMUAmSDYB9cdDmsky3whD26f6wMGsaPmbq7 bounty of about 25 bars ***)
L7076
Variable y : set
L7077
Variable z : set
L7078
Variable w : set
L7079
Variable u : set
L7080
Variable v : set
L7081
Hypothesis H0 : SNo x
L7082
Hypothesis H1 : (βˆ€x2 : set, x2 ∈ SNoS_ (SNoLev x) β†’ (βˆ€y2 : set, SNo y2 β†’ (βˆ€P : prop, (SNo (x2 * y2) β†’ (βˆ€z2 : set, z2 ∈ SNoL x2 β†’ (βˆ€w2 : set, w2 ∈ SNoL y2 β†’ (z2 * y2 + x2 * w2) < x2 * y2 + z2 * w2)) β†’ (βˆ€z2 : set, z2 ∈ SNoR x2 β†’ (βˆ€w2 : set, w2 ∈ SNoR y2 β†’ (z2 * y2 + x2 * w2) < x2 * y2 + z2 * w2)) β†’ (βˆ€z2 : set, z2 ∈ SNoL x2 β†’ (βˆ€w2 : set, w2 ∈ SNoR y2 β†’ (x2 * y2 + z2 * w2) < z2 * y2 + x2 * w2)) β†’ (βˆ€z2 : set, z2 ∈ SNoR x2 β†’ (βˆ€w2 : set, w2 ∈ SNoL y2 β†’ (x2 * y2 + z2 * w2) < z2 * y2 + x2 * w2)) β†’ P) β†’ P)))
L7083
Hypothesis H2 : SNo y
L7084
Hypothesis H3 : (βˆ€x2 : set, x2 ∈ SNoS_ (SNoLev y) β†’ (βˆ€P : prop, (SNo (x * x2) β†’ (βˆ€y2 : set, y2 ∈ SNoL x β†’ (βˆ€z2 : set, z2 ∈ SNoL x2 β†’ (y2 * x2 + x * z2) < x * x2 + y2 * z2)) β†’ (βˆ€y2 : set, y2 ∈ SNoR x β†’ (βˆ€z2 : set, z2 ∈ SNoR x2 β†’ (y2 * x2 + x * z2) < x * x2 + y2 * z2)) β†’ (βˆ€y2 : set, y2 ∈ SNoL x β†’ (βˆ€z2 : set, z2 ∈ SNoR x2 β†’ (x * x2 + y2 * z2) < y2 * x2 + x * z2)) β†’ (βˆ€y2 : set, y2 ∈ SNoR x β†’ (βˆ€z2 : set, z2 ∈ SNoL x2 β†’ (x * x2 + y2 * z2) < y2 * x2 + x * z2)) β†’ P) β†’ P))
L7085
Hypothesis H4 : (βˆ€x2 : set, x2 ∈ SNoL x β†’ (βˆ€y2 : set, SNo y2 β†’ SNo (x2 * y2)))
L7086
Hypothesis H5 : (βˆ€x2 : set, x2 ∈ SNoL x β†’ SNo (x2 * y))
L7087
Hypothesis H6 : z ∈ SNoL x
L7088
Hypothesis H7 : w ∈ SNoL y
L7089
Hypothesis H8 : SNo z
L7090
Hypothesis H9 : SNoLev z ∈ SNoLev x
L7091
Hypothesis H10 : z < x
L7092
Hypothesis H11 : SNo w
L7093
Hypothesis H12 : SNoLev w ∈ SNoLev y
L7094
Hypothesis H13 : u ∈ SNoL x
L7095
Hypothesis H14 : v ∈ SNoR y
L7096
Hypothesis H15 : SNo u
L7097
Hypothesis H16 : SNoLev u ∈ SNoLev x
L7098
Hypothesis H17 : u < x
L7099
Hypothesis H18 : SNo v
L7100
Hypothesis H19 : SNo (z * y)
L7101
Hypothesis H20 : SNo (x * w)
L7102
Hypothesis H21 : SNo (z * w)
L7103
Hypothesis H22 : SNo (u * y)
L7104
Hypothesis H24 : SNo (u * v)
L7105
Hypothesis H25 : SNo (z * v)
L7106
Hypothesis H26 : SNo (u * w)
L7107
Hypothesis H27 : w < v
L7108
Theorem. (Conj_mul_SNo_prop_1__87__23)
(βˆ€x2 : set, x2 ∈ SNoL x β†’ (x * w + x2 * v) < x2 * w + x * v) β†’ (z * y + x * w + u * v) < u * y + x * v + z * w
In Proofgold the corresponding term root is 05d18e... and proposition id is 24d364...
Proof:
Proof not loaded.
End of Section Conj_mul_SNo_prop_1__87__23
Beginning of Section Conj_mul_SNo_prop_1__88__25
L7114
Variable x : set
(*** Conj_mul_SNo_prop_1__88__25 TMV66ZpU4SAZNwpNqeUAqDwmSirFtgnxqVg bounty of about 25 bars ***)
L7115
Variable y : set
L7116
Variable z : set
L7117
Variable w : set
L7118
Variable u : set
L7119
Variable v : set
L7120
Hypothesis H0 : SNo x
L7121
Hypothesis H1 : (βˆ€x2 : set, x2 ∈ SNoS_ (SNoLev x) β†’ (βˆ€y2 : set, SNo y2 β†’ (βˆ€P : prop, (SNo (x2 * y2) β†’ (βˆ€z2 : set, z2 ∈ SNoL x2 β†’ (βˆ€w2 : set, w2 ∈ SNoL y2 β†’ (z2 * y2 + x2 * w2) < x2 * y2 + z2 * w2)) β†’ (βˆ€z2 : set, z2 ∈ SNoR x2 β†’ (βˆ€w2 : set, w2 ∈ SNoR y2 β†’ (z2 * y2 + x2 * w2) < x2 * y2 + z2 * w2)) β†’ (βˆ€z2 : set, z2 ∈ SNoL x2 β†’ (βˆ€w2 : set, w2 ∈ SNoR y2 β†’ (x2 * y2 + z2 * w2) < z2 * y2 + x2 * w2)) β†’ (βˆ€z2 : set, z2 ∈ SNoR x2 β†’ (βˆ€w2 : set, w2 ∈ SNoL y2 β†’ (x2 * y2 + z2 * w2) < z2 * y2 + x2 * w2)) β†’ P) β†’ P)))
L7122
Hypothesis H2 : SNo y
L7123
Hypothesis H3 : (βˆ€x2 : set, x2 ∈ SNoS_ (SNoLev y) β†’ (βˆ€P : prop, (SNo (x * x2) β†’ (βˆ€y2 : set, y2 ∈ SNoL x β†’ (βˆ€z2 : set, z2 ∈ SNoL x2 β†’ (y2 * x2 + x * z2) < x * x2 + y2 * z2)) β†’ (βˆ€y2 : set, y2 ∈ SNoR x β†’ (βˆ€z2 : set, z2 ∈ SNoR x2 β†’ (y2 * x2 + x * z2) < x * x2 + y2 * z2)) β†’ (βˆ€y2 : set, y2 ∈ SNoL x β†’ (βˆ€z2 : set, z2 ∈ SNoR x2 β†’ (x * x2 + y2 * z2) < y2 * x2 + x * z2)) β†’ (βˆ€y2 : set, y2 ∈ SNoR x β†’ (βˆ€z2 : set, z2 ∈ SNoL x2 β†’ (x * x2 + y2 * z2) < y2 * x2 + x * z2)) β†’ P) β†’ P))
L7124
Hypothesis H4 : (βˆ€x2 : set, x2 ∈ SNoL x β†’ (βˆ€y2 : set, SNo y2 β†’ SNo (x2 * y2)))
L7125
Hypothesis H5 : (βˆ€x2 : set, x2 ∈ SNoL x β†’ SNo (x2 * y))
L7126
Hypothesis H6 : z ∈ SNoL x
L7127
Hypothesis H7 : w ∈ SNoL y
L7128
Hypothesis H8 : SNo z
L7129
Hypothesis H9 : SNoLev z ∈ SNoLev x
L7130
Hypothesis H10 : z < x
L7131
Hypothesis H11 : SNo w
L7132
Hypothesis H12 : SNoLev w ∈ SNoLev y
L7133
Hypothesis H13 : w < y
L7134
Hypothesis H14 : u ∈ SNoL x
L7135
Hypothesis H15 : v ∈ SNoR y
L7136
Hypothesis H16 : SNo u
L7137
Hypothesis H17 : SNoLev u ∈ SNoLev x
L7138
Hypothesis H18 : u < x
L7139
Hypothesis H19 : SNo v
L7140
Hypothesis H20 : y < v
L7141
Hypothesis H21 : SNo (z * y)
L7142
Hypothesis H22 : SNo (x * w)
L7143
Hypothesis H23 : SNo (z * w)
L7144
Hypothesis H24 : SNo (u * y)
L7145
Hypothesis H26 : SNo (u * v)
L7146
Hypothesis H27 : SNo (z * v)
L7147
Hypothesis H28 : SNo (u * w)
L7148
Theorem. (Conj_mul_SNo_prop_1__88__25)
w < v β†’ (z * y + x * w + u * v) < u * y + x * v + z * w
In Proofgold the corresponding term root is 82032c... and proposition id is baf9f5...
Proof:
Proof not loaded.
End of Section Conj_mul_SNo_prop_1__88__25
Beginning of Section Conj_mul_SNo_prop_1__88__27
L7154
Variable x : set
(*** Conj_mul_SNo_prop_1__88__27 TMZiXDXeWQnPki8fYvTtoxwgFgyD8ZxNotb bounty of about 25 bars ***)
L7155
Variable y : set
L7156
Variable z : set
L7157
Variable w : set
L7158
Variable u : set
L7159
Variable v : set
L7160
Hypothesis H0 : SNo x
L7161
Hypothesis H1 : (βˆ€x2 : set, x2 ∈ SNoS_ (SNoLev x) β†’ (βˆ€y2 : set, SNo y2 β†’ (βˆ€P : prop, (SNo (x2 * y2) β†’ (βˆ€z2 : set, z2 ∈ SNoL x2 β†’ (βˆ€w2 : set, w2 ∈ SNoL y2 β†’ (z2 * y2 + x2 * w2) < x2 * y2 + z2 * w2)) β†’ (βˆ€z2 : set, z2 ∈ SNoR x2 β†’ (βˆ€w2 : set, w2 ∈ SNoR y2 β†’ (z2 * y2 + x2 * w2) < x2 * y2 + z2 * w2)) β†’ (βˆ€z2 : set, z2 ∈ SNoL x2 β†’ (βˆ€w2 : set, w2 ∈ SNoR y2 β†’ (x2 * y2 + z2 * w2) < z2 * y2 + x2 * w2)) β†’ (βˆ€z2 : set, z2 ∈ SNoR x2 β†’ (βˆ€w2 : set, w2 ∈ SNoL y2 β†’ (x2 * y2 + z2 * w2) < z2 * y2 + x2 * w2)) β†’ P) β†’ P)))
L7162
Hypothesis H2 : SNo y
L7163
Hypothesis H3 : (βˆ€x2 : set, x2 ∈ SNoS_ (SNoLev y) β†’ (βˆ€P : prop, (SNo (x * x2) β†’ (βˆ€y2 : set, y2 ∈ SNoL x β†’ (βˆ€z2 : set, z2 ∈ SNoL x2 β†’ (y2 * x2 + x * z2) < x * x2 + y2 * z2)) β†’ (βˆ€y2 : set, y2 ∈ SNoR x β†’ (βˆ€z2 : set, z2 ∈ SNoR x2 β†’ (y2 * x2 + x * z2) < x * x2 + y2 * z2)) β†’ (βˆ€y2 : set, y2 ∈ SNoL x β†’ (βˆ€z2 : set, z2 ∈ SNoR x2 β†’ (x * x2 + y2 * z2) < y2 * x2 + x * z2)) β†’ (βˆ€y2 : set, y2 ∈ SNoR x β†’ (βˆ€z2 : set, z2 ∈ SNoL x2 β†’ (x * x2 + y2 * z2) < y2 * x2 + x * z2)) β†’ P) β†’ P))
L7164
Hypothesis H4 : (βˆ€x2 : set, x2 ∈ SNoL x β†’ (βˆ€y2 : set, SNo y2 β†’ SNo (x2 * y2)))
L7165
Hypothesis H5 : (βˆ€x2 : set, x2 ∈ SNoL x β†’ SNo (x2 * y))
L7166
Hypothesis H6 : z ∈ SNoL x
L7167
Hypothesis H7 : w ∈ SNoL y
L7168
Hypothesis H8 : SNo z
L7169
Hypothesis H9 : SNoLev z ∈ SNoLev x
L7170
Hypothesis H10 : z < x
L7171
Hypothesis H11 : SNo w
L7172
Hypothesis H12 : SNoLev w ∈ SNoLev y
L7173
Hypothesis H13 : w < y
L7174
Hypothesis H14 : u ∈ SNoL x
L7175
Hypothesis H15 : v ∈ SNoR y
L7176
Hypothesis H16 : SNo u
L7177
Hypothesis H17 : SNoLev u ∈ SNoLev x
L7178
Hypothesis H18 : u < x
L7179
Hypothesis H19 : SNo v
L7180
Hypothesis H20 : y < v
L7181
Hypothesis H21 : SNo (z * y)
L7182
Hypothesis H22 : SNo (x * w)
L7183
Hypothesis H23 : SNo (z * w)
L7184
Hypothesis H24 : SNo (u * y)
L7185
Hypothesis H25 : SNo (x * v)
L7186
Hypothesis H26 : SNo (u * v)
L7187
Hypothesis H28 : SNo (u * w)
L7188
Theorem. (Conj_mul_SNo_prop_1__88__27)
w < v β†’ (z * y + x * w + u * v) < u * y + x * v + z * w
In Proofgold the corresponding term root is c6c093... and proposition id is 8c86eb...
Proof:
Proof not loaded.
End of Section Conj_mul_SNo_prop_1__88__27
Beginning of Section Conj_mul_SNo_prop_1__94__4
L7194
Variable x : set
(*** Conj_mul_SNo_prop_1__94__4 TMT4MRm2shfbo1gPkYRxAsjD3Btsd4JWiG9 bounty of about 25 bars ***)
L7195
Variable y : set
L7196
Variable z : set
L7197
Variable w : set
L7198
Hypothesis H0 : SNo y
L7199
Hypothesis H1 : (βˆ€u : set, u ∈ SNoL x β†’ (βˆ€v : set, v ∈ SNoL y β†’ u * y + x * v + - (u * v) ∈ z))
L7200
Hypothesis H2 : (βˆ€u : set, u ∈ SNoR x β†’ (βˆ€v : set, v ∈ SNoR y β†’ u * y + x * v + - (u * v) ∈ z))
L7201
Hypothesis H3 : (βˆ€u : set, u ∈ SNoL x β†’ (βˆ€v : set, v ∈ SNoR y β†’ u * y + x * v + - (u * v) ∈ w))
L7202
Hypothesis H5 : x * y = SNoCut z w
L7203
Hypothesis H6 : (βˆ€u : set, u ∈ SNoL x β†’ (βˆ€v : set, SNo v β†’ SNo (u * v)))
L7204
Hypothesis H7 : (βˆ€u : set, u ∈ SNoR x β†’ (βˆ€v : set, SNo v β†’ SNo (u * v)))
L7205
Hypothesis H8 : (βˆ€u : set, u ∈ SNoL y β†’ SNo (x * u))
L7206
Hypothesis H9 : (βˆ€u : set, u ∈ SNoR y β†’ SNo (x * u))
L7207
Hypothesis H10 : SNoCutP z w
L7208
Theorem. (Conj_mul_SNo_prop_1__94__4)
SNo (x * y) β†’ (βˆ€P : prop, (SNo (x * y) β†’ (βˆ€u : set, u ∈ SNoL x β†’ (βˆ€v : set, v ∈ SNoL y β†’ (u * y + x * v) < x * y + u * v)) β†’ (βˆ€u : set, u ∈ SNoR x β†’ (βˆ€v : set, v ∈ SNoR y β†’ (u * y + x * v) < x * y + u * v)) β†’ (βˆ€u : set, u ∈ SNoL x β†’ (βˆ€v : set, v ∈ SNoR y β†’ (x * y + u * v) < u * y + x * v)) β†’ (βˆ€u : set, u ∈ SNoR x β†’ (βˆ€v : set, v ∈ SNoL y β†’ (x * y + u * v) < u * y + x * v)) β†’ P) β†’ P)
In Proofgold the corresponding term root is 368141... and proposition id is a97363...
Proof:
Proof not loaded.
End of Section Conj_mul_SNo_prop_1__94__4
Beginning of Section Conj_mul_SNo_prop_1__96__3
L7214
Variable x : set
(*** Conj_mul_SNo_prop_1__96__3 TMdzKjxZ8BPSVoPLgJLGeLvhj2VBgqs5VbY bounty of about 25 bars ***)
L7215
Variable y : set
L7216
Variable z : set
L7217
Variable w : set
L7218
Hypothesis H0 : SNo x
L7219
Hypothesis H1 : (βˆ€u : set, u ∈ SNoS_ (SNoLev x) β†’ (βˆ€v : set, SNo v β†’ (βˆ€P : prop, (SNo (u * v) β†’ (βˆ€x2 : set, x2 ∈ SNoL u β†’ (βˆ€y2 : set, y2 ∈ SNoL v β†’ (x2 * v + u * y2) < u * v + x2 * y2)) β†’ (βˆ€x2 : set, x2 ∈ SNoR u β†’ (βˆ€y2 : set, y2 ∈ SNoR v β†’ (x2 * v + u * y2) < u * v + x2 * y2)) β†’ (βˆ€x2 : set, x2 ∈ SNoL u β†’ (βˆ€y2 : set, y2 ∈ SNoR v β†’ (u * v + x2 * y2) < x2 * v + u * y2)) β†’ (βˆ€x2 : set, x2 ∈ SNoR u β†’ (βˆ€y2 : set, y2 ∈ SNoL v β†’ (u * v + x2 * y2) < x2 * v + u * y2)) β†’ P) β†’ P)))
L7220
Hypothesis H2 : SNo y
L7221
Hypothesis H4 : (βˆ€u : set, u ∈ z β†’ (βˆ€P : prop, (βˆ€v : set, v ∈ SNoL x β†’ (βˆ€x2 : set, x2 ∈ SNoL y β†’ u = v * y + x * x2 + - (v * x2) β†’ P)) β†’ (βˆ€v : set, v ∈ SNoR x β†’ (βˆ€x2 : set, x2 ∈ SNoR y β†’ u = v * y + x * x2 + - (v * x2) β†’ P)) β†’ P))
L7222
Hypothesis H5 : (βˆ€u : set, u ∈ SNoL x β†’ (βˆ€v : set, v ∈ SNoL y β†’ u * y + x * v + - (u * v) ∈ z))
L7223
Hypothesis H6 : (βˆ€u : set, u ∈ SNoR x β†’ (βˆ€v : set, v ∈ SNoR y β†’ u * y + x * v + - (u * v) ∈ z))
L7224
Hypothesis H7 : (βˆ€u : set, u ∈ w β†’ (βˆ€P : prop, (βˆ€v : set, v ∈ SNoL x β†’ (βˆ€x2 : set, x2 ∈ SNoR y β†’ u = v * y + x * x2 + - (v * x2) β†’ P)) β†’ (βˆ€v : set, v ∈ SNoR x β†’ (βˆ€x2 : set, x2 ∈ SNoL y β†’ u = v * y + x * x2 + - (v * x2) β†’ P)) β†’ P))
L7225
Hypothesis H8 : (βˆ€u : set, u ∈ SNoL x β†’ (βˆ€v : set, v ∈ SNoR y β†’ u * y + x * v + - (u * v) ∈ w))
L7226
Hypothesis H9 : (βˆ€u : set, u ∈ SNoR x β†’ (βˆ€v : set, v ∈ SNoL y β†’ u * y + x * v + - (u * v) ∈ w))
L7227
Hypothesis H10 : x * y = SNoCut z w
L7228
Hypothesis H11 : (βˆ€u : set, u ∈ SNoL x β†’ (βˆ€v : set, SNo v β†’ SNo (u * v)))
L7229
Hypothesis H12 : (βˆ€u : set, u ∈ SNoR x β†’ (βˆ€v : set, SNo v β†’ SNo (u * v)))
L7230
Hypothesis H13 : (βˆ€u : set, u ∈ SNoL x β†’ SNo (u * y))
L7231
Hypothesis H14 : (βˆ€u : set, u ∈ SNoR x β†’ SNo (u * y))
L7232
Hypothesis H15 : (βˆ€u : set, u ∈ SNoL y β†’ SNo (x * u))
L7233
Theorem. (Conj_mul_SNo_prop_1__96__3)
(βˆ€u : set, u ∈ SNoR y β†’ SNo (x * u)) β†’ (βˆ€P : prop, (SNo (x * y) β†’ (βˆ€u : set, u ∈ SNoL x β†’ (βˆ€v : set, v ∈ SNoL y β†’ (u * y + x * v) < x * y + u * v)) β†’ (βˆ€u : set, u ∈ SNoR x β†’ (βˆ€v : set, v ∈ SNoR y β†’ (u * y + x * v) < x * y + u * v)) β†’ (βˆ€u : set, u ∈ SNoL x β†’ (βˆ€v : set, v ∈ SNoR y β†’ (x * y + u * v) < u * y + x * v)) β†’ (βˆ€u : set, u ∈ SNoR x β†’ (βˆ€v : set, v ∈ SNoL y β†’ (x * y + u * v) < u * y + x * v)) β†’ P) β†’ P)
In Proofgold the corresponding term root is f94351... and proposition id is f0dbaf...
Proof:
Proof not loaded.
End of Section Conj_mul_SNo_prop_1__96__3
Beginning of Section Conj_mul_SNo_prop_1__97__6
L7239
Variable x : set
(*** Conj_mul_SNo_prop_1__97__6 TMafv7YQUBjTo9jQPgaohTzYWX1CHFHA3FL bounty of about 25 bars ***)
L7240
Variable y : set
L7241
Variable z : set
L7242
Variable w : set
L7243
Hypothesis H0 : SNo x
L7244
Hypothesis H1 : (βˆ€u : set, u ∈ SNoS_ (SNoLev x) β†’ (βˆ€v : set, SNo v β†’ (βˆ€P : prop, (SNo (u * v) β†’ (βˆ€x2 : set, x2 ∈ SNoL u β†’ (βˆ€y2 : set, y2 ∈ SNoL v β†’ (x2 * v + u * y2) < u * v + x2 * y2)) β†’ (βˆ€x2 : set, x2 ∈ SNoR u β†’ (βˆ€y2 : set, y2 ∈ SNoR v β†’ (x2 * v + u * y2) < u * v + x2 * y2)) β†’ (βˆ€x2 : set, x2 ∈ SNoL u β†’ (βˆ€y2 : set, y2 ∈ SNoR v β†’ (u * v + x2 * y2) < x2 * v + u * y2)) β†’ (βˆ€x2 : set, x2 ∈ SNoR u β†’ (βˆ€y2 : set, y2 ∈ SNoL v β†’ (u * v + x2 * y2) < x2 * v + u * y2)) β†’ P) β†’ P)))
L7245
Hypothesis H2 : SNo y
L7246
Hypothesis H3 : (βˆ€u : set, u ∈ SNoS_ (SNoLev y) β†’ (βˆ€P : prop, (SNo (x * u) β†’ (βˆ€v : set, v ∈ SNoL x β†’ (βˆ€x2 : set, x2 ∈ SNoL u β†’ (v * u + x * x2) < x * u + v * x2)) β†’ (βˆ€v : set, v ∈ SNoR x β†’ (βˆ€x2 : set, x2 ∈ SNoR u β†’ (v * u + x * x2) < x * u + v * x2)) β†’ (βˆ€v : set, v ∈ SNoL x β†’ (βˆ€x2 : set, x2 ∈ SNoR u β†’ (x * u + v * x2) < v * u + x * x2)) β†’ (βˆ€v : set, v ∈ SNoR x β†’ (βˆ€x2 : set, x2 ∈ SNoL u β†’ (x * u + v * x2) < v * u + x * x2)) β†’ P) β†’ P))
L7247
Hypothesis H4 : (βˆ€u : set, u ∈ z β†’ (βˆ€P : prop, (βˆ€v : set, v ∈ SNoL x β†’ (βˆ€x2 : set, x2 ∈ SNoL y β†’ u = v * y + x * x2 + - (v * x2) β†’ P)) β†’ (βˆ€v : set, v ∈ SNoR x β†’ (βˆ€x2 : set, x2 ∈ SNoR y β†’ u = v * y + x * x2 + - (v * x2) β†’ P)) β†’ P))
L7248
Hypothesis H5 : (βˆ€u : set, u ∈ SNoL x β†’ (βˆ€v : set, v ∈ SNoL y β†’ u * y + x * v + - (u * v) ∈ z))
L7249
Hypothesis H7 : (βˆ€u : set, u ∈ w β†’ (βˆ€P : prop, (βˆ€v : set, v ∈ SNoL x β†’ (βˆ€x2 : set, x2 ∈ SNoR y β†’ u = v * y + x * x2 + - (v * x2) β†’ P)) β†’ (βˆ€v : set, v ∈ SNoR x β†’ (βˆ€x2 : set, x2 ∈ SNoL y β†’ u = v * y + x * x2 + - (v * x2) β†’ P)) β†’ P))
L7250
Hypothesis H8 : (βˆ€u : set, u ∈ SNoL x β†’ (βˆ€v : set, v ∈ SNoR y β†’ u * y + x * v + - (u * v) ∈ w))
L7251
Hypothesis H9 : (βˆ€u : set, u ∈ SNoR x β†’ (βˆ€v : set, v ∈ SNoL y β†’ u * y + x * v + - (u * v) ∈ w))
L7252
Hypothesis H10 : x * y = SNoCut z w
L7253
Hypothesis H11 : (βˆ€u : set, u ∈ SNoL x β†’ (βˆ€v : set, SNo v β†’ SNo (u * v)))
L7254
Hypothesis H12 : (βˆ€u : set, u ∈ SNoR x β†’ (βˆ€v : set, SNo v β†’ SNo (u * v)))
L7255
Hypothesis H13 : (βˆ€u : set, u ∈ SNoL x β†’ SNo (u * y))
L7256
Hypothesis H14 : (βˆ€u : set, u ∈ SNoR x β†’ SNo (u * y))
L7257
Theorem. (Conj_mul_SNo_prop_1__97__6)
(βˆ€u : set, u ∈ SNoL y β†’ SNo (x * u)) β†’ (βˆ€P : prop, (SNo (x * y) β†’ (βˆ€u : set, u ∈ SNoL x β†’ (βˆ€v : set, v ∈ SNoL y β†’ (u * y + x * v) < x * y + u * v)) β†’ (βˆ€u : set, u ∈ SNoR x β†’ (βˆ€v : set, v ∈ SNoR y β†’ (u * y + x * v) < x * y + u * v)) β†’ (βˆ€u : set, u ∈ SNoL x β†’ (βˆ€v : set, v ∈ SNoR y β†’ (x * y + u * v) < u * y + x * v)) β†’ (βˆ€u : set, u ∈ SNoR x β†’ (βˆ€v : set, v ∈ SNoL y β†’ (x * y + u * v) < u * y + x * v)) β†’ P) β†’ P)
In Proofgold the corresponding term root is deadc5... and proposition id is 6ed60f...
Proof:
Proof not loaded.
End of Section Conj_mul_SNo_prop_1__97__6
Beginning of Section Conj_mul_SNo_prop_1__98__11
L7263
Variable x : set
(*** Conj_mul_SNo_prop_1__98__11 TMUEwwJf8LwyD5ohrL6KfvCN2rtEiyNk9Xy bounty of about 25 bars ***)
L7264
Variable y : set
L7265
Variable z : set
L7266
Variable w : set
L7267
Hypothesis H0 : SNo x
L7268
Hypothesis H1 : (βˆ€u : set, u ∈ SNoS_ (SNoLev x) β†’ (βˆ€v : set, SNo v β†’ (βˆ€P : prop, (SNo (u * v) β†’ (βˆ€x2 : set, x2 ∈ SNoL u β†’ (βˆ€y2 : set, y2 ∈ SNoL v β†’ (x2 * v + u * y2) < u * v + x2 * y2)) β†’ (βˆ€x2 : set, x2 ∈ SNoR u β†’ (βˆ€y2 : set, y2 ∈ SNoR v β†’ (x2 * v + u * y2) < u * v + x2 * y2)) β†’ (βˆ€x2 : set, x2 ∈ SNoL u β†’ (βˆ€y2 : set, y2 ∈ SNoR v β†’ (u * v + x2 * y2) < x2 * v + u * y2)) β†’ (βˆ€x2 : set, x2 ∈ SNoR u β†’ (βˆ€y2 : set, y2 ∈ SNoL v β†’ (u * v + x2 * y2) < x2 * v + u * y2)) β†’ P) β†’ P)))
L7269
Hypothesis H2 : SNo y
L7270
Hypothesis H3 : (βˆ€u : set, u ∈ SNoS_ (SNoLev y) β†’ (βˆ€P : prop, (SNo (x * u) β†’ (βˆ€v : set, v ∈ SNoL x β†’ (βˆ€x2 : set, x2 ∈ SNoL u β†’ (v * u + x * x2) < x * u + v * x2)) β†’ (βˆ€v : set, v ∈ SNoR x β†’ (βˆ€x2 : set, x2 ∈ SNoR u β†’ (v * u + x * x2) < x * u + v * x2)) β†’ (βˆ€v : set, v ∈ SNoL x β†’ (βˆ€x2 : set, x2 ∈ SNoR u β†’ (x * u + v * x2) < v * u + x * x2)) β†’ (βˆ€v : set, v ∈ SNoR x β†’ (βˆ€x2 : set, x2 ∈ SNoL u β†’ (x * u + v * x2) < v * u + x * x2)) β†’ P) β†’ P))
L7271
Hypothesis H4 : (βˆ€u : set, u ∈ z β†’ (βˆ€P : prop, (βˆ€v : set, v ∈ SNoL x β†’ (βˆ€x2 : set, x2 ∈ SNoL y β†’ u = v * y + x * x2 + - (v * x2) β†’ P)) β†’ (βˆ€v : set, v ∈ SNoR x β†’ (βˆ€x2 : set, x2 ∈ SNoR y β†’ u = v * y + x * x2 + - (v * x2) β†’ P)) β†’ P))
L7272
Hypothesis H5 : (βˆ€u : set, u ∈ SNoL x β†’ (βˆ€v : set, v ∈ SNoL y β†’ u * y + x * v + - (u * v) ∈ z))
L7273
Hypothesis H6 : (βˆ€u : set, u ∈ SNoR x β†’ (βˆ€v : set, v ∈ SNoR y β†’ u * y + x * v + - (u * v) ∈ z))
L7274
Hypothesis H7 : (βˆ€u : set, u ∈ w β†’ (βˆ€P : prop, (βˆ€v : set, v ∈ SNoL x β†’ (βˆ€x2 : set, x2 ∈ SNoR y β†’ u = v * y + x * x2 + - (v * x2) β†’ P)) β†’ (βˆ€v : set, v ∈ SNoR x β†’ (βˆ€x2 : set, x2 ∈ SNoL y β†’ u = v * y + x * x2 + - (v * x2) β†’ P)) β†’ P))
L7275
Hypothesis H8 : (βˆ€u : set, u ∈ SNoL x β†’ (βˆ€v : set, v ∈ SNoR y β†’ u * y + x * v + - (u * v) ∈ w))
L7276
Hypothesis H9 : (βˆ€u : set, u ∈ SNoR x β†’ (βˆ€v : set, v ∈ SNoL y β†’ u * y + x * v + - (u * v) ∈ w))
L7277
Hypothesis H10 : x * y = SNoCut z w
L7278
Hypothesis H12 : (βˆ€u : set, u ∈ SNoR x β†’ (βˆ€v : set, SNo v β†’ SNo (u * v)))
L7279
Hypothesis H13 : (βˆ€u : set, u ∈ SNoL x β†’ SNo (u * y))
L7280
Theorem. (Conj_mul_SNo_prop_1__98__11)
(βˆ€u : set, u ∈ SNoR x β†’ SNo (u * y)) β†’ (βˆ€P : prop, (SNo (x * y) β†’ (βˆ€u : set, u ∈ SNoL x β†’ (βˆ€v : set, v ∈ SNoL y β†’ (u * y + x * v) < x * y + u * v)) β†’ (βˆ€u : set, u ∈ SNoR x β†’ (βˆ€v : set, v ∈ SNoR y β†’ (u * y + x * v) < x * y + u * v)) β†’ (βˆ€u : set, u ∈ SNoL x β†’ (βˆ€v : set, v ∈ SNoR y β†’ (x * y + u * v) < u * y + x * v)) β†’ (βˆ€u : set, u ∈ SNoR x β†’ (βˆ€v : set, v ∈ SNoL y β†’ (x * y + u * v) < u * y + x * v)) β†’ P) β†’ P)
In Proofgold the corresponding term root is 5e72f4... and proposition id is 5d32e3...
Proof:
Proof not loaded.
End of Section Conj_mul_SNo_prop_1__98__11
Beginning of Section Conj_mul_SNo_prop_1__101__4
L7286
Variable x : set
(*** Conj_mul_SNo_prop_1__101__4 TMFtEU3REsyooDgzQdBDus1qsuNqas1n9Wn bounty of about 25 bars ***)
L7287
Variable y : set
L7288
Variable z : set
L7289
Variable w : set
L7290
Hypothesis H0 : SNo x
L7291
Hypothesis H1 : (βˆ€u : set, u ∈ SNoS_ (SNoLev x) β†’ (βˆ€v : set, SNo v β†’ (βˆ€P : prop, (SNo (u * v) β†’ (βˆ€x2 : set, x2 ∈ SNoL u β†’ (βˆ€y2 : set, y2 ∈ SNoL v β†’ (x2 * v + u * y2) < u * v + x2 * y2)) β†’ (βˆ€x2 : set, x2 ∈ SNoR u β†’ (βˆ€y2 : set, y2 ∈ SNoR v β†’ (x2 * v + u * y2) < u * v + x2 * y2)) β†’ (βˆ€x2 : set, x2 ∈ SNoL u β†’ (βˆ€y2 : set, y2 ∈ SNoR v β†’ (u * v + x2 * y2) < x2 * v + u * y2)) β†’ (βˆ€x2 : set, x2 ∈ SNoR u β†’ (βˆ€y2 : set, y2 ∈ SNoL v β†’ (u * v + x2 * y2) < x2 * v + u * y2)) β†’ P) β†’ P)))
L7292
Hypothesis H2 : SNo y
L7293
Hypothesis H3 : (βˆ€u : set, u ∈ SNoS_ (SNoLev y) β†’ (βˆ€P : prop, (SNo (x * u) β†’ (βˆ€v : set, v ∈ SNoL x β†’ (βˆ€x2 : set, x2 ∈ SNoL u β†’ (v * u + x * x2) < x * u + v * x2)) β†’ (βˆ€v : set, v ∈ SNoR x β†’ (βˆ€x2 : set, x2 ∈ SNoR u β†’ (v * u + x * x2) < x * u + v * x2)) β†’ (βˆ€v : set, v ∈ SNoL x β†’ (βˆ€x2 : set, x2 ∈ SNoR u β†’ (x * u + v * x2) < v * u + x * x2)) β†’ (βˆ€v : set, v ∈ SNoR x β†’ (βˆ€x2 : set, x2 ∈ SNoL u β†’ (x * u + v * x2) < v * u + x * x2)) β†’ P) β†’ P))
L7294
Hypothesis H5 : (βˆ€u : set, u ∈ SNoL x β†’ (βˆ€v : set, v ∈ SNoL y β†’ u * y + x * v + - (u * v) ∈ z))
L7295
Hypothesis H6 : (βˆ€u : set, u ∈ SNoR x β†’ (βˆ€v : set, v ∈ SNoR y β†’ u * y + x * v + - (u * v) ∈ z))
L7296
Hypothesis H7 : (βˆ€u : set, u ∈ w β†’ (βˆ€P : prop, (βˆ€v : set, v ∈ SNoL x β†’ (βˆ€x2 : set, x2 ∈ SNoR y β†’ u = v * y + x * x2 + - (v * x2) β†’ P)) β†’ (βˆ€v : set, v ∈ SNoR x β†’ (βˆ€x2 : set, x2 ∈ SNoL y β†’ u = v * y + x * x2 + - (v * x2) β†’ P)) β†’ P))
L7297
Hypothesis H8 : (βˆ€u : set, u ∈ SNoL x β†’ (βˆ€v : set, v ∈ SNoR y β†’ u * y + x * v + - (u * v) ∈ w))
L7298
Hypothesis H9 : (βˆ€u : set, u ∈ SNoR x β†’ (βˆ€v : set, v ∈ SNoL y β†’ u * y + x * v + - (u * v) ∈ w))
L7299
Hypothesis H10 : x * y = SNoCut z w
L7300
Theorem. (Conj_mul_SNo_prop_1__101__4)
(βˆ€u : set, u ∈ SNoL x β†’ (βˆ€v : set, SNo v β†’ SNo (u * v))) β†’ (βˆ€P : prop, (SNo (x * y) β†’ (βˆ€u : set, u ∈ SNoL x β†’ (βˆ€v : set, v ∈ SNoL y β†’ (u * y + x * v) < x * y + u * v)) β†’ (βˆ€u : set, u ∈ SNoR x β†’ (βˆ€v : set, v ∈ SNoR y β†’ (u * y + x * v) < x * y + u * v)) β†’ (βˆ€u : set, u ∈ SNoL x β†’ (βˆ€v : set, v ∈ SNoR y β†’ (x * y + u * v) < u * y + x * v)) β†’ (βˆ€u : set, u ∈ SNoR x β†’ (βˆ€v : set, v ∈ SNoL y β†’ (x * y + u * v) < u * y + x * v)) β†’ P) β†’ P)
In Proofgold the corresponding term root is 41900c... and proposition id is 02e589...
Proof:
Proof not loaded.
End of Section Conj_mul_SNo_prop_1__101__4
Beginning of Section Conj_mul_SNo_eq_3__2__1
L7306
Variable x : set
(*** Conj_mul_SNo_eq_3__2__1 TMK1eztCy8yifbWE5UCyb9KKJZLv6MiKpYi bounty of about 25 bars ***)
L7307
Variable y : set
L7308
Variable z : set
L7309
Variable w : set
L7310
Variable u : set
L7311
Variable v : set
L7312
Variable x2 : set
L7313
Variable y2 : set
L7314
Hypothesis H0 : SNo x
L7315
Hypothesis H2 : (βˆ€z2 : set, z2 ∈ SNoR x β†’ (βˆ€w2 : set, w2 ∈ SNoR y β†’ (z2 * y + x * w2) < x * y + z2 * w2))
L7316
Hypothesis H3 : (βˆ€z2 : set, z2 ∈ SNoR x β†’ (βˆ€w2 : set, w2 ∈ SNoL y β†’ (x * y + z2 * w2) < z2 * y + x * w2))
L7317
Hypothesis H4 : u ∈ SNoR x
L7318
Hypothesis H5 : v ∈ SNoR y
L7319
Hypothesis H6 : z = u * y + x * v + - (u * v)
L7320
Hypothesis H7 : SNo (u * y)
L7321
Hypothesis H8 : SNo (x * v)
L7322
Hypothesis H9 : SNo (u * v)
L7323
Hypothesis H10 : x2 ∈ SNoR x
L7324
Hypothesis H11 : y2 ∈ SNoL y
L7325
Hypothesis H12 : w = x2 * y + x * y2 + - (x2 * y2)
L7326
Hypothesis H13 : SNo x2
L7327
Hypothesis H14 : SNo y2
L7328
Hypothesis H15 : SNo (x2 * y)
L7329
Theorem. (Conj_mul_SNo_eq_3__2__1)
SNo (x * y2) β†’ z < w
In Proofgold the corresponding term root is 8b3b52... and proposition id is c7d161...
Proof:
Proof not loaded.
End of Section Conj_mul_SNo_eq_3__2__1
Beginning of Section Conj_mul_SNo_eq_3__2__9
L7335
Variable x : set
(*** Conj_mul_SNo_eq_3__2__9 TMaEBcUwJZoTiPyBHhHaPxow6bK3FizSqqY bounty of about 25 bars ***)
L7336
Variable y : set
L7337
Variable z : set
L7338
Variable w : set
L7339
Variable u : set
L7340
Variable v : set
L7341
Variable x2 : set
L7342
Variable y2 : set
L7343
Hypothesis H0 : SNo x
L7344
Hypothesis H1 : SNo (x * y)
L7345
Hypothesis H2 : (βˆ€z2 : set, z2 ∈ SNoR x β†’ (βˆ€w2 : set, w2 ∈ SNoR y β†’ (z2 * y + x * w2) < x * y + z2 * w2))
L7346
Hypothesis H3 : (βˆ€z2 : set, z2 ∈ SNoR x β†’ (βˆ€w2 : set, w2 ∈ SNoL y β†’ (x * y + z2 * w2) < z2 * y + x * w2))
L7347
Hypothesis H4 : u ∈ SNoR x
L7348
Hypothesis H5 : v ∈ SNoR y
L7349
Hypothesis H6 : z = u * y + x * v + - (u * v)
L7350
Hypothesis H7 : SNo (u * y)
L7351
Hypothesis H8 : SNo (x * v)
L7352
Hypothesis H10 : x2 ∈ SNoR x
L7353
Hypothesis H11 : y2 ∈ SNoL y
L7354
Hypothesis H12 : w = x2 * y + x * y2 + - (x2 * y2)
L7355
Hypothesis H13 : SNo x2
L7356
Hypothesis H14 : SNo y2
L7357
Hypothesis H15 : SNo (x2 * y)
L7358
Theorem. (Conj_mul_SNo_eq_3__2__9)
SNo (x * y2) β†’ z < w
In Proofgold the corresponding term root is 127035... and proposition id is a939a3...
Proof:
Proof not loaded.
End of Section Conj_mul_SNo_eq_3__2__9
Beginning of Section Conj_mul_SNo_eq_3__3__10
L7364
Variable x : set
(*** Conj_mul_SNo_eq_3__3__10 TMNocM1yrjDUPfxtNpiHCx2JcEdRQpGfgdT bounty of about 25 bars ***)
L7365
Variable y : set
L7366
Variable z : set
L7367
Variable w : set
L7368
Variable u : set
L7369
Variable v : set
L7370
Variable x2 : set
L7371
Variable y2 : set
L7372
Hypothesis H0 : SNo x
L7373
Hypothesis H1 : SNo y
L7374
Hypothesis H2 : SNo (x * y)
L7375
Hypothesis H3 : (βˆ€z2 : set, z2 ∈ SNoR x β†’ (βˆ€w2 : set, w2 ∈ SNoR y β†’ (z2 * y + x * w2) < x * y + z2 * w2))
L7376
Hypothesis H4 : (βˆ€z2 : set, z2 ∈ SNoR x β†’ (βˆ€w2 : set, w2 ∈ SNoL y β†’ (x * y + z2 * w2) < z2 * y + x * w2))
L7377
Hypothesis H5 : u ∈ SNoR x
L7378
Hypothesis H6 : v ∈ SNoR y
L7379
Hypothesis H7 : z = u * y + x * v + - (u * v)
L7380
Hypothesis H8 : SNo (u * y)
L7381
Hypothesis H9 : SNo (x * v)
L7382
Hypothesis H11 : x2 ∈ SNoR x
L7383
Hypothesis H12 : y2 ∈ SNoL y
L7384
Hypothesis H13 : w = x2 * y + x * y2 + - (x2 * y2)
L7385
Hypothesis H14 : SNo x2
L7386
Hypothesis H15 : SNo y2
L7387
Theorem. (Conj_mul_SNo_eq_3__3__10)
SNo (x2 * y) β†’ z < w
In Proofgold the corresponding term root is c7650a... and proposition id is 997382...
Proof:
Proof not loaded.
End of Section Conj_mul_SNo_eq_3__3__10
Beginning of Section Conj_mul_SNo_eq_3__3__13
L7393
Variable x : set
(*** Conj_mul_SNo_eq_3__3__13 TMaMiaVSWy3YUYNBoYTRiiL7QT8ByWv2fm4 bounty of about 25 bars ***)
L7394
Variable y : set
L7395
Variable z : set
L7396
Variable w : set
L7397
Variable u : set
L7398
Variable v : set
L7399
Variable x2 : set
L7400
Variable y2 : set
L7401
Hypothesis H0 : SNo x
L7402
Hypothesis H1 : SNo y
L7403
Hypothesis H2 : SNo (x * y)
L7404
Hypothesis H3 : (βˆ€z2 : set, z2 ∈ SNoR x β†’ (βˆ€w2 : set, w2 ∈ SNoR y β†’ (z2 * y + x * w2) < x * y + z2 * w2))
L7405
Hypothesis H4 : (βˆ€z2 : set, z2 ∈ SNoR x β†’ (βˆ€w2 : set, w2 ∈ SNoL y β†’ (x * y + z2 * w2) < z2 * y + x * w2))
L7406
Hypothesis H5 : u ∈ SNoR x
L7407
Hypothesis H6 : v ∈ SNoR y
L7408
Hypothesis H7 : z = u * y + x * v + - (u * v)
L7409
Hypothesis H8 : SNo (u * y)
L7410
Hypothesis H9 : SNo (x * v)
L7411
Hypothesis H10 : SNo (u * v)
L7412
Hypothesis H11 : x2 ∈ SNoR x
L7413
Hypothesis H12 : y2 ∈ SNoL y
L7414
Hypothesis H14 : SNo x2
L7415
Hypothesis H15 : SNo y2
L7416
Theorem. (Conj_mul_SNo_eq_3__3__13)
SNo (x2 * y) β†’ z < w
In Proofgold the corresponding term root is 4486a6... and proposition id is 1bd8d1...
Proof:
Proof not loaded.
End of Section Conj_mul_SNo_eq_3__3__13
Beginning of Section Conj_mul_SNo_eq_3__7__14
L7422
Variable x : set
(*** Conj_mul_SNo_eq_3__7__14 TMEmvSygZRa2gu92SpLVLk9mq9P6ACPkKKS bounty of about 25 bars ***)
L7423
Variable y : set
L7424
Variable z : set
L7425
Variable w : set
L7426
Variable u : set
L7427
Variable v : set
L7428
Variable x2 : set
L7429
Hypothesis H0 : SNo x
L7430
Hypothesis H1 : SNo y
L7431
Hypothesis H2 : (βˆ€y2 : set, y2 ∈ z β†’ (βˆ€P : prop, (βˆ€z2 : set, z2 ∈ SNoL x β†’ (βˆ€w2 : set, w2 ∈ SNoR y β†’ y2 = z2 * y + x * w2 + - (z2 * w2) β†’ P)) β†’ (βˆ€z2 : set, z2 ∈ SNoR x β†’ (βˆ€w2 : set, w2 ∈ SNoL y β†’ y2 = z2 * y + x * w2 + - (z2 * w2) β†’ P)) β†’ P))
L7432
Hypothesis H3 : SNo (x * y)
L7433
Hypothesis H4 : (βˆ€y2 : set, y2 ∈ SNoR x β†’ (βˆ€z2 : set, z2 ∈ SNoR y β†’ (y2 * y + x * z2) < x * y + y2 * z2))
L7434
Hypothesis H5 : (βˆ€y2 : set, y2 ∈ SNoL x β†’ (βˆ€z2 : set, z2 ∈ SNoR y β†’ (x * y + y2 * z2) < y2 * y + x * z2))
L7435
Hypothesis H6 : (βˆ€y2 : set, y2 ∈ SNoR x β†’ (βˆ€z2 : set, z2 ∈ SNoL y β†’ (x * y + y2 * z2) < y2 * y + x * z2))
L7436
Hypothesis H7 : u ∈ z
L7437
Hypothesis H8 : v ∈ SNoR x
L7438
Hypothesis H9 : x2 ∈ SNoR y
L7439
Hypothesis H10 : w = v * y + x * x2 + - (v * x2)
L7440
Hypothesis H11 : SNo v
L7441
Hypothesis H12 : SNo x2
L7442
Hypothesis H13 : SNo (v * y)
L7443
Theorem. (Conj_mul_SNo_eq_3__7__14)
SNo (v * x2) β†’ w < u
In Proofgold the corresponding term root is 9cf6aa... and proposition id is 86f21d...
Proof:
Proof not loaded.
End of Section Conj_mul_SNo_eq_3__7__14
Beginning of Section Conj_mul_SNo_eq_3__8__0
L7449
Variable x : set
(*** Conj_mul_SNo_eq_3__8__0 TMNN93asp2pVJta9q1nyWeN24sxa36szYSk bounty of about 25 bars ***)
L7450
Variable y : set
L7451
Variable z : set
L7452
Variable w : set
L7453
Variable u : set
L7454
Variable v : set
L7455
Variable x2 : set
L7456
Hypothesis H1 : SNo y
L7457
Hypothesis H2 : (βˆ€y2 : set, y2 ∈ z β†’ (βˆ€P : prop, (βˆ€z2 : set, z2 ∈ SNoL x β†’ (βˆ€w2 : set, w2 ∈ SNoR y β†’ y2 = z2 * y + x * w2 + - (z2 * w2) β†’ P)) β†’ (βˆ€z2 : set, z2 ∈ SNoR x β†’ (βˆ€w2 : set, w2 ∈ SNoL y β†’ y2 = z2 * y + x * w2 + - (z2 * w2) β†’ P)) β†’ P))
L7458
Hypothesis H3 : SNo (x * y)
L7459
Hypothesis H4 : (βˆ€y2 : set, y2 ∈ SNoR x β†’ (βˆ€z2 : set, z2 ∈ SNoR y β†’ (y2 * y + x * z2) < x * y + y2 * z2))
L7460
Hypothesis H5 : (βˆ€y2 : set, y2 ∈ SNoL x β†’ (βˆ€z2 : set, z2 ∈ SNoR y β†’ (x * y + y2 * z2) < y2 * y + x * z2))
L7461
Hypothesis H6 : (βˆ€y2 : set, y2 ∈ SNoR x β†’ (βˆ€z2 : set, z2 ∈ SNoL y β†’ (x * y + y2 * z2) < y2 * y + x * z2))
L7462
Hypothesis H7 : u ∈ z
L7463
Hypothesis H8 : v ∈ SNoR x
L7464
Hypothesis H9 : x2 ∈ SNoR y
L7465
Hypothesis H10 : w = v * y + x * x2 + - (v * x2)
L7466
Hypothesis H11 : SNo v
L7467
Hypothesis H12 : SNo x2
L7468
Hypothesis H13 : SNo (v * y)
L7469
Theorem. (Conj_mul_SNo_eq_3__8__0)
SNo (x * x2) β†’ w < u
In Proofgold the corresponding term root is ea3b60... and proposition id is c639ee...
Proof:
Proof not loaded.
End of Section Conj_mul_SNo_eq_3__8__0
Beginning of Section Conj_mul_SNo_eq_3__8__2
L7475
Variable x : set
(*** Conj_mul_SNo_eq_3__8__2 TMSBS9FGwaXcQZKArwxx7hw79etCtumQmZ5 bounty of about 25 bars ***)
L7476
Variable y : set
L7477
Variable z : set
L7478
Variable w : set
L7479
Variable u : set
L7480
Variable v : set
L7481
Variable x2 : set
L7482
Hypothesis H0 : SNo x
L7483
Hypothesis H1 : SNo y
L7484
Hypothesis H3 : SNo (x * y)
L7485
Hypothesis H4 : (βˆ€y2 : set, y2 ∈ SNoR x β†’ (βˆ€z2 : set, z2 ∈ SNoR y β†’ (y2 * y + x * z2) < x * y + y2 * z2))
L7486
Hypothesis H5 : (βˆ€y2 : set, y2 ∈ SNoL x β†’ (βˆ€z2 : set, z2 ∈ SNoR y β†’ (x * y + y2 * z2) < y2 * y + x * z2))
L7487
Hypothesis H6 : (βˆ€y2 : set, y2 ∈ SNoR x β†’ (βˆ€z2 : set, z2 ∈ SNoL y β†’ (x * y + y2 * z2) < y2 * y + x * z2))
L7488
Hypothesis H7 : u ∈ z
L7489
Hypothesis H8 : v ∈ SNoR x
L7490
Hypothesis H9 : x2 ∈ SNoR y
L7491
Hypothesis H10 : w = v * y + x * x2 + - (v * x2)
L7492
Hypothesis H11 : SNo v
L7493
Hypothesis H12 : SNo x2
L7494
Hypothesis H13 : SNo (v * y)
L7495
Theorem. (Conj_mul_SNo_eq_3__8__2)
SNo (x * x2) β†’ w < u
In Proofgold the corresponding term root is 180f5e... and proposition id is d3ce1c...
Proof:
Proof not loaded.
End of Section Conj_mul_SNo_eq_3__8__2
Beginning of Section Conj_mul_SNo_eq_3__8__4
L7501
Variable x : set
(*** Conj_mul_SNo_eq_3__8__4 TMRAzhSVoBEfg3KDAhBNRU2PZRNqwePp4cx bounty of about 25 bars ***)
L7502
Variable y : set
L7503
Variable z : set
L7504
Variable w : set
L7505
Variable u : set
L7506
Variable v : set
L7507
Variable x2 : set
L7508
Hypothesis H0 : SNo x
L7509
Hypothesis H1 : SNo y
L7510
Hypothesis H2 : (βˆ€y2 : set, y2 ∈ z β†’ (βˆ€P : prop, (βˆ€z2 : set, z2 ∈ SNoL x β†’ (βˆ€w2 : set, w2 ∈ SNoR y β†’ y2 = z2 * y + x * w2 + - (z2 * w2) β†’ P)) β†’ (βˆ€z2 : set, z2 ∈ SNoR x β†’ (βˆ€w2 : set, w2 ∈ SNoL y β†’ y2 = z2 * y + x * w2 + - (z2 * w2) β†’ P)) β†’ P))
L7511
Hypothesis H3 : SNo (x * y)
L7512
Hypothesis H5 : (βˆ€y2 : set, y2 ∈ SNoL x β†’ (βˆ€z2 : set, z2 ∈ SNoR y β†’ (x * y + y2 * z2) < y2 * y + x * z2))
L7513
Hypothesis H6 : (βˆ€y2 : set, y2 ∈ SNoR x β†’ (βˆ€z2 : set, z2 ∈ SNoL y β†’ (x * y + y2 * z2) < y2 * y + x * z2))
L7514
Hypothesis H7 : u ∈ z
L7515
Hypothesis H8 : v ∈ SNoR x
L7516
Hypothesis H9 : x2 ∈ SNoR y
L7517
Hypothesis H10 : w = v * y + x * x2 + - (v * x2)
L7518
Hypothesis H11 : SNo v
L7519
Hypothesis H12 : SNo x2
L7520
Hypothesis H13 : SNo (v * y)
L7521
Theorem. (Conj_mul_SNo_eq_3__8__4)
SNo (x * x2) β†’ w < u
In Proofgold the corresponding term root is ef07fd... and proposition id is 0b9638...
Proof:
Proof not loaded.
End of Section Conj_mul_SNo_eq_3__8__4
Beginning of Section Conj_mul_SNo_eq_3__8__8
L7527
Variable x : set
(*** Conj_mul_SNo_eq_3__8__8 TMWFFbDwkDUG4mMndpMnUkMHYAGojVrAHC6 bounty of about 25 bars ***)
L7528
Variable y : set
L7529
Variable z : set
L7530
Variable w : set
L7531
Variable u : set
L7532
Variable v : set
L7533
Variable x2 : set
L7534
Hypothesis H0 : SNo x
L7535
Hypothesis H1 : SNo y
L7536
Hypothesis H2 : (βˆ€y2 : set, y2 ∈ z β†’ (βˆ€P : prop, (βˆ€z2 : set, z2 ∈ SNoL x β†’ (βˆ€w2 : set, w2 ∈ SNoR y β†’ y2 = z2 * y + x * w2 + - (z2 * w2) β†’ P)) β†’ (βˆ€z2 : set, z2 ∈ SNoR x β†’ (βˆ€w2 : set, w2 ∈ SNoL y β†’ y2 = z2 * y + x * w2 + - (z2 * w2) β†’ P)) β†’ P))
L7537
Hypothesis H3 : SNo (x * y)
L7538
Hypothesis H4 : (βˆ€y2 : set, y2 ∈ SNoR x β†’ (βˆ€z2 : set, z2 ∈ SNoR y β†’ (y2 * y + x * z2) < x * y + y2 * z2))
L7539
Hypothesis H5 : (βˆ€y2 : set, y2 ∈ SNoL x β†’ (βˆ€z2 : set, z2 ∈ SNoR y β†’ (x * y + y2 * z2) < y2 * y + x * z2))
L7540
Hypothesis H6 : (βˆ€y2 : set, y2 ∈ SNoR x β†’ (βˆ€z2 : set, z2 ∈ SNoL y β†’ (x * y + y2 * z2) < y2 * y + x * z2))
L7541
Hypothesis H7 : u ∈ z
L7542
Hypothesis H9 : x2 ∈ SNoR y
L7543
Hypothesis H10 : w = v * y + x * x2 + - (v * x2)
L7544
Hypothesis H11 : SNo v
L7545
Hypothesis H12 : SNo x2
L7546
Hypothesis H13 : SNo (v * y)
L7547
Theorem. (Conj_mul_SNo_eq_3__8__8)
SNo (x * x2) β†’ w < u
In Proofgold the corresponding term root is 0e17fe... and proposition id is 990c78...
Proof:
Proof not loaded.
End of Section Conj_mul_SNo_eq_3__8__8
Beginning of Section Conj_mul_SNo_eq_3__8__12
L7553
Variable x : set
(*** Conj_mul_SNo_eq_3__8__12 TMSJGbUy1ePEHeFxZ2iC2xe36goVLctVivc bounty of about 25 bars ***)
L7554
Variable y : set
L7555
Variable z : set
L7556
Variable w : set
L7557
Variable u : set
L7558
Variable v : set
L7559
Variable x2 : set
L7560
Hypothesis H0 : SNo x
L7561
Hypothesis H1 : SNo y
L7562
Hypothesis H2 : (βˆ€y2 : set, y2 ∈ z β†’ (βˆ€P : prop, (βˆ€z2 : set, z2 ∈ SNoL x β†’ (βˆ€w2 : set, w2 ∈ SNoR y β†’ y2 = z2 * y + x * w2 + - (z2 * w2) β†’ P)) β†’ (βˆ€z2 : set, z2 ∈ SNoR x β†’ (βˆ€w2 : set, w2 ∈ SNoL y β†’ y2 = z2 * y + x * w2 + - (z2 * w2) β†’ P)) β†’ P))
L7563
Hypothesis H3 : SNo (x * y)
L7564
Hypothesis H4 : (βˆ€y2 : set, y2 ∈ SNoR x β†’ (βˆ€z2 : set, z2 ∈ SNoR y β†’ (y2 * y + x * z2) < x * y + y2 * z2))
L7565
Hypothesis H5 : (βˆ€y2 : set, y2 ∈ SNoL x β†’ (βˆ€z2 : set, z2 ∈ SNoR y β†’ (x * y + y2 * z2) < y2 * y + x * z2))
L7566
Hypothesis H6 : (βˆ€y2 : set, y2 ∈ SNoR x β†’ (βˆ€z2 : set, z2 ∈ SNoL y β†’ (x * y + y2 * z2) < y2 * y + x * z2))
L7567
Hypothesis H7 : u ∈ z
L7568
Hypothesis H8 : v ∈ SNoR x
L7569
Hypothesis H9 : x2 ∈ SNoR y
L7570
Hypothesis H10 : w = v * y + x * x2 + - (v * x2)
L7571
Hypothesis H11 : SNo v
L7572
Hypothesis H13 : SNo (v * y)
L7573
Theorem. (Conj_mul_SNo_eq_3__8__12)
SNo (x * x2) β†’ w < u
In Proofgold the corresponding term root is f1814e... and proposition id is d39549...
Proof:
Proof not loaded.
End of Section Conj_mul_SNo_eq_3__8__12
Beginning of Section Conj_mul_SNo_eq_3__10__3
L7579
Variable x : set
(*** Conj_mul_SNo_eq_3__10__3 TMKsvDDvK4YHLBv3omHWg4pMuWhx6wNTnLG bounty of about 25 bars ***)
L7580
Variable y : set
L7581
Variable z : set
L7582
Variable w : set
L7583
Variable u : set
L7584
Variable v : set
L7585
Variable x2 : set
L7586
Variable y2 : set
L7587
Hypothesis H0 : SNo (x * y)
L7588
Hypothesis H1 : (βˆ€z2 : set, z2 ∈ SNoL x β†’ (βˆ€w2 : set, w2 ∈ SNoL y β†’ (z2 * y + x * w2) < x * y + z2 * w2))
L7589
Hypothesis H2 : (βˆ€z2 : set, z2 ∈ SNoR x β†’ (βˆ€w2 : set, w2 ∈ SNoL y β†’ (x * y + z2 * w2) < z2 * y + x * w2))
L7590
Hypothesis H4 : v ∈ SNoL y
L7591
Hypothesis H5 : z = u * y + x * v + - (u * v)
L7592
Hypothesis H6 : SNo (u * y)
L7593
Hypothesis H7 : SNo (x * v)
L7594
Hypothesis H8 : SNo (u * v)
L7595
Hypothesis H9 : x2 ∈ SNoR x
L7596
Hypothesis H10 : y2 ∈ SNoL y
L7597
Hypothesis H11 : w = x2 * y + x * y2 + - (x2 * y2)
L7598
Hypothesis H12 : SNo x2
L7599
Hypothesis H13 : SNo y2
L7600
Hypothesis H14 : SNo (x2 * y)
L7601
Hypothesis H15 : SNo (x * y2)
L7602
Theorem. (Conj_mul_SNo_eq_3__10__3)
SNo (x2 * y2) β†’ z < w
In Proofgold the corresponding term root is b1bdb6... and proposition id is 53ac42...
Proof:
Proof not loaded.
End of Section Conj_mul_SNo_eq_3__10__3
Beginning of Section Conj_mul_SNo_eq_3__10__14
L7608
Variable x : set
(*** Conj_mul_SNo_eq_3__10__14 TMYVjbuHWrAWyzb1kjW8bR8xy7yUtKrbrSq bounty of about 25 bars ***)
L7609
Variable y : set
L7610
Variable z : set
L7611
Variable w : set
L7612
Variable u : set
L7613
Variable v : set
L7614
Variable x2 : set
L7615
Variable y2 : set
L7616
Hypothesis H0 : SNo (x * y)
L7617
Hypothesis H1 : (βˆ€z2 : set, z2 ∈ SNoL x β†’ (βˆ€w2 : set, w2 ∈ SNoL y β†’ (z2 * y + x * w2) < x * y + z2 * w2))
L7618
Hypothesis H2 : (βˆ€z2 : set, z2 ∈ SNoR x β†’ (βˆ€w2 : set, w2 ∈ SNoL y β†’ (x * y + z2 * w2) < z2 * y + x * w2))
L7619
Hypothesis H3 : u ∈ SNoL x
L7620
Hypothesis H4 : v ∈ SNoL y
L7621
Hypothesis H5 : z = u * y + x * v + - (u * v)
L7622
Hypothesis H6 : SNo (u * y)
L7623
Hypothesis H7 : SNo (x * v)
L7624
Hypothesis H8 : SNo (u * v)
L7625
Hypothesis H9 : x2 ∈ SNoR x
L7626
Hypothesis H10 : y2 ∈ SNoL y
L7627
Hypothesis H11 : w = x2 * y + x * y2 + - (x2 * y2)
L7628
Hypothesis H12 : SNo x2
L7629
Hypothesis H13 : SNo y2
L7630
Hypothesis H15 : SNo (x * y2)
L7631
Theorem. (Conj_mul_SNo_eq_3__10__14)
SNo (x2 * y2) β†’ z < w
In Proofgold the corresponding term root is 75e00a... and proposition id is dcc053...
Proof:
Proof not loaded.
End of Section Conj_mul_SNo_eq_3__10__14
Beginning of Section Conj_mul_SNo_eq_3__10__15
L7637
Variable x : set
(*** Conj_mul_SNo_eq_3__10__15 TMUfct8UDntKCpzcvHRbuu9XigsCnDgrdS2 bounty of about 25 bars ***)
L7638
Variable y : set
L7639
Variable z : set
L7640
Variable w : set
L7641
Variable u : set
L7642
Variable v : set
L7643
Variable x2 : set
L7644
Variable y2 : set
L7645
Hypothesis H0 : SNo (x * y)
L7646
Hypothesis H1 : (βˆ€z2 : set, z2 ∈ SNoL x β†’ (βˆ€w2 : set, w2 ∈ SNoL y β†’ (z2 * y + x * w2) < x * y + z2 * w2))
L7647
Hypothesis H2 : (βˆ€z2 : set, z2 ∈ SNoR x β†’ (βˆ€w2 : set, w2 ∈ SNoL y β†’ (x * y + z2 * w2) < z2 * y + x * w2))
L7648
Hypothesis H3 : u ∈ SNoL x
L7649
Hypothesis H4 : v ∈ SNoL y
L7650
Hypothesis H5 : z = u * y + x * v + - (u * v)
L7651
Hypothesis H6 : SNo (u * y)
L7652
Hypothesis H7 : SNo (x * v)
L7653
Hypothesis H8 : SNo (u * v)
L7654
Hypothesis H9 : x2 ∈ SNoR x
L7655
Hypothesis H10 : y2 ∈ SNoL y
L7656
Hypothesis H11 : w = x2 * y + x * y2 + - (x2 * y2)
L7657
Hypothesis H12 : SNo x2
L7658
Hypothesis H13 : SNo y2
L7659
Hypothesis H14 : SNo (x2 * y)
L7660
Theorem. (Conj_mul_SNo_eq_3__10__15)
SNo (x2 * y2) β†’ z < w
In Proofgold the corresponding term root is 368e3e... and proposition id is 01c710...
Proof:
Proof not loaded.
End of Section Conj_mul_SNo_eq_3__10__15
Beginning of Section Conj_mul_SNo_eq_3__12__1
L7666
Variable x : set
(*** Conj_mul_SNo_eq_3__12__1 TMYdHaeRkktrsPxk6n7A4woGKqtCA5DbjAv bounty of about 25 bars ***)
L7667
Variable y : set
L7668
Variable z : set
L7669
Variable w : set
L7670
Variable u : set
L7671
Variable v : set
L7672
Variable x2 : set
L7673
Variable y2 : set
L7674
Hypothesis H0 : SNo x
L7675
Hypothesis H2 : SNo (x * y)
L7676
Hypothesis H3 : (βˆ€z2 : set, z2 ∈ SNoL x β†’ (βˆ€w2 : set, w2 ∈ SNoL y β†’ (z2 * y + x * w2) < x * y + z2 * w2))
L7677
Hypothesis H4 : (βˆ€z2 : set, z2 ∈ SNoR x β†’ (βˆ€w2 : set, w2 ∈ SNoL y β†’ (x * y + z2 * w2) < z2 * y + x * w2))
L7678
Hypothesis H5 : u ∈ SNoL x
L7679
Hypothesis H6 : v ∈ SNoL y
L7680
Hypothesis H7 : z = u * y + x * v + - (u * v)
L7681
Hypothesis H8 : SNo (u * y)
L7682
Hypothesis H9 : SNo (x * v)
L7683
Hypothesis H10 : SNo (u * v)
L7684
Hypothesis H11 : x2 ∈ SNoR x
L7685
Hypothesis H12 : y2 ∈ SNoL y
L7686
Hypothesis H13 : w = x2 * y + x * y2 + - (x2 * y2)
L7687
Hypothesis H14 : SNo x2
L7688
Hypothesis H15 : SNo y2
L7689
Theorem. (Conj_mul_SNo_eq_3__12__1)
SNo (x2 * y) β†’ z < w
In Proofgold the corresponding term root is 36a280... and proposition id is 19f188...
Proof:
Proof not loaded.
End of Section Conj_mul_SNo_eq_3__12__1
Beginning of Section Conj_mul_SNo_eq_3__12__15
L7695
Variable x : set
(*** Conj_mul_SNo_eq_3__12__15 TMZiaDRwP1qNiaLb1YJDN4UjSQTtAQty6sN bounty of about 25 bars ***)
L7696
Variable y : set
L7697
Variable z : set
L7698
Variable w : set
L7699
Variable u : set
L7700
Variable v : set
L7701
Variable x2 : set
L7702
Variable y2 : set
L7703
Hypothesis H0 : SNo x
L7704
Hypothesis H1 : SNo y
L7705
Hypothesis H2 : SNo (x * y)
L7706
Hypothesis H3 : (βˆ€z2 : set, z2 ∈ SNoL x β†’ (βˆ€w2 : set, w2 ∈ SNoL y β†’ (z2 * y + x * w2) < x * y + z2 * w2))
L7707
Hypothesis H4 : (βˆ€z2 : set, z2 ∈ SNoR x β†’ (βˆ€w2 : set, w2 ∈ SNoL y β†’ (x * y + z2 * w2) < z2 * y + x * w2))
L7708
Hypothesis H5 : u ∈ SNoL x
L7709
Hypothesis H6 : v ∈ SNoL y
L7710
Hypothesis H7 : z = u * y + x * v + - (u * v)
L7711
Hypothesis H8 : SNo (u * y)
L7712
Hypothesis H9 : SNo (x * v)
L7713
Hypothesis H10 : SNo (u * v)
L7714
Hypothesis H11 : x2 ∈ SNoR x
L7715
Hypothesis H12 : y2 ∈ SNoL y
L7716
Hypothesis H13 : w = x2 * y + x * y2 + - (x2 * y2)
L7717
Hypothesis H14 : SNo x2
L7718
Theorem. (Conj_mul_SNo_eq_3__12__15)
SNo (x2 * y) β†’ z < w
In Proofgold the corresponding term root is 2aa34d... and proposition id is 4294d1...
Proof:
Proof not loaded.
End of Section Conj_mul_SNo_eq_3__12__15
Beginning of Section Conj_mul_SNo_eq_3__13__5
L7724
Variable x : set
(*** Conj_mul_SNo_eq_3__13__5 TMMbGuWQ45HwJVcLBXKYCyugCCegQ8vWJsG bounty of about 25 bars ***)
L7725
Variable y : set
L7726
Variable z : set
L7727
Variable w : set
L7728
Variable u : set
L7729
Variable v : set
L7730
Variable x2 : set
L7731
Variable y2 : set
L7732
Hypothesis H0 : SNo (x * y)
L7733
Hypothesis H1 : (βˆ€z2 : set, z2 ∈ SNoL x β†’ (βˆ€w2 : set, w2 ∈ SNoL y β†’ (z2 * y + x * w2) < x * y + z2 * w2))
L7734
Hypothesis H2 : (βˆ€z2 : set, z2 ∈ SNoL x β†’ (βˆ€w2 : set, w2 ∈ SNoR y β†’ (x * y + z2 * w2) < z2 * y + x * w2))
L7735
Hypothesis H3 : u ∈ SNoL x
L7736
Hypothesis H4 : v ∈ SNoL y
L7737
Hypothesis H6 : SNo (u * y)
L7738
Hypothesis H7 : SNo (x * v)
L7739
Hypothesis H8 : SNo (u * v)
L7740
Hypothesis H9 : x2 ∈ SNoL x
L7741
Hypothesis H10 : y2 ∈ SNoR y
L7742
Hypothesis H11 : w = x2 * y + x * y2 + - (x2 * y2)
L7743
Hypothesis H12 : SNo x2
L7744
Hypothesis H13 : SNo y2
L7745
Hypothesis H14 : SNo (x2 * y)
L7746
Hypothesis H15 : SNo (x * y2)
L7747
Theorem. (Conj_mul_SNo_eq_3__13__5)
SNo (x2 * y2) β†’ z < w
In Proofgold the corresponding term root is 642c26... and proposition id is dfa1ad...
Proof:
Proof not loaded.
End of Section Conj_mul_SNo_eq_3__13__5
Beginning of Section Conj_mul_SNo_eq_3__13__8
L7753
Variable x : set
(*** Conj_mul_SNo_eq_3__13__8 TMWkPZpWnKZXabLr71UwwMnLeGYQaQfXdjv bounty of about 25 bars ***)
L7754
Variable y : set
L7755
Variable z : set
L7756
Variable w : set
L7757
Variable u : set
L7758
Variable v : set
L7759
Variable x2 : set
L7760
Variable y2 : set
L7761
Hypothesis H0 : SNo (x * y)
L7762
Hypothesis H1 : (βˆ€z2 : set, z2 ∈ SNoL x β†’ (βˆ€w2 : set, w2 ∈ SNoL y β†’ (z2 * y + x * w2) < x * y + z2 * w2))
L7763
Hypothesis H2 : (βˆ€z2 : set, z2 ∈ SNoL x β†’ (βˆ€w2 : set, w2 ∈ SNoR y β†’ (x * y + z2 * w2) < z2 * y + x * w2))
L7764
Hypothesis H3 : u ∈ SNoL x
L7765
Hypothesis H4 : v ∈ SNoL y
L7766
Hypothesis H5 : z = u * y + x * v + - (u * v)
L7767
Hypothesis H6 : SNo (u * y)
L7768
Hypothesis H7 : SNo (x * v)
L7769
Hypothesis H9 : x2 ∈ SNoL x
L7770
Hypothesis H10 : y2 ∈ SNoR y
L7771
Hypothesis H11 : w = x2 * y + x * y2 + - (x2 * y2)
L7772
Hypothesis H12 : SNo x2
L7773
Hypothesis H13 : SNo y2
L7774
Hypothesis H14 : SNo (x2 * y)
L7775
Hypothesis H15 : SNo (x * y2)
L7776
Theorem. (Conj_mul_SNo_eq_3__13__8)
SNo (x2 * y2) β†’ z < w
In Proofgold the corresponding term root is ca7533... and proposition id is aec889...
Proof:
Proof not loaded.
End of Section Conj_mul_SNo_eq_3__13__8
Beginning of Section Conj_mul_SNo_eq_3__13__13
L7782
Variable x : set
(*** Conj_mul_SNo_eq_3__13__13 TMKFRhjzHfTAFTitp9GfiNhxntNugQfqjcM bounty of about 25 bars ***)
L7783
Variable y : set
L7784
Variable z : set
L7785
Variable w : set
L7786
Variable u : set
L7787
Variable v : set
L7788
Variable x2 : set
L7789
Variable y2 : set
L7790
Hypothesis H0 : SNo (x * y)
L7791
Hypothesis H1 : (βˆ€z2 : set, z2 ∈ SNoL x β†’ (βˆ€w2 : set, w2 ∈ SNoL y β†’ (z2 * y + x * w2) < x * y + z2 * w2))
L7792
Hypothesis H2 : (βˆ€z2 : set, z2 ∈ SNoL x β†’ (βˆ€w2 : set, w2 ∈ SNoR y β†’ (x * y + z2 * w2) < z2 * y + x * w2))
L7793
Hypothesis H3 : u ∈ SNoL x
L7794
Hypothesis H4 : v ∈ SNoL y
L7795
Hypothesis H5 : z = u * y + x * v + - (u * v)
L7796
Hypothesis H6 : SNo (u * y)
L7797
Hypothesis H7 : SNo (x * v)
L7798
Hypothesis H8 : SNo (u * v)
L7799
Hypothesis H9 : x2 ∈ SNoL x
L7800
Hypothesis H10 : y2 ∈ SNoR y
L7801
Hypothesis H11 : w = x2 * y + x * y2 + - (x2 * y2)
L7802
Hypothesis H12 : SNo x2
L7803
Hypothesis H14 : SNo (x2 * y)
L7804
Hypothesis H15 : SNo (x * y2)
L7805
Theorem. (Conj_mul_SNo_eq_3__13__13)
SNo (x2 * y2) β†’ z < w
In Proofgold the corresponding term root is fa9019... and proposition id is b6f698...
Proof:
Proof not loaded.
End of Section Conj_mul_SNo_eq_3__13__13
Beginning of Section Conj_mul_SNo_eq_3__13__14
L7811
Variable x : set
(*** Conj_mul_SNo_eq_3__13__14 TMMyK8wbLBj3TZtB7tnp3qxCXwX8L8ouRQC bounty of about 25 bars ***)
L7812
Variable y : set
L7813
Variable z : set
L7814
Variable w : set
L7815
Variable u : set
L7816
Variable v : set
L7817
Variable x2 : set
L7818
Variable y2 : set
L7819
Hypothesis H0 : SNo (x * y)
L7820
Hypothesis H1 : (βˆ€z2 : set, z2 ∈ SNoL x β†’ (βˆ€w2 : set, w2 ∈ SNoL y β†’ (z2 * y + x * w2) < x * y + z2 * w2))
L7821
Hypothesis H2 : (βˆ€z2 : set, z2 ∈ SNoL x β†’ (βˆ€w2 : set, w2 ∈ SNoR y β†’ (x * y + z2 * w2) < z2 * y + x * w2))
L7822
Hypothesis H3 : u ∈ SNoL x
L7823
Hypothesis H4 : v ∈ SNoL y
L7824
Hypothesis H5 : z = u * y + x * v + - (u * v)
L7825
Hypothesis H6 : SNo (u * y)
L7826
Hypothesis H7 : SNo (x * v)
L7827
Hypothesis H8 : SNo (u * v)
L7828
Hypothesis H9 : x2 ∈ SNoL x
L7829
Hypothesis H10 : y2 ∈ SNoR y
L7830
Hypothesis H11 : w = x2 * y + x * y2 + - (x2 * y2)
L7831
Hypothesis H12 : SNo x2
L7832
Hypothesis H13 : SNo y2
L7833
Hypothesis H15 : SNo (x * y2)
L7834
Theorem. (Conj_mul_SNo_eq_3__13__14)
SNo (x2 * y2) β†’ z < w
In Proofgold the corresponding term root is c53cd1... and proposition id is 358ee1...
Proof:
Proof not loaded.
End of Section Conj_mul_SNo_eq_3__13__14
Beginning of Section Conj_mul_SNo_eq_3__14__2
L7840
Variable x : set
(*** Conj_mul_SNo_eq_3__14__2 TMRTwNfxynUmbQptUgmLPY6iurQErMBDgz1 bounty of about 25 bars ***)
L7841
Variable y : set
L7842
Variable z : set
L7843
Variable w : set
L7844
Variable u : set
L7845
Variable v : set
L7846
Variable x2 : set
L7847
Variable y2 : set
L7848
Hypothesis H0 : SNo x
L7849
Hypothesis H1 : SNo (x * y)
L7850
Hypothesis H3 : (βˆ€z2 : set, z2 ∈ SNoL x β†’ (βˆ€w2 : set, w2 ∈ SNoR y β†’ (x * y + z2 * w2) < z2 * y + x * w2))
L7851
Hypothesis H4 : u ∈ SNoL x
L7852
Hypothesis H5 : v ∈ SNoL y
L7853
Hypothesis H6 : z = u * y + x * v + - (u * v)
L7854
Hypothesis H7 : SNo (u * y)
L7855
Hypothesis H8 : SNo (x * v)
L7856
Hypothesis H9 : SNo (u * v)
L7857
Hypothesis H10 : x2 ∈ SNoL x
L7858
Hypothesis H11 : y2 ∈ SNoR y
L7859
Hypothesis H12 : w = x2 * y + x * y2 + - (x2 * y2)
L7860
Hypothesis H13 : SNo x2
L7861
Hypothesis H14 : SNo y2
L7862
Hypothesis H15 : SNo (x2 * y)
L7863
Theorem. (Conj_mul_SNo_eq_3__14__2)
SNo (x * y2) β†’ z < w
In Proofgold the corresponding term root is 2f63ea... and proposition id is 8f07b4...
Proof:
Proof not loaded.
End of Section Conj_mul_SNo_eq_3__14__2
Beginning of Section Conj_mul_SNo_eq_3__14__12
L7869
Variable x : set
(*** Conj_mul_SNo_eq_3__14__12 TMVH2x4Ko1vr7fX5i71a7WLXVzpDEmyVn1c bounty of about 25 bars ***)
L7870
Variable y : set
L7871
Variable z : set
L7872
Variable w : set
L7873
Variable u : set
L7874
Variable v : set
L7875
Variable x2 : set
L7876
Variable y2 : set
L7877
Hypothesis H0 : SNo x
L7878
Hypothesis H1 : SNo (x * y)
L7879
Hypothesis H2 : (βˆ€z2 : set, z2 ∈ SNoL x β†’ (βˆ€w2 : set, w2 ∈ SNoL y β†’ (z2 * y + x * w2) < x * y + z2 * w2))
L7880
Hypothesis H3 : (βˆ€z2 : set, z2 ∈ SNoL x β†’ (βˆ€w2 : set, w2 ∈ SNoR y β†’ (x * y + z2 * w2) < z2 * y + x * w2))
L7881
Hypothesis H4 : u ∈ SNoL x
L7882
Hypothesis H5 : v ∈ SNoL y
L7883
Hypothesis H6 : z = u * y + x * v + - (u * v)
L7884
Hypothesis H7 : SNo (u * y)
L7885
Hypothesis H8 : SNo (x * v)
L7886
Hypothesis H9 : SNo (u * v)
L7887
Hypothesis H10 : x2 ∈ SNoL x
L7888
Hypothesis H11 : y2 ∈ SNoR y
L7889
Hypothesis H13 : SNo x2
L7890
Hypothesis H14 : SNo y2
L7891
Hypothesis H15 : SNo (x2 * y)
L7892
Theorem. (Conj_mul_SNo_eq_3__14__12)
SNo (x * y2) β†’ z < w
In Proofgold the corresponding term root is e67c6e... and proposition id is 99613b...
Proof:
Proof not loaded.
End of Section Conj_mul_SNo_eq_3__14__12
Beginning of Section Conj_mul_SNo_eq_3__15__4
L7898
Variable x : set
(*** Conj_mul_SNo_eq_3__15__4 TMLhVonSxp18GuKBNKPG3WDe6RQqQq1oXg8 bounty of about 25 bars ***)
L7899
Variable y : set
L7900
Variable z : set
L7901
Variable w : set
L7902
Variable u : set
L7903
Variable v : set
L7904
Variable x2 : set
L7905
Variable y2 : set
L7906
Hypothesis H0 : SNo x
L7907
Hypothesis H1 : SNo y
L7908
Hypothesis H2 : SNo (x * y)
L7909
Hypothesis H3 : (βˆ€z2 : set, z2 ∈ SNoL x β†’ (βˆ€w2 : set, w2 ∈ SNoL y β†’ (z2 * y + x * w2) < x * y + z2 * w2))
L7910
Hypothesis H5 : u ∈ SNoL x
L7911
Hypothesis H6 : v ∈ SNoL y
L7912
Hypothesis H7 : z = u * y + x * v + - (u * v)
L7913
Hypothesis H8 : SNo (u * y)
L7914
Hypothesis H9 : SNo (x * v)
L7915
Hypothesis H10 : SNo (u * v)
L7916
Hypothesis H11 : x2 ∈ SNoL x
L7917
Hypothesis H12 : y2 ∈ SNoR y
L7918
Hypothesis H13 : w = x2 * y + x * y2 + - (x2 * y2)
L7919
Hypothesis H14 : SNo x2
L7920
Hypothesis H15 : SNo y2
L7921
Theorem. (Conj_mul_SNo_eq_3__15__4)
SNo (x2 * y) β†’ z < w
In Proofgold the corresponding term root is 780f22... and proposition id is dee5fa...
Proof:
Proof not loaded.
End of Section Conj_mul_SNo_eq_3__15__4
Beginning of Section Conj_mul_SNo_eq_3__15__15
L7927
Variable x : set
(*** Conj_mul_SNo_eq_3__15__15 TMTcgBRXxjLL51kiyKj3KnerSTbJHq72GKj bounty of about 25 bars ***)
L7928
Variable y : set
L7929
Variable z : set
L7930
Variable w : set
L7931
Variable u : set
L7932
Variable v : set
L7933
Variable x2 : set
L7934
Variable y2 : set
L7935
Hypothesis H0 : SNo x
L7936
Hypothesis H1 : SNo y
L7937
Hypothesis H2 : SNo (x * y)
L7938
Hypothesis H3 : (βˆ€z2 : set, z2 ∈ SNoL x β†’ (βˆ€w2 : set, w2 ∈ SNoL y β†’ (z2 * y + x * w2) < x * y + z2 * w2))
L7939
Hypothesis H4 : (βˆ€z2 : set, z2 ∈ SNoL x β†’ (βˆ€w2 : set, w2 ∈ SNoR y β†’ (x * y + z2 * w2) < z2 * y + x * w2))
L7940
Hypothesis H5 : u ∈ SNoL x
L7941
Hypothesis H6 : v ∈ SNoL y
L7942
Hypothesis H7 : z = u * y + x * v + - (u * v)
L7943
Hypothesis H8 : SNo (u * y)
L7944
Hypothesis H9 : SNo (x * v)
L7945
Hypothesis H10 : SNo (u * v)
L7946
Hypothesis H11 : x2 ∈ SNoL x
L7947
Hypothesis H12 : y2 ∈ SNoR y
L7948
Hypothesis H13 : w = x2 * y + x * y2 + - (x2 * y2)
L7949
Hypothesis H14 : SNo x2
L7950
Theorem. (Conj_mul_SNo_eq_3__15__15)
SNo (x2 * y) β†’ z < w
In Proofgold the corresponding term root is 790473... and proposition id is 5e60fb...
Proof:
Proof not loaded.
End of Section Conj_mul_SNo_eq_3__15__15
Beginning of Section Conj_mul_SNo_Lt__8__12
L7956
Variable x : set
(*** Conj_mul_SNo_Lt__8__12 TMUi2N8CmfnNhkR7dEWRmqEzMtgQvLbreXr bounty of about 25 bars ***)
L7957
Variable y : set
L7958
Variable z : set
L7959
Variable w : set
L7960
Variable u : set
L7961
Hypothesis H0 : SNo z
L7962
Hypothesis H1 : SNo (x * y)
L7963
Hypothesis H2 : SNo (z * y)
L7964
Hypothesis H3 : (βˆ€v : set, v ∈ SNoR z β†’ (βˆ€x2 : set, x2 ∈ SNoL y β†’ (z * y + v * x2) < v * y + z * x2))
L7965
Hypothesis H4 : SNo (x * w)
L7966
Hypothesis H5 : SNo (z * w)
L7967
Hypothesis H6 : (βˆ€v : set, v ∈ SNoR z β†’ (βˆ€x2 : set, x2 ∈ SNoR w β†’ (v * w + z * x2) < z * w + v * x2))
L7968
Hypothesis H7 : SNo (z * y + x * w)
L7969
Hypothesis H8 : SNo (x * y + z * w)
L7970
Hypothesis H9 : x ∈ SNoR z
L7971
Hypothesis H10 : SNo u
L7972
Hypothesis H11 : u ∈ SNoL y
L7973
Hypothesis H13 : SNo (x * u)
L7974
Theorem. (Conj_mul_SNo_Lt__8__12)
SNo (z * u) β†’ (z * y + x * w) < x * y + z * w
In Proofgold the corresponding term root is 2d7b2b... and proposition id is 5603ed...
Proof:
Proof not loaded.
End of Section Conj_mul_SNo_Lt__8__12
Beginning of Section Conj_mul_SNo_Lt__10__5
L7980
Variable x : set
(*** Conj_mul_SNo_Lt__10__5 TMJmsWtTQdmtNAJTrWXUpD5NzyxcqrKaTma bounty of about 25 bars ***)
L7981
Variable y : set
L7982
Variable z : set
L7983
Variable w : set
L7984
Variable u : set
L7985
Hypothesis H0 : SNo x
L7986
Hypothesis H1 : SNo z
L7987
Hypothesis H2 : SNo w
L7988
Hypothesis H3 : SNo (x * y)
L7989
Hypothesis H4 : SNo (z * y)
L7990
Hypothesis H6 : SNo (x * w)
L7991
Hypothesis H7 : SNo (z * w)
L7992
Hypothesis H8 : (βˆ€v : set, v ∈ SNoR z β†’ (βˆ€x2 : set, x2 ∈ SNoR w β†’ (v * w + z * x2) < z * w + v * x2))
L7993
Hypothesis H9 : SNo (z * y + x * w)
L7994
Hypothesis H10 : SNo (x * y + z * w)
L7995
Hypothesis H11 : x ∈ SNoR z
L7996
Hypothesis H12 : SNo u
L7997
Hypothesis H13 : w < u
L7998
Hypothesis H14 : SNoLev u ∈ SNoLev w
L7999
Hypothesis H15 : u ∈ SNoL y
L8000
Theorem. (Conj_mul_SNo_Lt__10__5)
u ∈ SNoR w β†’ (z * y + x * w) < x * y + z * w
In Proofgold the corresponding term root is f5a4cd... and proposition id is 316fb5...
Proof:
Proof not loaded.
End of Section Conj_mul_SNo_Lt__10__5
Beginning of Section Conj_mul_SNo_Lt__13__5
L8006
Variable x : set
(*** Conj_mul_SNo_Lt__13__5 TMW2oTjFewTeCQwyz5MpBnxT2gGL6FkZxyn bounty of about 25 bars ***)
L8007
Variable y : set
L8008
Variable z : set
L8009
Variable w : set
L8010
Hypothesis H0 : SNo y
L8011
Hypothesis H1 : SNo w
L8012
Hypothesis H2 : w < y
L8013
Hypothesis H3 : SNo (x * y)
L8014
Hypothesis H4 : SNo (z * y)
L8015
Hypothesis H6 : (βˆ€u : set, u ∈ SNoL x β†’ (βˆ€v : set, v ∈ SNoR w β†’ (x * w + u * v) < u * w + x * v))
L8016
Hypothesis H7 : SNo (z * w)
L8017
Hypothesis H8 : z ∈ SNoL x
L8018
Hypothesis H9 : SNoLev y ∈ SNoLev w
L8019
Theorem. (Conj_mul_SNo_Lt__13__5)
y ∈ SNoR w β†’ (z * y + x * w) < x * y + z * w
In Proofgold the corresponding term root is b5ecb7... and proposition id is c61b68...
Proof:
Proof not loaded.
End of Section Conj_mul_SNo_Lt__13__5
Beginning of Section Conj_mul_SNo_Lt__15__6
L8025
Variable x : set
(*** Conj_mul_SNo_Lt__15__6 TMLp5puwZgkm8MWLwQ3479bZy3z7SMHtAPh bounty of about 25 bars ***)
L8026
Variable y : set
L8027
Variable z : set
L8028
Variable w : set
L8029
Variable u : set
L8030
Hypothesis H0 : SNo (x * y)
L8031
Hypothesis H1 : SNo (z * y)
L8032
Hypothesis H2 : SNo (x * w)
L8033
Hypothesis H3 : (βˆ€v : set, v ∈ SNoL x β†’ (βˆ€x2 : set, x2 ∈ SNoR w β†’ (x * w + v * x2) < v * w + x * x2))
L8034
Hypothesis H4 : SNo (z * w)
L8035
Hypothesis H5 : SNo (z * y + x * w)
L8036
Hypothesis H7 : z ∈ SNoL x
L8037
Hypothesis H8 : u ∈ SNoR w
L8038
Hypothesis H9 : SNo (x * u)
L8039
Hypothesis H10 : SNo (z * u)
L8040
Hypothesis H11 : SNo (z * y + x * u)
L8041
Hypothesis H12 : SNo (z * w + x * u)
L8042
Hypothesis H13 : SNo (x * y + z * u)
L8043
Hypothesis H14 : SNo (x * w + z * u)
L8044
Hypothesis H15 : (z * y + x * u) < x * y + z * u
L8045
Theorem. (Conj_mul_SNo_Lt__15__6)
(x * w + z * u) < z * w + x * u β†’ (z * y + x * w) < x * y + z * w
In Proofgold the corresponding term root is 49ef97... and proposition id is f0c12d...
Proof:
Proof not loaded.
End of Section Conj_mul_SNo_Lt__15__6
Beginning of Section Conj_mul_SNo_Lt__18__9
L8051
Variable x : set
(*** Conj_mul_SNo_Lt__18__9 TMWviGyBXZCryZU5DPM4KrvMCFb3FdRNNYj bounty of about 25 bars ***)
L8052
Variable y : set
L8053
Variable z : set
L8054
Variable w : set
L8055
Variable u : set
L8056
Hypothesis H0 : SNo (x * y)
L8057
Hypothesis H1 : (βˆ€v : set, v ∈ SNoL x β†’ (βˆ€x2 : set, x2 ∈ SNoL y β†’ (v * y + x * x2) < x * y + v * x2))
L8058
Hypothesis H2 : SNo (z * y)
L8059
Hypothesis H3 : SNo (x * w)
L8060
Hypothesis H4 : (βˆ€v : set, v ∈ SNoL x β†’ (βˆ€x2 : set, x2 ∈ SNoR w β†’ (x * w + v * x2) < v * w + x * x2))
L8061
Hypothesis H5 : SNo (z * w)
L8062
Hypothesis H6 : SNo (z * y + x * w)
L8063
Hypothesis H7 : SNo (x * y + z * w)
L8064
Hypothesis H8 : z ∈ SNoL x
L8065
Hypothesis H10 : u ∈ SNoR w
L8066
Hypothesis H11 : SNo (x * u)
L8067
Hypothesis H12 : SNo (z * u)
L8068
Hypothesis H13 : SNo (z * y + x * u)
L8069
Hypothesis H14 : SNo (z * w + x * u)
L8070
Theorem. (Conj_mul_SNo_Lt__18__9)
SNo (x * y + z * u) β†’ (z * y + x * w) < x * y + z * w
In Proofgold the corresponding term root is 0e5768... and proposition id is aac590...
Proof:
Proof not loaded.
End of Section Conj_mul_SNo_Lt__18__9
Beginning of Section Conj_mul_SNo_Lt__19__2
L8076
Variable x : set
(*** Conj_mul_SNo_Lt__19__2 TMUXoNKxu7hQtWawTtRVjrToq3C7wFSNUZG bounty of about 25 bars ***)
L8077
Variable y : set
L8078
Variable z : set
L8079
Variable w : set
L8080
Variable u : set
L8081
Hypothesis H0 : SNo (x * y)
L8082
Hypothesis H1 : (βˆ€v : set, v ∈ SNoL x β†’ (βˆ€x2 : set, x2 ∈ SNoL y β†’ (v * y + x * x2) < x * y + v * x2))
L8083
Hypothesis H3 : SNo (x * w)
L8084
Hypothesis H4 : (βˆ€v : set, v ∈ SNoL x β†’ (βˆ€x2 : set, x2 ∈ SNoR w β†’ (x * w + v * x2) < v * w + x * x2))
L8085
Hypothesis H5 : SNo (z * w)
L8086
Hypothesis H6 : SNo (z * y + x * w)
L8087
Hypothesis H7 : SNo (x * y + z * w)
L8088
Hypothesis H8 : z ∈ SNoL x
L8089
Hypothesis H9 : u ∈ SNoL y
L8090
Hypothesis H10 : u ∈ SNoR w
L8091
Hypothesis H11 : SNo (x * u)
L8092
Hypothesis H12 : SNo (z * u)
L8093
Hypothesis H13 : SNo (z * y + x * u)
L8094
Theorem. (Conj_mul_SNo_Lt__19__2)
SNo (z * w + x * u) β†’ (z * y + x * w) < x * y + z * w
In Proofgold the corresponding term root is e0318a... and proposition id is ae601e...
Proof:
Proof not loaded.
End of Section Conj_mul_SNo_Lt__19__2