Beginning of Section Conj_mul_SNo_assoc_lem2__104__20
(*** $I sig/Nov2021ConjPreamble.mgs ***)
L4
Variable g : (set → (set → set))
(*** Conj_mul_SNo_assoc_lem2__104__20 TMK6u5DjJQh1yirxpZGirezK2TB3pfkedXk bounty of about 25 bars ***)
L5
Variable x : set
L6
Variable y : set
L7
Variable z : set
L8
Variable w : set
L9
Variable u : set
L10
Hypothesis H0 : (∀v : set, ∀x2 : set, SNo v → SNo x2 → SNo (g v x2))
L11
Hypothesis H1 : (∀v : set, ∀x2 : set, ∀y2 : set, SNo v → SNo x2 → SNo y2 → g v (x2 + y2) = g v x2 + g v y2)
L12
Hypothesis H2 : (∀v : set, ∀x2 : set, ∀y2 : set, SNo v → SNo x2 → SNo y2 → g (v + x2) y2 = g v y2 + g x2 y2)
L13
Hypothesis H3 : (∀v : set, ∀x2 : set, SNo v → SNo x2 → (∀y2 : set, y2 ∈ SNoL (g v x2) → (∀P : prop, (∀z2 : set, z2 ∈ SNoL v → (∀w2 : set, w2 ∈ SNoL x2 → (y2 + g z2 w2) ≤ g z2 x2 + g v w2 → P)) → (∀z2 : set, z2 ∈ SNoR v → (∀w2 : set, w2 ∈ SNoR x2 → (y2 + g z2 w2) ≤ g z2 x2 + g v w2 → P)) → P)))
L14
Hypothesis H4 : (∀v : set, ∀x2 : set, SNo v → SNo x2 → (∀y2 : set, y2 ∈ SNoR (g v x2) → (∀P : prop, (∀z2 : set, z2 ∈ SNoL v → (∀w2 : set, w2 ∈ SNoR x2 → (g z2 x2 + g v w2) ≤ y2 + g z2 w2 → P)) → (∀z2 : set, z2 ∈ SNoR v → (∀w2 : set, w2 ∈ SNoL x2 → (g z2 x2 + g v w2) ≤ y2 + g z2 w2 → P)) → P)))
L15
Hypothesis H5 : (∀v : set, ∀x2 : set, ∀y2 : set, ∀z2 : set, SNo v → SNo x2 → SNo y2 → SNo z2 → y2 < v → z2 < x2 → (g y2 x2 + g v z2) < g v x2 + g y2 z2)
L16
Hypothesis H6 : (∀v : set, ∀x2 : set, ∀y2 : set, ∀z2 : set, SNo v → SNo x2 → SNo y2 → SNo z2 → y2 ≤ v → z2 ≤ x2 → (g y2 x2 + g v z2) ≤ g v x2 + g y2 z2)
L17
Hypothesis H7 : SNo x
L18
Hypothesis H8 : SNo y
L19
Hypothesis H9 : SNo z
L20
Hypothesis H10 : (∀v : set, v ∈ SNoS_ (SNoLev x) → g v (g y z) = g (g v y) z)
L21
Hypothesis H11 : (∀v : set, v ∈ SNoS_ (SNoLev y) → g x (g v z) = g (g x v) z)
L22
Hypothesis H12 : (∀v : set, v ∈ SNoS_ (SNoLev z) → g x (g y v) = g (g x y) v)
L23
Hypothesis H13 : (∀v : set, v ∈ SNoS_ (SNoLev x) → (∀x2 : set, x2 ∈ SNoS_ (SNoLev y) → g v (g x2 z) = g (g v x2) z))
L24
Hypothesis H14 : (∀v : set, v ∈ SNoS_ (SNoLev x) → (∀x2 : set, x2 ∈ SNoS_ (SNoLev z) → g v (g y x2) = g (g v y) x2))
L25
Hypothesis H15 : (∀v : set, v ∈ SNoS_ (SNoLev y) → (∀x2 : set, x2 ∈ SNoS_ (SNoLev z) → g x (g v x2) = g (g x v) x2))
L26
Hypothesis H16 : (∀v : set, v ∈ SNoS_ (SNoLev x) → (∀x2 : set, x2 ∈ SNoS_ (SNoLev y) → (∀y2 : set, y2 ∈ SNoS_ (SNoLev z) → g v (g x2 y2) = g (g v x2) y2)))
L27
Hypothesis H17 : (∀v : set, v ∈ w → (∀P : prop, (∀x2 : set, x2 ∈ SNoL x → (∀y2 : set, y2 ∈ SNoR (g y z) → v = g x2 (g y z) + g x y2 + - (g x2 y2) → P)) → (∀x2 : set, x2 ∈ SNoR x → (∀y2 : set, y2 ∈ SNoL (g y z) → v = g x2 (g y z) + g x y2 + - (g x2 y2) → P)) → P))
L28
Hypothesis H18 : u ∈ w
L29
Hypothesis H19 : SNo (g x y)
L30
Hypothesis H21 : SNo (g (g x y) z)
L31
Theorem. (Conj_mul_SNo_assoc_lem2__104__20)
(∀v : set, v ∈ SNoS_ (SNoLev x) → (∀x2 : set, SNo x2 → (∀y2 : set, y2 ∈ SNoS_ (SNoLev y) → (∀z2 : set, z2 ∈ SNoS_ (SNoLev z) → u = g v (g y z) + g x x2 + - (g v x2) → (g x (g y2 z + g y z2) + g v (x2 + g y2 z2)) ≤ g v (g y2 z + g y z2) + g x (x2 + g y2 z2) → (g (g x y + g v y2) z + g (g v y + g x y2) z2) < g (g v y + g x y2) z + g (g x y + g v y2) z2 → g (g x y) z < u)))) → g (g x y) z < u
In Proofgold the corresponding term root is 693a2b... and proposition id is b9c7d0...
Proof:
Proof not loaded.
End of Section Conj_mul_SNo_assoc_lem2__104__20
Beginning of Section Conj_mul_SNo_assoc_lem2__105__3
L37
Variable g : (set → (set → set))
(*** Conj_mul_SNo_assoc_lem2__105__3 TMPYWVCR1T5YtSVDRtVo4wu6uTCVE1KYosx bounty of about 25 bars ***)
L38
Variable x : set
L39
Variable y : set
L40
Variable z : set
L41
Variable w : set
L42
Variable u : set
L43
Hypothesis H0 : (∀v : set, ∀x2 : set, SNo v → SNo x2 → SNo (g v x2))
L44
Hypothesis H1 : (∀v : set, ∀x2 : set, ∀y2 : set, SNo v → SNo x2 → SNo y2 → g v (x2 + y2) = g v x2 + g v y2)
L45
Hypothesis H2 : (∀v : set, ∀x2 : set, ∀y2 : set, SNo v → SNo x2 → SNo y2 → g (v + x2) y2 = g v y2 + g x2 y2)
L46
Hypothesis H4 : (∀v : set, ∀x2 : set, SNo v → SNo x2 → (∀y2 : set, y2 ∈ SNoR (g v x2) → (∀P : prop, (∀z2 : set, z2 ∈ SNoL v → (∀w2 : set, w2 ∈ SNoR x2 → (g z2 x2 + g v w2) ≤ y2 + g z2 w2 → P)) → (∀z2 : set, z2 ∈ SNoR v → (∀w2 : set, w2 ∈ SNoL x2 → (g z2 x2 + g v w2) ≤ y2 + g z2 w2 → P)) → P)))
L47
Hypothesis H5 : (∀v : set, ∀x2 : set, ∀y2 : set, ∀z2 : set, SNo v → SNo x2 → SNo y2 → SNo z2 → y2 < v → z2 < x2 → (g y2 x2 + g v z2) < g v x2 + g y2 z2)
L48
Hypothesis H6 : (∀v : set, ∀x2 : set, ∀y2 : set, ∀z2 : set, SNo v → SNo x2 → SNo y2 → SNo z2 → y2 ≤ v → z2 ≤ x2 → (g y2 x2 + g v z2) ≤ g v x2 + g y2 z2)
L49
Hypothesis H7 : SNo x
L50
Hypothesis H8 : SNo y
L51
Hypothesis H9 : SNo z
L52
Hypothesis H10 : (∀v : set, v ∈ SNoS_ (SNoLev x) → g v (g y z) = g (g v y) z)
L53
Hypothesis H11 : (∀v : set, v ∈ SNoS_ (SNoLev y) → g x (g v z) = g (g x v) z)
L54
Hypothesis H12 : (∀v : set, v ∈ SNoS_ (SNoLev z) → g x (g y v) = g (g x y) v)
L55
Hypothesis H13 : (∀v : set, v ∈ SNoS_ (SNoLev x) → (∀x2 : set, x2 ∈ SNoS_ (SNoLev y) → g v (g x2 z) = g (g v x2) z))
L56
Hypothesis H14 : (∀v : set, v ∈ SNoS_ (SNoLev x) → (∀x2 : set, x2 ∈ SNoS_ (SNoLev z) → g v (g y x2) = g (g v y) x2))
L57
Hypothesis H15 : (∀v : set, v ∈ SNoS_ (SNoLev y) → (∀x2 : set, x2 ∈ SNoS_ (SNoLev z) → g x (g v x2) = g (g x v) x2))
L58
Hypothesis H16 : (∀v : set, v ∈ SNoS_ (SNoLev x) → (∀x2 : set, x2 ∈ SNoS_ (SNoLev y) → (∀y2 : set, y2 ∈ SNoS_ (SNoLev z) → g v (g x2 y2) = g (g v x2) y2)))
L59
Hypothesis H17 : (∀v : set, v ∈ w → (∀P : prop, (∀x2 : set, x2 ∈ SNoL x → (∀y2 : set, y2 ∈ SNoR (g y z) → v = g x2 (g y z) + g x y2 + - (g x2 y2) → P)) → (∀x2 : set, x2 ∈ SNoR x → (∀y2 : set, y2 ∈ SNoL (g y z) → v = g x2 (g y z) + g x y2 + - (g x2 y2) → P)) → P))
L60
Hypothesis H18 : u ∈ w
L61
Hypothesis H19 : SNo (g x y)
L62
Hypothesis H20 : SNo (g y z)
L63
Theorem. (Conj_mul_SNo_assoc_lem2__105__3)
SNo (g (g x y) z) → g (g x y) z < u
In Proofgold the corresponding term root is 5c7298... and proposition id is 965044...
Proof:
Proof not loaded.
End of Section Conj_mul_SNo_assoc_lem2__105__3
Beginning of Section Conj_mul_SNo_assoc_lem2__105__5
L69
Variable g : (set → (set → set))
(*** Conj_mul_SNo_assoc_lem2__105__5 TMYuBvd6eZcAY8qsc2DMHcEB7nrCiXYsEKi bounty of about 25 bars ***)
L70
Variable x : set
L71
Variable y : set
L72
Variable z : set
L73
Variable w : set
L74
Variable u : set
L75
Hypothesis H0 : (∀v : set, ∀x2 : set, SNo v → SNo x2 → SNo (g v x2))
L76
Hypothesis H1 : (∀v : set, ∀x2 : set, ∀y2 : set, SNo v → SNo x2 → SNo y2 → g v (x2 + y2) = g v x2 + g v y2)
L77
Hypothesis H2 : (∀v : set, ∀x2 : set, ∀y2 : set, SNo v → SNo x2 → SNo y2 → g (v + x2) y2 = g v y2 + g x2 y2)
L78
Hypothesis H3 : (∀v : set, ∀x2 : set, SNo v → SNo x2 → (∀y2 : set, y2 ∈ SNoL (g v x2) → (∀P : prop, (∀z2 : set, z2 ∈ SNoL v → (∀w2 : set, w2 ∈ SNoL x2 → (y2 + g z2 w2) ≤ g z2 x2 + g v w2 → P)) → (∀z2 : set, z2 ∈ SNoR v → (∀w2 : set, w2 ∈ SNoR x2 → (y2 + g z2 w2) ≤ g z2 x2 + g v w2 → P)) → P)))
L79
Hypothesis H4 : (∀v : set, ∀x2 : set, SNo v → SNo x2 → (∀y2 : set, y2 ∈ SNoR (g v x2) → (∀P : prop, (∀z2 : set, z2 ∈ SNoL v → (∀w2 : set, w2 ∈ SNoR x2 → (g z2 x2 + g v w2) ≤ y2 + g z2 w2 → P)) → (∀z2 : set, z2 ∈ SNoR v → (∀w2 : set, w2 ∈ SNoL x2 → (g z2 x2 + g v w2) ≤ y2 + g z2 w2 → P)) → P)))
L80
Hypothesis H6 : (∀v : set, ∀x2 : set, ∀y2 : set, ∀z2 : set, SNo v → SNo x2 → SNo y2 → SNo z2 → y2 ≤ v → z2 ≤ x2 → (g y2 x2 + g v z2) ≤ g v x2 + g y2 z2)
L81
Hypothesis H7 : SNo x
L82
Hypothesis H8 : SNo y
L83
Hypothesis H9 : SNo z
L84
Hypothesis H10 : (∀v : set, v ∈ SNoS_ (SNoLev x) → g v (g y z) = g (g v y) z)
L85
Hypothesis H11 : (∀v : set, v ∈ SNoS_ (SNoLev y) → g x (g v z) = g (g x v) z)
L86
Hypothesis H12 : (∀v : set, v ∈ SNoS_ (SNoLev z) → g x (g y v) = g (g x y) v)
L87
Hypothesis H13 : (∀v : set, v ∈ SNoS_ (SNoLev x) → (∀x2 : set, x2 ∈ SNoS_ (SNoLev y) → g v (g x2 z) = g (g v x2) z))
L88
Hypothesis H14 : (∀v : set, v ∈ SNoS_ (SNoLev x) → (∀x2 : set, x2 ∈ SNoS_ (SNoLev z) → g v (g y x2) = g (g v y) x2))
L89
Hypothesis H15 : (∀v : set, v ∈ SNoS_ (SNoLev y) → (∀x2 : set, x2 ∈ SNoS_ (SNoLev z) → g x (g v x2) = g (g x v) x2))
L90
Hypothesis H16 : (∀v : set, v ∈ SNoS_ (SNoLev x) → (∀x2 : set, x2 ∈ SNoS_ (SNoLev y) → (∀y2 : set, y2 ∈ SNoS_ (SNoLev z) → g v (g x2 y2) = g (g v x2) y2)))
L91
Hypothesis H17 : (∀v : set, v ∈ w → (∀P : prop, (∀x2 : set, x2 ∈ SNoL x → (∀y2 : set, y2 ∈ SNoR (g y z) → v = g x2 (g y z) + g x y2 + - (g x2 y2) → P)) → (∀x2 : set, x2 ∈ SNoR x → (∀y2 : set, y2 ∈ SNoL (g y z) → v = g x2 (g y z) + g x y2 + - (g x2 y2) → P)) → P))
L92
Hypothesis H18 : u ∈ w
L93
Hypothesis H19 : SNo (g x y)
L94
Hypothesis H20 : SNo (g y z)
L95
Theorem. (Conj_mul_SNo_assoc_lem2__105__5)
SNo (g (g x y) z) → g (g x y) z < u
In Proofgold the corresponding term root is cb8452... and proposition id is d1a905...
Proof:
Proof not loaded.
End of Section Conj_mul_SNo_assoc_lem2__105__5
Beginning of Section Conj_mul_SNo_assoc_lem2__105__8
L101
Variable g : (set → (set → set))
(*** Conj_mul_SNo_assoc_lem2__105__8 TMWAfofFQMt7amBFC2oFkmZg7hpKA41Jmhi bounty of about 25 bars ***)
L102
Variable x : set
L103
Variable y : set
L104
Variable z : set
L105
Variable w : set
L106
Variable u : set
L107
Hypothesis H0 : (∀v : set, ∀x2 : set, SNo v → SNo x2 → SNo (g v x2))
L108
Hypothesis H1 : (∀v : set, ∀x2 : set, ∀y2 : set, SNo v → SNo x2 → SNo y2 → g v (x2 + y2) = g v x2 + g v y2)
L109
Hypothesis H2 : (∀v : set, ∀x2 : set, ∀y2 : set, SNo v → SNo x2 → SNo y2 → g (v + x2) y2 = g v y2 + g x2 y2)
L110
Hypothesis H3 : (∀v : set, ∀x2 : set, SNo v → SNo x2 → (∀y2 : set, y2 ∈ SNoL (g v x2) → (∀P : prop, (∀z2 : set, z2 ∈ SNoL v → (∀w2 : set, w2 ∈ SNoL x2 → (y2 + g z2 w2) ≤ g z2 x2 + g v w2 → P)) → (∀z2 : set, z2 ∈ SNoR v → (∀w2 : set, w2 ∈ SNoR x2 → (y2 + g z2 w2) ≤ g z2 x2 + g v w2 → P)) → P)))
L111
Hypothesis H4 : (∀v : set, ∀x2 : set, SNo v → SNo x2 → (∀y2 : set, y2 ∈ SNoR (g v x2) → (∀P : prop, (∀z2 : set, z2 ∈ SNoL v → (∀w2 : set, w2 ∈ SNoR x2 → (g z2 x2 + g v w2) ≤ y2 + g z2 w2 → P)) → (∀z2 : set, z2 ∈ SNoR v → (∀w2 : set, w2 ∈ SNoL x2 → (g z2 x2 + g v w2) ≤ y2 + g z2 w2 → P)) → P)))
L112
Hypothesis H5 : (∀v : set, ∀x2 : set, ∀y2 : set, ∀z2 : set, SNo v → SNo x2 → SNo y2 → SNo z2 → y2 < v → z2 < x2 → (g y2 x2 + g v z2) < g v x2 + g y2 z2)
L113
Hypothesis H6 : (∀v : set, ∀x2 : set, ∀y2 : set, ∀z2 : set, SNo v → SNo x2 → SNo y2 → SNo z2 → y2 ≤ v → z2 ≤ x2 → (g y2 x2 + g v z2) ≤ g v x2 + g y2 z2)
L114
Hypothesis H7 : SNo x
L115
Hypothesis H9 : SNo z
L116
Hypothesis H10 : (∀v : set, v ∈ SNoS_ (SNoLev x) → g v (g y z) = g (g v y) z)
L117
Hypothesis H11 : (∀v : set, v ∈ SNoS_ (SNoLev y) → g x (g v z) = g (g x v) z)
L118
Hypothesis H12 : (∀v : set, v ∈ SNoS_ (SNoLev z) → g x (g y v) = g (g x y) v)
L119
Hypothesis H13 : (∀v : set, v ∈ SNoS_ (SNoLev x) → (∀x2 : set, x2 ∈ SNoS_ (SNoLev y) → g v (g x2 z) = g (g v x2) z))
L120
Hypothesis H14 : (∀v : set, v ∈ SNoS_ (SNoLev x) → (∀x2 : set, x2 ∈ SNoS_ (SNoLev z) → g v (g y x2) = g (g v y) x2))
L121
Hypothesis H15 : (∀v : set, v ∈ SNoS_ (SNoLev y) → (∀x2 : set, x2 ∈ SNoS_ (SNoLev z) → g x (g v x2) = g (g x v) x2))
L122
Hypothesis H16 : (∀v : set, v ∈ SNoS_ (SNoLev x) → (∀x2 : set, x2 ∈ SNoS_ (SNoLev y) → (∀y2 : set, y2 ∈ SNoS_ (SNoLev z) → g v (g x2 y2) = g (g v x2) y2)))
L123
Hypothesis H17 : (∀v : set, v ∈ w → (∀P : prop, (∀x2 : set, x2 ∈ SNoL x → (∀y2 : set, y2 ∈ SNoR (g y z) → v = g x2 (g y z) + g x y2 + - (g x2 y2) → P)) → (∀x2 : set, x2 ∈ SNoR x → (∀y2 : set, y2 ∈ SNoL (g y z) → v = g x2 (g y z) + g x y2 + - (g x2 y2) → P)) → P))
L124
Hypothesis H18 : u ∈ w
L125
Hypothesis H19 : SNo (g x y)
L126
Hypothesis H20 : SNo (g y z)
L127
Theorem. (Conj_mul_SNo_assoc_lem2__105__8)
SNo (g (g x y) z) → g (g x y) z < u
In Proofgold the corresponding term root is 669fc5... and proposition id is ee6650...
Proof:
Proof not loaded.
End of Section Conj_mul_SNo_assoc_lem2__105__8
Beginning of Section Conj_mul_SNo_assoc_lem2__106__8
L133
Variable g : (set → (set → set))
(*** Conj_mul_SNo_assoc_lem2__106__8 TMWMdmixDcm5Dz1fmjipp2yRtdXkQnTATBY bounty of about 25 bars ***)
L134
Variable x : set
L135
Variable y : set
L136
Variable z : set
L137
Variable w : set
L138
Variable u : set
L139
Hypothesis H0 : (∀v : set, ∀x2 : set, SNo v → SNo x2 → SNo (g v x2))
L140
Hypothesis H1 : (∀v : set, ∀x2 : set, ∀y2 : set, SNo v → SNo x2 → SNo y2 → g v (x2 + y2) = g v x2 + g v y2)
L141
Hypothesis H2 : (∀v : set, ∀x2 : set, ∀y2 : set, SNo v → SNo x2 → SNo y2 → g (v + x2) y2 = g v y2 + g x2 y2)
L142
Hypothesis H3 : (∀v : set, ∀x2 : set, SNo v → SNo x2 → (∀y2 : set, y2 ∈ SNoL (g v x2) → (∀P : prop, (∀z2 : set, z2 ∈ SNoL v → (∀w2 : set, w2 ∈ SNoL x2 → (y2 + g z2 w2) ≤ g z2 x2 + g v w2 → P)) → (∀z2 : set, z2 ∈ SNoR v → (∀w2 : set, w2 ∈ SNoR x2 → (y2 + g z2 w2) ≤ g z2 x2 + g v w2 → P)) → P)))
L143
Hypothesis H4 : (∀v : set, ∀x2 : set, SNo v → SNo x2 → (∀y2 : set, y2 ∈ SNoR (g v x2) → (∀P : prop, (∀z2 : set, z2 ∈ SNoL v → (∀w2 : set, w2 ∈ SNoR x2 → (g z2 x2 + g v w2) ≤ y2 + g z2 w2 → P)) → (∀z2 : set, z2 ∈ SNoR v → (∀w2 : set, w2 ∈ SNoL x2 → (g z2 x2 + g v w2) ≤ y2 + g z2 w2 → P)) → P)))
L144
Hypothesis H5 : (∀v : set, ∀x2 : set, ∀y2 : set, ∀z2 : set, SNo v → SNo x2 → SNo y2 → SNo z2 → y2 < v → z2 < x2 → (g y2 x2 + g v z2) < g v x2 + g y2 z2)
L145
Hypothesis H6 : (∀v : set, ∀x2 : set, ∀y2 : set, ∀z2 : set, SNo v → SNo x2 → SNo y2 → SNo z2 → y2 ≤ v → z2 ≤ x2 → (g y2 x2 + g v z2) ≤ g v x2 + g y2 z2)
L146
Hypothesis H7 : SNo x
L147
Hypothesis H9 : SNo z
L148
Hypothesis H10 : (∀v : set, v ∈ SNoS_ (SNoLev x) → g v (g y z) = g (g v y) z)
L149
Hypothesis H11 : (∀v : set, v ∈ SNoS_ (SNoLev y) → g x (g v z) = g (g x v) z)
L150
Hypothesis H12 : (∀v : set, v ∈ SNoS_ (SNoLev z) → g x (g y v) = g (g x y) v)
L151
Hypothesis H13 : (∀v : set, v ∈ SNoS_ (SNoLev x) → (∀x2 : set, x2 ∈ SNoS_ (SNoLev y) → g v (g x2 z) = g (g v x2) z))
L152
Hypothesis H14 : (∀v : set, v ∈ SNoS_ (SNoLev x) → (∀x2 : set, x2 ∈ SNoS_ (SNoLev z) → g v (g y x2) = g (g v y) x2))
L153
Hypothesis H15 : (∀v : set, v ∈ SNoS_ (SNoLev y) → (∀x2 : set, x2 ∈ SNoS_ (SNoLev z) → g x (g v x2) = g (g x v) x2))
L154
Hypothesis H16 : (∀v : set, v ∈ SNoS_ (SNoLev x) → (∀x2 : set, x2 ∈ SNoS_ (SNoLev y) → (∀y2 : set, y2 ∈ SNoS_ (SNoLev z) → g v (g x2 y2) = g (g v x2) y2)))
L155
Hypothesis H17 : (∀v : set, v ∈ w → (∀P : prop, (∀x2 : set, x2 ∈ SNoL x → (∀y2 : set, y2 ∈ SNoR (g y z) → v = g x2 (g y z) + g x y2 + - (g x2 y2) → P)) → (∀x2 : set, x2 ∈ SNoR x → (∀y2 : set, y2 ∈ SNoL (g y z) → v = g x2 (g y z) + g x y2 + - (g x2 y2) → P)) → P))
L156
Hypothesis H18 : u ∈ w
L157
Hypothesis H19 : SNo (g x y)
L158
Theorem. (Conj_mul_SNo_assoc_lem2__106__8)
SNo (g y z) → g (g x y) z < u
In Proofgold the corresponding term root is 27723a... and proposition id is 150469...
Proof:
Proof not loaded.
End of Section Conj_mul_SNo_assoc_lem2__106__8
Beginning of Section Conj_mul_SNo_assoc_lem2__107__3
L164
Variable g : (set → (set → set))
(*** Conj_mul_SNo_assoc_lem2__107__3 TMMwe7swEjYDLfswiADW22LntB3yMrmiYBa bounty of about 25 bars ***)
L165
Variable x : set
L166
Variable y : set
L167
Variable z : set
L168
Variable w : set
L169
Variable u : set
L170
Hypothesis H0 : (∀v : set, ∀x2 : set, SNo v → SNo x2 → SNo (g v x2))
L171
Hypothesis H1 : (∀v : set, ∀x2 : set, ∀y2 : set, SNo v → SNo x2 → SNo y2 → g v (x2 + y2) = g v x2 + g v y2)
L172
Hypothesis H2 : (∀v : set, ∀x2 : set, ∀y2 : set, SNo v → SNo x2 → SNo y2 → g (v + x2) y2 = g v y2 + g x2 y2)
L173
Hypothesis H4 : (∀v : set, ∀x2 : set, SNo v → SNo x2 → (∀y2 : set, y2 ∈ SNoR (g v x2) → (∀P : prop, (∀z2 : set, z2 ∈ SNoL v → (∀w2 : set, w2 ∈ SNoR x2 → (g z2 x2 + g v w2) ≤ y2 + g z2 w2 → P)) → (∀z2 : set, z2 ∈ SNoR v → (∀w2 : set, w2 ∈ SNoL x2 → (g z2 x2 + g v w2) ≤ y2 + g z2 w2 → P)) → P)))
L174
Hypothesis H5 : (∀v : set, ∀x2 : set, ∀y2 : set, ∀z2 : set, SNo v → SNo x2 → SNo y2 → SNo z2 → y2 < v → z2 < x2 → (g y2 x2 + g v z2) < g v x2 + g y2 z2)
L175
Hypothesis H6 : (∀v : set, ∀x2 : set, ∀y2 : set, ∀z2 : set, SNo v → SNo x2 → SNo y2 → SNo z2 → y2 ≤ v → z2 ≤ x2 → (g y2 x2 + g v z2) ≤ g v x2 + g y2 z2)
L176
Hypothesis H7 : SNo x
L177
Hypothesis H8 : SNo y
L178
Hypothesis H9 : SNo z
L179
Hypothesis H10 : (∀v : set, v ∈ SNoS_ (SNoLev x) → g v (g y z) = g (g v y) z)
L180
Hypothesis H11 : (∀v : set, v ∈ SNoS_ (SNoLev y) → g x (g v z) = g (g x v) z)
L181
Hypothesis H12 : (∀v : set, v ∈ SNoS_ (SNoLev z) → g x (g y v) = g (g x y) v)
L182
Hypothesis H13 : (∀v : set, v ∈ SNoS_ (SNoLev x) → (∀x2 : set, x2 ∈ SNoS_ (SNoLev y) → g v (g x2 z) = g (g v x2) z))
L183
Hypothesis H14 : (∀v : set, v ∈ SNoS_ (SNoLev x) → (∀x2 : set, x2 ∈ SNoS_ (SNoLev z) → g v (g y x2) = g (g v y) x2))
L184
Hypothesis H15 : (∀v : set, v ∈ SNoS_ (SNoLev y) → (∀x2 : set, x2 ∈ SNoS_ (SNoLev z) → g x (g v x2) = g (g x v) x2))
L185
Hypothesis H16 : (∀v : set, v ∈ SNoS_ (SNoLev x) → (∀x2 : set, x2 ∈ SNoS_ (SNoLev y) → (∀y2 : set, y2 ∈ SNoS_ (SNoLev z) → g v (g x2 y2) = g (g v x2) y2)))
L186
Hypothesis H17 : (∀v : set, v ∈ w → (∀P : prop, (∀x2 : set, x2 ∈ SNoL x → (∀y2 : set, y2 ∈ SNoR (g y z) → v = g x2 (g y z) + g x y2 + - (g x2 y2) → P)) → (∀x2 : set, x2 ∈ SNoR x → (∀y2 : set, y2 ∈ SNoL (g y z) → v = g x2 (g y z) + g x y2 + - (g x2 y2) → P)) → P))
L187
Hypothesis H18 : u ∈ w
L188
Theorem. (Conj_mul_SNo_assoc_lem2__107__3)
SNo (g x y) → g (g x y) z < u
In Proofgold the corresponding term root is 3e53df... and proposition id is 4451aa...
Proof:
Proof not loaded.
End of Section Conj_mul_SNo_assoc_lem2__107__3
Beginning of Section Conj_mul_SNo_assoc__1__15
L194
Variable x : set
(*** Conj_mul_SNo_assoc__1__15 TMYr9xoXoDRbqpZkV5pui5VVJR3DXoy3KPs bounty of about 25 bars ***)
L195
Variable y : set
L196
Variable z : set
L197
Variable w : set
L198
Variable u : set
L199
Variable v : set
L200
Variable x2 : set
L201
Hypothesis H0 : SNo x
L202
Hypothesis H1 : SNo y
L203
Hypothesis H2 : SNo z
L204
Hypothesis H3 : (∀y2 : set, y2 ∈ SNoS_ (SNoLev x) → y2 * y * z = (y2 * y) * z)
L205
Hypothesis H4 : (∀y2 : set, y2 ∈ SNoS_ (SNoLev y) → x * y2 * z = (x * y2) * z)
L206
Hypothesis H5 : (∀y2 : set, y2 ∈ SNoS_ (SNoLev z) → x * y * y2 = (x * y) * y2)
L207
Hypothesis H6 : (∀y2 : set, y2 ∈ SNoS_ (SNoLev x) → (∀z2 : set, z2 ∈ SNoS_ (SNoLev y) → y2 * z2 * z = (y2 * z2) * z))
L208
Hypothesis H7 : (∀y2 : set, y2 ∈ SNoS_ (SNoLev x) → (∀z2 : set, z2 ∈ SNoS_ (SNoLev z) → y2 * y * z2 = (y2 * y) * z2))
L209
Hypothesis H8 : (∀y2 : set, y2 ∈ SNoS_ (SNoLev y) → (∀z2 : set, z2 ∈ SNoS_ (SNoLev z) → x * y2 * z2 = (x * y2) * z2))
L210
Hypothesis H9 : (∀y2 : set, y2 ∈ SNoS_ (SNoLev x) → (∀z2 : set, z2 ∈ SNoS_ (SNoLev y) → (∀w2 : set, w2 ∈ SNoS_ (SNoLev z) → y2 * z2 * w2 = (y2 * z2) * w2)))
L211
Hypothesis H10 : SNo (x * y)
L212
Hypothesis H11 : SNoCutP w u
L213
Hypothesis H12 : (∀y2 : set, y2 ∈ w → (∀P : prop, (∀z2 : set, z2 ∈ SNoL x → (∀w2 : set, w2 ∈ SNoL (y * z) → y2 = z2 * y * z + x * w2 + - (z2 * w2) → P)) → (∀z2 : set, z2 ∈ SNoR x → (∀w2 : set, w2 ∈ SNoR (y * z) → y2 = z2 * y * z + x * w2 + - (z2 * w2) → P)) → P))
L214
Hypothesis H13 : (∀y2 : set, y2 ∈ u → (∀P : prop, (∀z2 : set, z2 ∈ SNoL x → (∀w2 : set, w2 ∈ SNoR (y * z) → y2 = z2 * y * z + x * w2 + - (z2 * w2) → P)) → (∀z2 : set, z2 ∈ SNoR x → (∀w2 : set, w2 ∈ SNoL (y * z) → y2 = z2 * y * z + x * w2 + - (z2 * w2) → P)) → P))
L215
Hypothesis H14 : x * y * z = SNoCut w u
L216
Hypothesis H16 : (∀y2 : set, y2 ∈ v → (∀P : prop, (∀z2 : set, z2 ∈ SNoL (x * y) → (∀w2 : set, w2 ∈ SNoL z → y2 = z2 * z + (x * y) * w2 + - (z2 * w2) → P)) → (∀z2 : set, z2 ∈ SNoR (x * y) → (∀w2 : set, w2 ∈ SNoR z → y2 = z2 * z + (x * y) * w2 + - (z2 * w2) → P)) → P))
L217
Hypothesis H17 : (∀y2 : set, y2 ∈ x2 → (∀P : prop, (∀z2 : set, z2 ∈ SNoL (x * y) → (∀w2 : set, w2 ∈ SNoR z → y2 = z2 * z + (x * y) * w2 + - (z2 * w2) → P)) → (∀z2 : set, z2 ∈ SNoR (x * y) → (∀w2 : set, w2 ∈ SNoL z → y2 = z2 * z + (x * y) * w2 + - (z2 * w2) → P)) → P))
L218
Hypothesis H18 : (x * y) * z = SNoCut v x2
L219
Hypothesis H19 : (∀y2 : set, ∀z2 : set, SNo y2 → SNo z2 → (∀w2 : set, w2 ∈ SNoL (z2 * y2) → (∀P : prop, (∀u2 : set, u2 ∈ SNoL y2 → (∀v2 : set, v2 ∈ SNoL z2 → (w2 + v2 * u2) ≤ z2 * u2 + v2 * y2 → P)) → (∀u2 : set, u2 ∈ SNoR y2 → (∀v2 : set, v2 ∈ SNoR z2 → (w2 + v2 * u2) ≤ z2 * u2 + v2 * y2 → P)) → P)))
L220
Hypothesis H20 : (∀y2 : set, ∀z2 : set, SNo y2 → SNo z2 → (∀w2 : set, w2 ∈ SNoR (z2 * y2) → (∀P : prop, (∀u2 : set, u2 ∈ SNoL y2 → (∀v2 : set, v2 ∈ SNoR z2 → (z2 * u2 + v2 * y2) ≤ w2 + v2 * u2 → P)) → (∀u2 : set, u2 ∈ SNoR y2 → (∀v2 : set, v2 ∈ SNoL z2 → (z2 * u2 + v2 * y2) ≤ w2 + v2 * u2 → P)) → P)))
L221
Hypothesis H21 : (∀y2 : set, ∀z2 : set, ∀w2 : set, ∀u2 : set, SNo y2 → SNo z2 → SNo w2 → SNo u2 → w2 < y2 → u2 < z2 → (z2 * w2 + u2 * y2) < z2 * y2 + u2 * w2)
L222
Hypothesis H22 : (∀y2 : set, ∀z2 : set, ∀w2 : set, ∀u2 : set, SNo y2 → SNo z2 → SNo w2 → SNo u2 → w2 ≤ y2 → u2 ≤ z2 → (z2 * w2 + u2 * y2) ≤ z2 * y2 + u2 * w2)
L223
Hypothesis H23 : (∀y2 : set, y2 ∈ SNoS_ (SNoLev x) → (y2 * y) * z = y2 * y * z)
L224
Hypothesis H24 : (∀y2 : set, y2 ∈ SNoS_ (SNoLev y) → (x * y2) * z = x * y2 * z)
L225
Hypothesis H25 : (∀y2 : set, y2 ∈ SNoS_ (SNoLev z) → (x * y) * y2 = x * y * y2)
L226
Hypothesis H26 : (∀y2 : set, y2 ∈ SNoS_ (SNoLev y) → (∀z2 : set, z2 ∈ SNoS_ (SNoLev x) → (z2 * y2) * z = z2 * y2 * z))
L227
Hypothesis H27 : (∀y2 : set, y2 ∈ SNoS_ (SNoLev z) → (∀z2 : set, z2 ∈ SNoS_ (SNoLev x) → (z2 * y) * y2 = z2 * y * y2))
L228
Hypothesis H28 : (∀y2 : set, y2 ∈ SNoS_ (SNoLev z) → (∀z2 : set, z2 ∈ SNoS_ (SNoLev y) → (x * z2) * y2 = x * z2 * y2))
L229
Theorem. (Conj_mul_SNo_assoc__1__15)
(∀y2 : set, y2 ∈ SNoS_ (SNoLev z) → (∀z2 : set, z2 ∈ SNoS_ (SNoLev y) → (∀w2 : set, w2 ∈ SNoS_ (SNoLev x) → (w2 * z2) * y2 = w2 * z2 * y2))) → SNoCut w u = SNoCut v x2
In Proofgold the corresponding term root is 6a2887... and proposition id is dc0c38...
Proof:
Proof not loaded.
End of Section Conj_mul_SNo_assoc__1__15
Beginning of Section Conj_mul_SNo_assoc__1__25
L235
Variable x : set
(*** Conj_mul_SNo_assoc__1__25 TMciSyPjj9S6v3rpBPVGhU2FWHQyr1vgEn2 bounty of about 25 bars ***)
L236
Variable y : set
L237
Variable z : set
L238
Variable w : set
L239
Variable u : set
L240
Variable v : set
L241
Variable x2 : set
L242
Hypothesis H0 : SNo x
L243
Hypothesis H1 : SNo y
L244
Hypothesis H2 : SNo z
L245
Hypothesis H3 : (∀y2 : set, y2 ∈ SNoS_ (SNoLev x) → y2 * y * z = (y2 * y) * z)
L246
Hypothesis H4 : (∀y2 : set, y2 ∈ SNoS_ (SNoLev y) → x * y2 * z = (x * y2) * z)
L247
Hypothesis H5 : (∀y2 : set, y2 ∈ SNoS_ (SNoLev z) → x * y * y2 = (x * y) * y2)
L248
Hypothesis H6 : (∀y2 : set, y2 ∈ SNoS_ (SNoLev x) → (∀z2 : set, z2 ∈ SNoS_ (SNoLev y) → y2 * z2 * z = (y2 * z2) * z))
L249
Hypothesis H7 : (∀y2 : set, y2 ∈ SNoS_ (SNoLev x) → (∀z2 : set, z2 ∈ SNoS_ (SNoLev z) → y2 * y * z2 = (y2 * y) * z2))
L250
Hypothesis H8 : (∀y2 : set, y2 ∈ SNoS_ (SNoLev y) → (∀z2 : set, z2 ∈ SNoS_ (SNoLev z) → x * y2 * z2 = (x * y2) * z2))
L251
Hypothesis H9 : (∀y2 : set, y2 ∈ SNoS_ (SNoLev x) → (∀z2 : set, z2 ∈ SNoS_ (SNoLev y) → (∀w2 : set, w2 ∈ SNoS_ (SNoLev z) → y2 * z2 * w2 = (y2 * z2) * w2)))
L252
Hypothesis H10 : SNo (x * y)
L253
Hypothesis H11 : SNoCutP w u
L254
Hypothesis H12 : (∀y2 : set, y2 ∈ w → (∀P : prop, (∀z2 : set, z2 ∈ SNoL x → (∀w2 : set, w2 ∈ SNoL (y * z) → y2 = z2 * y * z + x * w2 + - (z2 * w2) → P)) → (∀z2 : set, z2 ∈ SNoR x → (∀w2 : set, w2 ∈ SNoR (y * z) → y2 = z2 * y * z + x * w2 + - (z2 * w2) → P)) → P))
L255
Hypothesis H13 : (∀y2 : set, y2 ∈ u → (∀P : prop, (∀z2 : set, z2 ∈ SNoL x → (∀w2 : set, w2 ∈ SNoR (y * z) → y2 = z2 * y * z + x * w2 + - (z2 * w2) → P)) → (∀z2 : set, z2 ∈ SNoR x → (∀w2 : set, w2 ∈ SNoL (y * z) → y2 = z2 * y * z + x * w2 + - (z2 * w2) → P)) → P))
L256
Hypothesis H14 : x * y * z = SNoCut w u
L257
Hypothesis H15 : SNoCutP v x2
L258
Hypothesis H16 : (∀y2 : set, y2 ∈ v → (∀P : prop, (∀z2 : set, z2 ∈ SNoL (x * y) → (∀w2 : set, w2 ∈ SNoL z → y2 = z2 * z + (x * y) * w2 + - (z2 * w2) → P)) → (∀z2 : set, z2 ∈ SNoR (x * y) → (∀w2 : set, w2 ∈ SNoR z → y2 = z2 * z + (x * y) * w2 + - (z2 * w2) → P)) → P))
L259
Hypothesis H17 : (∀y2 : set, y2 ∈ x2 → (∀P : prop, (∀z2 : set, z2 ∈ SNoL (x * y) → (∀w2 : set, w2 ∈ SNoR z → y2 = z2 * z + (x * y) * w2 + - (z2 * w2) → P)) → (∀z2 : set, z2 ∈ SNoR (x * y) → (∀w2 : set, w2 ∈ SNoL z → y2 = z2 * z + (x * y) * w2 + - (z2 * w2) → P)) → P))
L260
Hypothesis H18 : (x * y) * z = SNoCut v x2
L261
Hypothesis H19 : (∀y2 : set, ∀z2 : set, SNo y2 → SNo z2 → (∀w2 : set, w2 ∈ SNoL (z2 * y2) → (∀P : prop, (∀u2 : set, u2 ∈ SNoL y2 → (∀v2 : set, v2 ∈ SNoL z2 → (w2 + v2 * u2) ≤ z2 * u2 + v2 * y2 → P)) → (∀u2 : set, u2 ∈ SNoR y2 → (∀v2 : set, v2 ∈ SNoR z2 → (w2 + v2 * u2) ≤ z2 * u2 + v2 * y2 → P)) → P)))
L262
Hypothesis H20 : (∀y2 : set, ∀z2 : set, SNo y2 → SNo z2 → (∀w2 : set, w2 ∈ SNoR (z2 * y2) → (∀P : prop, (∀u2 : set, u2 ∈ SNoL y2 → (∀v2 : set, v2 ∈ SNoR z2 → (z2 * u2 + v2 * y2) ≤ w2 + v2 * u2 → P)) → (∀u2 : set, u2 ∈ SNoR y2 → (∀v2 : set, v2 ∈ SNoL z2 → (z2 * u2 + v2 * y2) ≤ w2 + v2 * u2 → P)) → P)))
L263
Hypothesis H21 : (∀y2 : set, ∀z2 : set, ∀w2 : set, ∀u2 : set, SNo y2 → SNo z2 → SNo w2 → SNo u2 → w2 < y2 → u2 < z2 → (z2 * w2 + u2 * y2) < z2 * y2 + u2 * w2)
L264
Hypothesis H22 : (∀y2 : set, ∀z2 : set, ∀w2 : set, ∀u2 : set, SNo y2 → SNo z2 → SNo w2 → SNo u2 → w2 ≤ y2 → u2 ≤ z2 → (z2 * w2 + u2 * y2) ≤ z2 * y2 + u2 * w2)
L265
Hypothesis H23 : (∀y2 : set, y2 ∈ SNoS_ (SNoLev x) → (y2 * y) * z = y2 * y * z)
L266
Hypothesis H24 : (∀y2 : set, y2 ∈ SNoS_ (SNoLev y) → (x * y2) * z = x * y2 * z)
L267
Hypothesis H26 : (∀y2 : set, y2 ∈ SNoS_ (SNoLev y) → (∀z2 : set, z2 ∈ SNoS_ (SNoLev x) → (z2 * y2) * z = z2 * y2 * z))
L268
Hypothesis H27 : (∀y2 : set, y2 ∈ SNoS_ (SNoLev z) → (∀z2 : set, z2 ∈ SNoS_ (SNoLev x) → (z2 * y) * y2 = z2 * y * y2))
L269
Hypothesis H28 : (∀y2 : set, y2 ∈ SNoS_ (SNoLev z) → (∀z2 : set, z2 ∈ SNoS_ (SNoLev y) → (x * z2) * y2 = x * z2 * y2))
L270
Theorem. (Conj_mul_SNo_assoc__1__25)
(∀y2 : set, y2 ∈ SNoS_ (SNoLev z) → (∀z2 : set, z2 ∈ SNoS_ (SNoLev y) → (∀w2 : set, w2 ∈ SNoS_ (SNoLev x) → (w2 * z2) * y2 = w2 * z2 * y2))) → SNoCut w u = SNoCut v x2
In Proofgold the corresponding term root is fd9921... and proposition id is c6d298...
Proof:
Proof not loaded.
End of Section Conj_mul_SNo_assoc__1__25
Beginning of Section Conj_mul_SNo_assoc__3__5
L276
Variable x : set
(*** Conj_mul_SNo_assoc__3__5 TMQUgb6TcmJVwmvSxEvVMfXvkNcUbYVdDqM bounty of about 25 bars ***)
L277
Variable y : set
L278
Variable z : set
L279
Variable w : set
L280
Variable u : set
L281
Variable v : set
L282
Variable x2 : set
L283
Hypothesis H0 : SNo x
L284
Hypothesis H1 : SNo y
L285
Hypothesis H2 : SNo z
L286
Hypothesis H3 : (∀y2 : set, y2 ∈ SNoS_ (SNoLev x) → y2 * y * z = (y2 * y) * z)
L287
Hypothesis H4 : (∀y2 : set, y2 ∈ SNoS_ (SNoLev y) → x * y2 * z = (x * y2) * z)
L288
Hypothesis H6 : (∀y2 : set, y2 ∈ SNoS_ (SNoLev x) → (∀z2 : set, z2 ∈ SNoS_ (SNoLev y) → y2 * z2 * z = (y2 * z2) * z))
L289
Hypothesis H7 : (∀y2 : set, y2 ∈ SNoS_ (SNoLev x) → (∀z2 : set, z2 ∈ SNoS_ (SNoLev z) → y2 * y * z2 = (y2 * y) * z2))
L290
Hypothesis H8 : (∀y2 : set, y2 ∈ SNoS_ (SNoLev y) → (∀z2 : set, z2 ∈ SNoS_ (SNoLev z) → x * y2 * z2 = (x * y2) * z2))
L291
Hypothesis H9 : (∀y2 : set, y2 ∈ SNoS_ (SNoLev x) → (∀z2 : set, z2 ∈ SNoS_ (SNoLev y) → (∀w2 : set, w2 ∈ SNoS_ (SNoLev z) → y2 * z2 * w2 = (y2 * z2) * w2)))
L292
Hypothesis H10 : SNo (x * y)
L293
Hypothesis H11 : SNoCutP w u
L294
Hypothesis H12 : (∀y2 : set, y2 ∈ w → (∀P : prop, (∀z2 : set, z2 ∈ SNoL x → (∀w2 : set, w2 ∈ SNoL (y * z) → y2 = z2 * y * z + x * w2 + - (z2 * w2) → P)) → (∀z2 : set, z2 ∈ SNoR x → (∀w2 : set, w2 ∈ SNoR (y * z) → y2 = z2 * y * z + x * w2 + - (z2 * w2) → P)) → P))
L295
Hypothesis H13 : (∀y2 : set, y2 ∈ u → (∀P : prop, (∀z2 : set, z2 ∈ SNoL x → (∀w2 : set, w2 ∈ SNoR (y * z) → y2 = z2 * y * z + x * w2 + - (z2 * w2) → P)) → (∀z2 : set, z2 ∈ SNoR x → (∀w2 : set, w2 ∈ SNoL (y * z) → y2 = z2 * y * z + x * w2 + - (z2 * w2) → P)) → P))
L296
Hypothesis H14 : x * y * z = SNoCut w u
L297
Hypothesis H15 : SNoCutP v x2
L298
Hypothesis H16 : (∀y2 : set, y2 ∈ v → (∀P : prop, (∀z2 : set, z2 ∈ SNoL (x * y) → (∀w2 : set, w2 ∈ SNoL z → y2 = z2 * z + (x * y) * w2 + - (z2 * w2) → P)) → (∀z2 : set, z2 ∈ SNoR (x * y) → (∀w2 : set, w2 ∈ SNoR z → y2 = z2 * z + (x * y) * w2 + - (z2 * w2) → P)) → P))
L299
Hypothesis H17 : (∀y2 : set, y2 ∈ x2 → (∀P : prop, (∀z2 : set, z2 ∈ SNoL (x * y) → (∀w2 : set, w2 ∈ SNoR z → y2 = z2 * z + (x * y) * w2 + - (z2 * w2) → P)) → (∀z2 : set, z2 ∈ SNoR (x * y) → (∀w2 : set, w2 ∈ SNoL z → y2 = z2 * z + (x * y) * w2 + - (z2 * w2) → P)) → P))
L300
Hypothesis H18 : (x * y) * z = SNoCut v x2
L301
Hypothesis H19 : (∀y2 : set, ∀z2 : set, SNo y2 → SNo z2 → (∀w2 : set, w2 ∈ SNoL (z2 * y2) → (∀P : prop, (∀u2 : set, u2 ∈ SNoL y2 → (∀v2 : set, v2 ∈ SNoL z2 → (w2 + v2 * u2) ≤ z2 * u2 + v2 * y2 → P)) → (∀u2 : set, u2 ∈ SNoR y2 → (∀v2 : set, v2 ∈ SNoR z2 → (w2 + v2 * u2) ≤ z2 * u2 + v2 * y2 → P)) → P)))
L302
Hypothesis H20 : (∀y2 : set, ∀z2 : set, SNo y2 → SNo z2 → (∀w2 : set, w2 ∈ SNoR (z2 * y2) → (∀P : prop, (∀u2 : set, u2 ∈ SNoL y2 → (∀v2 : set, v2 ∈ SNoR z2 → (z2 * u2 + v2 * y2) ≤ w2 + v2 * u2 → P)) → (∀u2 : set, u2 ∈ SNoR y2 → (∀v2 : set, v2 ∈ SNoL z2 → (z2 * u2 + v2 * y2) ≤ w2 + v2 * u2 → P)) → P)))
L303
Hypothesis H21 : (∀y2 : set, ∀z2 : set, ∀w2 : set, ∀u2 : set, SNo y2 → SNo z2 → SNo w2 → SNo u2 → w2 < y2 → u2 < z2 → (z2 * w2 + u2 * y2) < z2 * y2 + u2 * w2)
L304
Hypothesis H22 : (∀y2 : set, ∀z2 : set, ∀w2 : set, ∀u2 : set, SNo y2 → SNo z2 → SNo w2 → SNo u2 → w2 ≤ y2 → u2 ≤ z2 → (z2 * w2 + u2 * y2) ≤ z2 * y2 + u2 * w2)
L305
Hypothesis H23 : (∀y2 : set, y2 ∈ SNoS_ (SNoLev x) → (y2 * y) * z = y2 * y * z)
L306
Hypothesis H24 : (∀y2 : set, y2 ∈ SNoS_ (SNoLev y) → (x * y2) * z = x * y2 * z)
L307
Hypothesis H25 : (∀y2 : set, y2 ∈ SNoS_ (SNoLev z) → (x * y) * y2 = x * y * y2)
L308
Hypothesis H26 : (∀y2 : set, y2 ∈ SNoS_ (SNoLev y) → (∀z2 : set, z2 ∈ SNoS_ (SNoLev x) → (z2 * y2) * z = z2 * y2 * z))
L309
Theorem. (Conj_mul_SNo_assoc__3__5)
(∀y2 : set, y2 ∈ SNoS_ (SNoLev z) → (∀z2 : set, z2 ∈ SNoS_ (SNoLev x) → (z2 * y) * y2 = z2 * y * y2)) → SNoCut w u = SNoCut v x2
In Proofgold the corresponding term root is a6dd6e... and proposition id is ddf5e6...
Proof:
Proof not loaded.
End of Section Conj_mul_SNo_assoc__3__5
Beginning of Section Conj_mul_SNo_assoc__5__18
L315
Variable x : set
(*** Conj_mul_SNo_assoc__5__18 TMTJHPauSJMxB7bdMQJUnr5s2Cf3dffVG5Y bounty of about 25 bars ***)
L316
Variable y : set
L317
Variable z : set
L318
Variable w : set
L319
Variable u : set
L320
Variable v : set
L321
Variable x2 : set
L322
Hypothesis H0 : SNo x
L323
Hypothesis H1 : SNo y
L324
Hypothesis H2 : SNo z
L325
Hypothesis H3 : (∀y2 : set, y2 ∈ SNoS_ (SNoLev x) → y2 * y * z = (y2 * y) * z)
L326
Hypothesis H4 : (∀y2 : set, y2 ∈ SNoS_ (SNoLev y) → x * y2 * z = (x * y2) * z)
L327
Hypothesis H5 : (∀y2 : set, y2 ∈ SNoS_ (SNoLev z) → x * y * y2 = (x * y) * y2)
L328
Hypothesis H6 : (∀y2 : set, y2 ∈ SNoS_ (SNoLev x) → (∀z2 : set, z2 ∈ SNoS_ (SNoLev y) → y2 * z2 * z = (y2 * z2) * z))
L329
Hypothesis H7 : (∀y2 : set, y2 ∈ SNoS_ (SNoLev x) → (∀z2 : set, z2 ∈ SNoS_ (SNoLev z) → y2 * y * z2 = (y2 * y) * z2))
L330
Hypothesis H8 : (∀y2 : set, y2 ∈ SNoS_ (SNoLev y) → (∀z2 : set, z2 ∈ SNoS_ (SNoLev z) → x * y2 * z2 = (x * y2) * z2))
L331
Hypothesis H9 : (∀y2 : set, y2 ∈ SNoS_ (SNoLev x) → (∀z2 : set, z2 ∈ SNoS_ (SNoLev y) → (∀w2 : set, w2 ∈ SNoS_ (SNoLev z) → y2 * z2 * w2 = (y2 * z2) * w2)))
L332
Hypothesis H10 : SNo (x * y)
L333
Hypothesis H11 : SNoCutP w u
L334
Hypothesis H12 : (∀y2 : set, y2 ∈ w → (∀P : prop, (∀z2 : set, z2 ∈ SNoL x → (∀w2 : set, w2 ∈ SNoL (y * z) → y2 = z2 * y * z + x * w2 + - (z2 * w2) → P)) → (∀z2 : set, z2 ∈ SNoR x → (∀w2 : set, w2 ∈ SNoR (y * z) → y2 = z2 * y * z + x * w2 + - (z2 * w2) → P)) → P))
L335
Hypothesis H13 : (∀y2 : set, y2 ∈ u → (∀P : prop, (∀z2 : set, z2 ∈ SNoL x → (∀w2 : set, w2 ∈ SNoR (y * z) → y2 = z2 * y * z + x * w2 + - (z2 * w2) → P)) → (∀z2 : set, z2 ∈ SNoR x → (∀w2 : set, w2 ∈ SNoL (y * z) → y2 = z2 * y * z + x * w2 + - (z2 * w2) → P)) → P))
L336
Hypothesis H14 : x * y * z = SNoCut w u
L337
Hypothesis H15 : SNoCutP v x2
L338
Hypothesis H16 : (∀y2 : set, y2 ∈ v → (∀P : prop, (∀z2 : set, z2 ∈ SNoL (x * y) → (∀w2 : set, w2 ∈ SNoL z → y2 = z2 * z + (x * y) * w2 + - (z2 * w2) → P)) → (∀z2 : set, z2 ∈ SNoR (x * y) → (∀w2 : set, w2 ∈ SNoR z → y2 = z2 * z + (x * y) * w2 + - (z2 * w2) → P)) → P))
L339
Hypothesis H17 : (∀y2 : set, y2 ∈ x2 → (∀P : prop, (∀z2 : set, z2 ∈ SNoL (x * y) → (∀w2 : set, w2 ∈ SNoR z → y2 = z2 * z + (x * y) * w2 + - (z2 * w2) → P)) → (∀z2 : set, z2 ∈ SNoR (x * y) → (∀w2 : set, w2 ∈ SNoL z → y2 = z2 * z + (x * y) * w2 + - (z2 * w2) → P)) → P))
L340
Hypothesis H19 : (∀y2 : set, ∀z2 : set, SNo y2 → SNo z2 → (∀w2 : set, w2 ∈ SNoL (z2 * y2) → (∀P : prop, (∀u2 : set, u2 ∈ SNoL y2 → (∀v2 : set, v2 ∈ SNoL z2 → (w2 + v2 * u2) ≤ z2 * u2 + v2 * y2 → P)) → (∀u2 : set, u2 ∈ SNoR y2 → (∀v2 : set, v2 ∈ SNoR z2 → (w2 + v2 * u2) ≤ z2 * u2 + v2 * y2 → P)) → P)))
L341
Hypothesis H20 : (∀y2 : set, ∀z2 : set, SNo y2 → SNo z2 → (∀w2 : set, w2 ∈ SNoR (z2 * y2) → (∀P : prop, (∀u2 : set, u2 ∈ SNoL y2 → (∀v2 : set, v2 ∈ SNoR z2 → (z2 * u2 + v2 * y2) ≤ w2 + v2 * u2 → P)) → (∀u2 : set, u2 ∈ SNoR y2 → (∀v2 : set, v2 ∈ SNoL z2 → (z2 * u2 + v2 * y2) ≤ w2 + v2 * u2 → P)) → P)))
L342
Hypothesis H21 : (∀y2 : set, ∀z2 : set, ∀w2 : set, ∀u2 : set, SNo y2 → SNo z2 → SNo w2 → SNo u2 → w2 < y2 → u2 < z2 → (z2 * w2 + u2 * y2) < z2 * y2 + u2 * w2)
L343
Hypothesis H22 : (∀y2 : set, ∀z2 : set, ∀w2 : set, ∀u2 : set, SNo y2 → SNo z2 → SNo w2 → SNo u2 → w2 ≤ y2 → u2 ≤ z2 → (z2 * w2 + u2 * y2) ≤ z2 * y2 + u2 * w2)
L344
Hypothesis H23 : (∀y2 : set, y2 ∈ SNoS_ (SNoLev x) → (y2 * y) * z = y2 * y * z)
L345
Hypothesis H24 : (∀y2 : set, y2 ∈ SNoS_ (SNoLev y) → (x * y2) * z = x * y2 * z)
L346
Theorem. (Conj_mul_SNo_assoc__5__18)
(∀y2 : set, y2 ∈ SNoS_ (SNoLev z) → (x * y) * y2 = x * y * y2) → SNoCut w u = SNoCut v x2
In Proofgold the corresponding term root is ea8c36... and proposition id is b25922...
Proof:
Proof not loaded.
End of Section Conj_mul_SNo_assoc__5__18
Beginning of Section Conj_mul_SNo_assoc__5__19
L352
Variable x : set
(*** Conj_mul_SNo_assoc__5__19 TMNztdDcVGaSPMMX6bKr84ERzSu2b632gx6 bounty of about 25 bars ***)
L353
Variable y : set
L354
Variable z : set
L355
Variable w : set
L356
Variable u : set
L357
Variable v : set
L358
Variable x2 : set
L359
Hypothesis H0 : SNo x
L360
Hypothesis H1 : SNo y
L361
Hypothesis H2 : SNo z
L362
Hypothesis H3 : (∀y2 : set, y2 ∈ SNoS_ (SNoLev x) → y2 * y * z = (y2 * y) * z)
L363
Hypothesis H4 : (∀y2 : set, y2 ∈ SNoS_ (SNoLev y) → x * y2 * z = (x * y2) * z)
L364
Hypothesis H5 : (∀y2 : set, y2 ∈ SNoS_ (SNoLev z) → x * y * y2 = (x * y) * y2)
L365
Hypothesis H6 : (∀y2 : set, y2 ∈ SNoS_ (SNoLev x) → (∀z2 : set, z2 ∈ SNoS_ (SNoLev y) → y2 * z2 * z = (y2 * z2) * z))
L366
Hypothesis H7 : (∀y2 : set, y2 ∈ SNoS_ (SNoLev x) → (∀z2 : set, z2 ∈ SNoS_ (SNoLev z) → y2 * y * z2 = (y2 * y) * z2))
L367
Hypothesis H8 : (∀y2 : set, y2 ∈ SNoS_ (SNoLev y) → (∀z2 : set, z2 ∈ SNoS_ (SNoLev z) → x * y2 * z2 = (x * y2) * z2))
L368
Hypothesis H9 : (∀y2 : set, y2 ∈ SNoS_ (SNoLev x) → (∀z2 : set, z2 ∈ SNoS_ (SNoLev y) → (∀w2 : set, w2 ∈ SNoS_ (SNoLev z) → y2 * z2 * w2 = (y2 * z2) * w2)))
L369
Hypothesis H10 : SNo (x * y)
L370
Hypothesis H11 : SNoCutP w u
L371
Hypothesis H12 : (∀y2 : set, y2 ∈ w → (∀P : prop, (∀z2 : set, z2 ∈ SNoL x → (∀w2 : set, w2 ∈ SNoL (y * z) → y2 = z2 * y * z + x * w2 + - (z2 * w2) → P)) → (∀z2 : set, z2 ∈ SNoR x → (∀w2 : set, w2 ∈ SNoR (y * z) → y2 = z2 * y * z + x * w2 + - (z2 * w2) → P)) → P))
L372
Hypothesis H13 : (∀y2 : set, y2 ∈ u → (∀P : prop, (∀z2 : set, z2 ∈ SNoL x → (∀w2 : set, w2 ∈ SNoR (y * z) → y2 = z2 * y * z + x * w2 + - (z2 * w2) → P)) → (∀z2 : set, z2 ∈ SNoR x → (∀w2 : set, w2 ∈ SNoL (y * z) → y2 = z2 * y * z + x * w2 + - (z2 * w2) → P)) → P))
L373
Hypothesis H14 : x * y * z = SNoCut w u
L374
Hypothesis H15 : SNoCutP v x2
L375
Hypothesis H16 : (∀y2 : set, y2 ∈ v → (∀P : prop, (∀z2 : set, z2 ∈ SNoL (x * y) → (∀w2 : set, w2 ∈ SNoL z → y2 = z2 * z + (x * y) * w2 + - (z2 * w2) → P)) → (∀z2 : set, z2 ∈ SNoR (x * y) → (∀w2 : set, w2 ∈ SNoR z → y2 = z2 * z + (x * y) * w2 + - (z2 * w2) → P)) → P))
L376
Hypothesis H17 : (∀y2 : set, y2 ∈ x2 → (∀P : prop, (∀z2 : set, z2 ∈ SNoL (x * y) → (∀w2 : set, w2 ∈ SNoR z → y2 = z2 * z + (x * y) * w2 + - (z2 * w2) → P)) → (∀z2 : set, z2 ∈ SNoR (x * y) → (∀w2 : set, w2 ∈ SNoL z → y2 = z2 * z + (x * y) * w2 + - (z2 * w2) → P)) → P))
L377
Hypothesis H18 : (x * y) * z = SNoCut v x2
L378
Hypothesis H20 : (∀y2 : set, ∀z2 : set, SNo y2 → SNo z2 → (∀w2 : set, w2 ∈ SNoR (z2 * y2) → (∀P : prop, (∀u2 : set, u2 ∈ SNoL y2 → (∀v2 : set, v2 ∈ SNoR z2 → (z2 * u2 + v2 * y2) ≤ w2 + v2 * u2 → P)) → (∀u2 : set, u2 ∈ SNoR y2 → (∀v2 : set, v2 ∈ SNoL z2 → (z2 * u2 + v2 * y2) ≤ w2 + v2 * u2 → P)) → P)))
L379
Hypothesis H21 : (∀y2 : set, ∀z2 : set, ∀w2 : set, ∀u2 : set, SNo y2 → SNo z2 → SNo w2 → SNo u2 → w2 < y2 → u2 < z2 → (z2 * w2 + u2 * y2) < z2 * y2 + u2 * w2)
L380
Hypothesis H22 : (∀y2 : set, ∀z2 : set, ∀w2 : set, ∀u2 : set, SNo y2 → SNo z2 → SNo w2 → SNo u2 → w2 ≤ y2 → u2 ≤ z2 → (z2 * w2 + u2 * y2) ≤ z2 * y2 + u2 * w2)
L381
Hypothesis H23 : (∀y2 : set, y2 ∈ SNoS_ (SNoLev x) → (y2 * y) * z = y2 * y * z)
L382
Hypothesis H24 : (∀y2 : set, y2 ∈ SNoS_ (SNoLev y) → (x * y2) * z = x * y2 * z)
L383
Theorem. (Conj_mul_SNo_assoc__5__19)
(∀y2 : set, y2 ∈ SNoS_ (SNoLev z) → (x * y) * y2 = x * y * y2) → SNoCut w u = SNoCut v x2
In Proofgold the corresponding term root is 3b2b20... and proposition id is 89f3a7...
Proof:
Proof not loaded.
End of Section Conj_mul_SNo_assoc__5__19
Beginning of Section Conj_mul_SNo_assoc__5__23
L389
Variable x : set
(*** Conj_mul_SNo_assoc__5__23 TMbMicSqa1Uo7gbRjiR5xbf7AUPsvxvfJ1f bounty of about 25 bars ***)
L390
Variable y : set
L391
Variable z : set
L392
Variable w : set
L393
Variable u : set
L394
Variable v : set
L395
Variable x2 : set
L396
Hypothesis H0 : SNo x
L397
Hypothesis H1 : SNo y
L398
Hypothesis H2 : SNo z
L399
Hypothesis H3 : (∀y2 : set, y2 ∈ SNoS_ (SNoLev x) → y2 * y * z = (y2 * y) * z)
L400
Hypothesis H4 : (∀y2 : set, y2 ∈ SNoS_ (SNoLev y) → x * y2 * z = (x * y2) * z)
L401
Hypothesis H5 : (∀y2 : set, y2 ∈ SNoS_ (SNoLev z) → x * y * y2 = (x * y) * y2)
L402
Hypothesis H6 : (∀y2 : set, y2 ∈ SNoS_ (SNoLev x) → (∀z2 : set, z2 ∈ SNoS_ (SNoLev y) → y2 * z2 * z = (y2 * z2) * z))
L403
Hypothesis H7 : (∀y2 : set, y2 ∈ SNoS_ (SNoLev x) → (∀z2 : set, z2 ∈ SNoS_ (SNoLev z) → y2 * y * z2 = (y2 * y) * z2))
L404
Hypothesis H8 : (∀y2 : set, y2 ∈ SNoS_ (SNoLev y) → (∀z2 : set, z2 ∈ SNoS_ (SNoLev z) → x * y2 * z2 = (x * y2) * z2))
L405
Hypothesis H9 : (∀y2 : set, y2 ∈ SNoS_ (SNoLev x) → (∀z2 : set, z2 ∈ SNoS_ (SNoLev y) → (∀w2 : set, w2 ∈ SNoS_ (SNoLev z) → y2 * z2 * w2 = (y2 * z2) * w2)))
L406
Hypothesis H10 : SNo (x * y)
L407
Hypothesis H11 : SNoCutP w u
L408
Hypothesis H12 : (∀y2 : set, y2 ∈ w → (∀P : prop, (∀z2 : set, z2 ∈ SNoL x → (∀w2 : set, w2 ∈ SNoL (y * z) → y2 = z2 * y * z + x * w2 + - (z2 * w2) → P)) → (∀z2 : set, z2 ∈ SNoR x → (∀w2 : set, w2 ∈ SNoR (y * z) → y2 = z2 * y * z + x * w2 + - (z2 * w2) → P)) → P))
L409
Hypothesis H13 : (∀y2 : set, y2 ∈ u → (∀P : prop, (∀z2 : set, z2 ∈ SNoL x → (∀w2 : set, w2 ∈ SNoR (y * z) → y2 = z2 * y * z + x * w2 + - (z2 * w2) → P)) → (∀z2 : set, z2 ∈ SNoR x → (∀w2 : set, w2 ∈ SNoL (y * z) → y2 = z2 * y * z + x * w2 + - (z2 * w2) → P)) → P))
L410
Hypothesis H14 : x * y * z = SNoCut w u
L411
Hypothesis H15 : SNoCutP v x2
L412
Hypothesis H16 : (∀y2 : set, y2 ∈ v → (∀P : prop, (∀z2 : set, z2 ∈ SNoL (x * y) → (∀w2 : set, w2 ∈ SNoL z → y2 = z2 * z + (x * y) * w2 + - (z2 * w2) → P)) → (∀z2 : set, z2 ∈ SNoR (x * y) → (∀w2 : set, w2 ∈ SNoR z → y2 = z2 * z + (x * y) * w2 + - (z2 * w2) → P)) → P))
L413
Hypothesis H17 : (∀y2 : set, y2 ∈ x2 → (∀P : prop, (∀z2 : set, z2 ∈ SNoL (x * y) → (∀w2 : set, w2 ∈ SNoR z → y2 = z2 * z + (x * y) * w2 + - (z2 * w2) → P)) → (∀z2 : set, z2 ∈ SNoR (x * y) → (∀w2 : set, w2 ∈ SNoL z → y2 = z2 * z + (x * y) * w2 + - (z2 * w2) → P)) → P))
L414
Hypothesis H18 : (x * y) * z = SNoCut v x2
L415
Hypothesis H19 : (∀y2 : set, ∀z2 : set, SNo y2 → SNo z2 → (∀w2 : set, w2 ∈ SNoL (z2 * y2) → (∀P : prop, (∀u2 : set, u2 ∈ SNoL y2 → (∀v2 : set, v2 ∈ SNoL z2 → (w2 + v2 * u2) ≤ z2 * u2 + v2 * y2 → P)) → (∀u2 : set, u2 ∈ SNoR y2 → (∀v2 : set, v2 ∈ SNoR z2 → (w2 + v2 * u2) ≤ z2 * u2 + v2 * y2 → P)) → P)))
L416
Hypothesis H20 : (∀y2 : set, ∀z2 : set, SNo y2 → SNo z2 → (∀w2 : set, w2 ∈ SNoR (z2 * y2) → (∀P : prop, (∀u2 : set, u2 ∈ SNoL y2 → (∀v2 : set, v2 ∈ SNoR z2 → (z2 * u2 + v2 * y2) ≤ w2 + v2 * u2 → P)) → (∀u2 : set, u2 ∈ SNoR y2 → (∀v2 : set, v2 ∈ SNoL z2 → (z2 * u2 + v2 * y2) ≤ w2 + v2 * u2 → P)) → P)))
L417
Hypothesis H21 : (∀y2 : set, ∀z2 : set, ∀w2 : set, ∀u2 : set, SNo y2 → SNo z2 → SNo w2 → SNo u2 → w2 < y2 → u2 < z2 → (z2 * w2 + u2 * y2) < z2 * y2 + u2 * w2)
L418
Hypothesis H22 : (∀y2 : set, ∀z2 : set, ∀w2 : set, ∀u2 : set, SNo y2 → SNo z2 → SNo w2 → SNo u2 → w2 ≤ y2 → u2 ≤ z2 → (z2 * w2 + u2 * y2) ≤ z2 * y2 + u2 * w2)
L419
Hypothesis H24 : (∀y2 : set, y2 ∈ SNoS_ (SNoLev y) → (x * y2) * z = x * y2 * z)
L420
Theorem. (Conj_mul_SNo_assoc__5__23)
(∀y2 : set, y2 ∈ SNoS_ (SNoLev z) → (x * y) * y2 = x * y * y2) → SNoCut w u = SNoCut v x2
In Proofgold the corresponding term root is 0a42a7... and proposition id is 0d6264...
Proof:
Proof not loaded.
End of Section Conj_mul_SNo_assoc__5__23
Beginning of Section Conj_mul_SNo_assoc__6__20
L426
Variable x : set
(*** Conj_mul_SNo_assoc__6__20 TMUNaL9yMpQFfhyTtYMgLWL4xc5GGgf2Cud bounty of about 25 bars ***)
L427
Variable y : set
L428
Variable z : set
L429
Variable w : set
L430
Variable u : set
L431
Variable v : set
L432
Variable x2 : set
L433
Hypothesis H0 : SNo x
L434
Hypothesis H1 : SNo y
L435
Hypothesis H2 : SNo z
L436
Hypothesis H3 : (∀y2 : set, y2 ∈ SNoS_ (SNoLev x) → y2 * y * z = (y2 * y) * z)
L437
Hypothesis H4 : (∀y2 : set, y2 ∈ SNoS_ (SNoLev y) → x * y2 * z = (x * y2) * z)
L438
Hypothesis H5 : (∀y2 : set, y2 ∈ SNoS_ (SNoLev z) → x * y * y2 = (x * y) * y2)
L439
Hypothesis H6 : (∀y2 : set, y2 ∈ SNoS_ (SNoLev x) → (∀z2 : set, z2 ∈ SNoS_ (SNoLev y) → y2 * z2 * z = (y2 * z2) * z))
L440
Hypothesis H7 : (∀y2 : set, y2 ∈ SNoS_ (SNoLev x) → (∀z2 : set, z2 ∈ SNoS_ (SNoLev z) → y2 * y * z2 = (y2 * y) * z2))
L441
Hypothesis H8 : (∀y2 : set, y2 ∈ SNoS_ (SNoLev y) → (∀z2 : set, z2 ∈ SNoS_ (SNoLev z) → x * y2 * z2 = (x * y2) * z2))
L442
Hypothesis H9 : (∀y2 : set, y2 ∈ SNoS_ (SNoLev x) → (∀z2 : set, z2 ∈ SNoS_ (SNoLev y) → (∀w2 : set, w2 ∈ SNoS_ (SNoLev z) → y2 * z2 * w2 = (y2 * z2) * w2)))
L443
Hypothesis H10 : SNo (x * y)
L444
Hypothesis H11 : SNoCutP w u
L445
Hypothesis H12 : (∀y2 : set, y2 ∈ w → (∀P : prop, (∀z2 : set, z2 ∈ SNoL x → (∀w2 : set, w2 ∈ SNoL (y * z) → y2 = z2 * y * z + x * w2 + - (z2 * w2) → P)) → (∀z2 : set, z2 ∈ SNoR x → (∀w2 : set, w2 ∈ SNoR (y * z) → y2 = z2 * y * z + x * w2 + - (z2 * w2) → P)) → P))
L446
Hypothesis H13 : (∀y2 : set, y2 ∈ u → (∀P : prop, (∀z2 : set, z2 ∈ SNoL x → (∀w2 : set, w2 ∈ SNoR (y * z) → y2 = z2 * y * z + x * w2 + - (z2 * w2) → P)) → (∀z2 : set, z2 ∈ SNoR x → (∀w2 : set, w2 ∈ SNoL (y * z) → y2 = z2 * y * z + x * w2 + - (z2 * w2) → P)) → P))
L447
Hypothesis H14 : x * y * z = SNoCut w u
L448
Hypothesis H15 : SNoCutP v x2
L449
Hypothesis H16 : (∀y2 : set, y2 ∈ v → (∀P : prop, (∀z2 : set, z2 ∈ SNoL (x * y) → (∀w2 : set, w2 ∈ SNoL z → y2 = z2 * z + (x * y) * w2 + - (z2 * w2) → P)) → (∀z2 : set, z2 ∈ SNoR (x * y) → (∀w2 : set, w2 ∈ SNoR z → y2 = z2 * z + (x * y) * w2 + - (z2 * w2) → P)) → P))
L450
Hypothesis H17 : (∀y2 : set, y2 ∈ x2 → (∀P : prop, (∀z2 : set, z2 ∈ SNoL (x * y) → (∀w2 : set, w2 ∈ SNoR z → y2 = z2 * z + (x * y) * w2 + - (z2 * w2) → P)) → (∀z2 : set, z2 ∈ SNoR (x * y) → (∀w2 : set, w2 ∈ SNoL z → y2 = z2 * z + (x * y) * w2 + - (z2 * w2) → P)) → P))
L451
Hypothesis H18 : (x * y) * z = SNoCut v x2
L452
Hypothesis H19 : (∀y2 : set, ∀z2 : set, SNo y2 → SNo z2 → (∀w2 : set, w2 ∈ SNoL (z2 * y2) → (∀P : prop, (∀u2 : set, u2 ∈ SNoL y2 → (∀v2 : set, v2 ∈ SNoL z2 → (w2 + v2 * u2) ≤ z2 * u2 + v2 * y2 → P)) → (∀u2 : set, u2 ∈ SNoR y2 → (∀v2 : set, v2 ∈ SNoR z2 → (w2 + v2 * u2) ≤ z2 * u2 + v2 * y2 → P)) → P)))
L453
Hypothesis H21 : (∀y2 : set, ∀z2 : set, ∀w2 : set, ∀u2 : set, SNo y2 → SNo z2 → SNo w2 → SNo u2 → w2 < y2 → u2 < z2 → (z2 * w2 + u2 * y2) < z2 * y2 + u2 * w2)
L454
Hypothesis H22 : (∀y2 : set, ∀z2 : set, ∀w2 : set, ∀u2 : set, SNo y2 → SNo z2 → SNo w2 → SNo u2 → w2 ≤ y2 → u2 ≤ z2 → (z2 * w2 + u2 * y2) ≤ z2 * y2 + u2 * w2)
L455
Hypothesis H23 : (∀y2 : set, y2 ∈ SNoS_ (SNoLev x) → (y2 * y) * z = y2 * y * z)
L456
Theorem. (Conj_mul_SNo_assoc__6__20)
(∀y2 : set, y2 ∈ SNoS_ (SNoLev y) → (x * y2) * z = x * y2 * z) → SNoCut w u = SNoCut v x2
In Proofgold the corresponding term root is e28d7f... and proposition id is b78cf1...
Proof:
Proof not loaded.
End of Section Conj_mul_SNo_assoc__6__20
Beginning of Section Conj_mul_SNo_assoc__7__6
L462
Variable x : set
(*** Conj_mul_SNo_assoc__7__6 TMWH7AsFtkApTcN8EzUYXLVhBhntNeWD3Ky bounty of about 25 bars ***)
L463
Variable y : set
L464
Variable z : set
L465
Variable w : set
L466
Variable u : set
L467
Variable v : set
L468
Variable x2 : set
L469
Hypothesis H0 : SNo x
L470
Hypothesis H1 : SNo y
L471
Hypothesis H2 : SNo z
L472
Hypothesis H3 : (∀y2 : set, y2 ∈ SNoS_ (SNoLev x) → y2 * y * z = (y2 * y) * z)
L473
Hypothesis H4 : (∀y2 : set, y2 ∈ SNoS_ (SNoLev y) → x * y2 * z = (x * y2) * z)
L474
Hypothesis H5 : (∀y2 : set, y2 ∈ SNoS_ (SNoLev z) → x * y * y2 = (x * y) * y2)
L475
Hypothesis H7 : (∀y2 : set, y2 ∈ SNoS_ (SNoLev x) → (∀z2 : set, z2 ∈ SNoS_ (SNoLev z) → y2 * y * z2 = (y2 * y) * z2))
L476
Hypothesis H8 : (∀y2 : set, y2 ∈ SNoS_ (SNoLev y) → (∀z2 : set, z2 ∈ SNoS_ (SNoLev z) → x * y2 * z2 = (x * y2) * z2))
L477
Hypothesis H9 : (∀y2 : set, y2 ∈ SNoS_ (SNoLev x) → (∀z2 : set, z2 ∈ SNoS_ (SNoLev y) → (∀w2 : set, w2 ∈ SNoS_ (SNoLev z) → y2 * z2 * w2 = (y2 * z2) * w2)))
L478
Hypothesis H10 : SNo (x * y)
L479
Hypothesis H11 : SNoCutP w u
L480
Hypothesis H12 : (∀y2 : set, y2 ∈ w → (∀P : prop, (∀z2 : set, z2 ∈ SNoL x → (∀w2 : set, w2 ∈ SNoL (y * z) → y2 = z2 * y * z + x * w2 + - (z2 * w2) → P)) → (∀z2 : set, z2 ∈ SNoR x → (∀w2 : set, w2 ∈ SNoR (y * z) → y2 = z2 * y * z + x * w2 + - (z2 * w2) → P)) → P))
L481
Hypothesis H13 : (∀y2 : set, y2 ∈ u → (∀P : prop, (∀z2 : set, z2 ∈ SNoL x → (∀w2 : set, w2 ∈ SNoR (y * z) → y2 = z2 * y * z + x * w2 + - (z2 * w2) → P)) → (∀z2 : set, z2 ∈ SNoR x → (∀w2 : set, w2 ∈ SNoL (y * z) → y2 = z2 * y * z + x * w2 + - (z2 * w2) → P)) → P))
L482
Hypothesis H14 : x * y * z = SNoCut w u
L483
Hypothesis H15 : SNoCutP v x2
L484
Hypothesis H16 : (∀y2 : set, y2 ∈ v → (∀P : prop, (∀z2 : set, z2 ∈ SNoL (x * y) → (∀w2 : set, w2 ∈ SNoL z → y2 = z2 * z + (x * y) * w2 + - (z2 * w2) → P)) → (∀z2 : set, z2 ∈ SNoR (x * y) → (∀w2 : set, w2 ∈ SNoR z → y2 = z2 * z + (x * y) * w2 + - (z2 * w2) → P)) → P))
L485
Hypothesis H17 : (∀y2 : set, y2 ∈ x2 → (∀P : prop, (∀z2 : set, z2 ∈ SNoL (x * y) → (∀w2 : set, w2 ∈ SNoR z → y2 = z2 * z + (x * y) * w2 + - (z2 * w2) → P)) → (∀z2 : set, z2 ∈ SNoR (x * y) → (∀w2 : set, w2 ∈ SNoL z → y2 = z2 * z + (x * y) * w2 + - (z2 * w2) → P)) → P))
L486
Hypothesis H18 : (x * y) * z = SNoCut v x2
L487
Hypothesis H19 : (∀y2 : set, ∀z2 : set, SNo y2 → SNo z2 → (∀w2 : set, w2 ∈ SNoL (z2 * y2) → (∀P : prop, (∀u2 : set, u2 ∈ SNoL y2 → (∀v2 : set, v2 ∈ SNoL z2 → (w2 + v2 * u2) ≤ z2 * u2 + v2 * y2 → P)) → (∀u2 : set, u2 ∈ SNoR y2 → (∀v2 : set, v2 ∈ SNoR z2 → (w2 + v2 * u2) ≤ z2 * u2 + v2 * y2 → P)) → P)))
L488
Hypothesis H20 : (∀y2 : set, ∀z2 : set, SNo y2 → SNo z2 → (∀w2 : set, w2 ∈ SNoR (z2 * y2) → (∀P : prop, (∀u2 : set, u2 ∈ SNoL y2 → (∀v2 : set, v2 ∈ SNoR z2 → (z2 * u2 + v2 * y2) ≤ w2 + v2 * u2 → P)) → (∀u2 : set, u2 ∈ SNoR y2 → (∀v2 : set, v2 ∈ SNoL z2 → (z2 * u2 + v2 * y2) ≤ w2 + v2 * u2 → P)) → P)))
L489
Hypothesis H21 : (∀y2 : set, ∀z2 : set, ∀w2 : set, ∀u2 : set, SNo y2 → SNo z2 → SNo w2 → SNo u2 → w2 < y2 → u2 < z2 → (z2 * w2 + u2 * y2) < z2 * y2 + u2 * w2)
L490
Hypothesis H22 : (∀y2 : set, ∀z2 : set, ∀w2 : set, ∀u2 : set, SNo y2 → SNo z2 → SNo w2 → SNo u2 → w2 ≤ y2 → u2 ≤ z2 → (z2 * w2 + u2 * y2) ≤ z2 * y2 + u2 * w2)
L491
Theorem. (Conj_mul_SNo_assoc__7__6)
(∀y2 : set, y2 ∈ SNoS_ (SNoLev x) → (y2 * y) * z = y2 * y * z) → SNoCut w u = SNoCut v x2
In Proofgold the corresponding term root is 1b0729... and proposition id is a400c6...
Proof:
Proof not loaded.
End of Section Conj_mul_SNo_assoc__7__6
Beginning of Section Conj_mul_SNo_assoc__9__0
L497
Variable x : set
(*** Conj_mul_SNo_assoc__9__0 TMMrsfdC6Ecnmmc13uAWMra4X6T2amXWdz5 bounty of about 25 bars ***)
L498
Variable y : set
L499
Variable z : set
L500
Variable w : set
L501
Variable u : set
L502
Variable v : set
L503
Variable x2 : set
L504
Hypothesis H1 : SNo y
L505
Hypothesis H2 : SNo z
L506
Hypothesis H3 : (∀y2 : set, y2 ∈ SNoS_ (SNoLev x) → y2 * y * z = (y2 * y) * z)
L507
Hypothesis H4 : (∀y2 : set, y2 ∈ SNoS_ (SNoLev y) → x * y2 * z = (x * y2) * z)
L508
Hypothesis H5 : (∀y2 : set, y2 ∈ SNoS_ (SNoLev z) → x * y * y2 = (x * y) * y2)
L509
Hypothesis H6 : (∀y2 : set, y2 ∈ SNoS_ (SNoLev x) → (∀z2 : set, z2 ∈ SNoS_ (SNoLev y) → y2 * z2 * z = (y2 * z2) * z))
L510
Hypothesis H7 : (∀y2 : set, y2 ∈ SNoS_ (SNoLev x) → (∀z2 : set, z2 ∈ SNoS_ (SNoLev z) → y2 * y * z2 = (y2 * y) * z2))
L511
Hypothesis H8 : (∀y2 : set, y2 ∈ SNoS_ (SNoLev y) → (∀z2 : set, z2 ∈ SNoS_ (SNoLev z) → x * y2 * z2 = (x * y2) * z2))
L512
Hypothesis H9 : (∀y2 : set, y2 ∈ SNoS_ (SNoLev x) → (∀z2 : set, z2 ∈ SNoS_ (SNoLev y) → (∀w2 : set, w2 ∈ SNoS_ (SNoLev z) → y2 * z2 * w2 = (y2 * z2) * w2)))
L513
Hypothesis H10 : SNo (x * y)
L514
Hypothesis H11 : SNoCutP w u
L515
Hypothesis H12 : (∀y2 : set, y2 ∈ w → (∀P : prop, (∀z2 : set, z2 ∈ SNoL x → (∀w2 : set, w2 ∈ SNoL (y * z) → y2 = z2 * y * z + x * w2 + - (z2 * w2) → P)) → (∀z2 : set, z2 ∈ SNoR x → (∀w2 : set, w2 ∈ SNoR (y * z) → y2 = z2 * y * z + x * w2 + - (z2 * w2) → P)) → P))
L516
Hypothesis H13 : (∀y2 : set, y2 ∈ u → (∀P : prop, (∀z2 : set, z2 ∈ SNoL x → (∀w2 : set, w2 ∈ SNoR (y * z) → y2 = z2 * y * z + x * w2 + - (z2 * w2) → P)) → (∀z2 : set, z2 ∈ SNoR x → (∀w2 : set, w2 ∈ SNoL (y * z) → y2 = z2 * y * z + x * w2 + - (z2 * w2) → P)) → P))
L517
Hypothesis H14 : x * y * z = SNoCut w u
L518
Hypothesis H15 : SNoCutP v x2
L519
Hypothesis H16 : (∀y2 : set, y2 ∈ v → (∀P : prop, (∀z2 : set, z2 ∈ SNoL (x * y) → (∀w2 : set, w2 ∈ SNoL z → y2 = z2 * z + (x * y) * w2 + - (z2 * w2) → P)) → (∀z2 : set, z2 ∈ SNoR (x * y) → (∀w2 : set, w2 ∈ SNoR z → y2 = z2 * z + (x * y) * w2 + - (z2 * w2) → P)) → P))
L520
Hypothesis H17 : (∀y2 : set, y2 ∈ x2 → (∀P : prop, (∀z2 : set, z2 ∈ SNoL (x * y) → (∀w2 : set, w2 ∈ SNoR z → y2 = z2 * z + (x * y) * w2 + - (z2 * w2) → P)) → (∀z2 : set, z2 ∈ SNoR (x * y) → (∀w2 : set, w2 ∈ SNoL z → y2 = z2 * z + (x * y) * w2 + - (z2 * w2) → P)) → P))
L521
Hypothesis H18 : (x * y) * z = SNoCut v x2
L522
Hypothesis H19 : (∀y2 : set, ∀z2 : set, SNo y2 → SNo z2 → (∀w2 : set, w2 ∈ SNoL (z2 * y2) → (∀P : prop, (∀u2 : set, u2 ∈ SNoL y2 → (∀v2 : set, v2 ∈ SNoL z2 → (w2 + v2 * u2) ≤ z2 * u2 + v2 * y2 → P)) → (∀u2 : set, u2 ∈ SNoR y2 → (∀v2 : set, v2 ∈ SNoR z2 → (w2 + v2 * u2) ≤ z2 * u2 + v2 * y2 → P)) → P)))
L523
Hypothesis H20 : (∀y2 : set, ∀z2 : set, SNo y2 → SNo z2 → (∀w2 : set, w2 ∈ SNoR (z2 * y2) → (∀P : prop, (∀u2 : set, u2 ∈ SNoL y2 → (∀v2 : set, v2 ∈ SNoR z2 → (z2 * u2 + v2 * y2) ≤ w2 + v2 * u2 → P)) → (∀u2 : set, u2 ∈ SNoR y2 → (∀v2 : set, v2 ∈ SNoL z2 → (z2 * u2 + v2 * y2) ≤ w2 + v2 * u2 → P)) → P)))
L524
Theorem. (Conj_mul_SNo_assoc__9__0)
(∀y2 : set, ∀z2 : set, ∀w2 : set, ∀u2 : set, SNo y2 → SNo z2 → SNo w2 → SNo u2 → w2 < y2 → u2 < z2 → (z2 * w2 + u2 * y2) < z2 * y2 + u2 * w2) → SNoCut w u = SNoCut v x2
In Proofgold the corresponding term root is 2ba8fb... and proposition id is d9bef4...
Proof:
Proof not loaded.
End of Section Conj_mul_SNo_assoc__9__0
Beginning of Section Conj_mul_SNo_assoc__10__13
L530
Variable x : set
(*** Conj_mul_SNo_assoc__10__13 TMQkYpM9GqnCKrvVJi2ANVvjjpPYJ69YWgB bounty of about 25 bars ***)
L531
Variable y : set
L532
Variable z : set
L533
Variable w : set
L534
Variable u : set
L535
Variable v : set
L536
Variable x2 : set
L537
Hypothesis H0 : SNo x
L538
Hypothesis H1 : SNo y
L539
Hypothesis H2 : SNo z
L540
Hypothesis H3 : (∀y2 : set, y2 ∈ SNoS_ (SNoLev x) → y2 * y * z = (y2 * y) * z)
L541
Hypothesis H4 : (∀y2 : set, y2 ∈ SNoS_ (SNoLev y) → x * y2 * z = (x * y2) * z)
L542
Hypothesis H5 : (∀y2 : set, y2 ∈ SNoS_ (SNoLev z) → x * y * y2 = (x * y) * y2)
L543
Hypothesis H6 : (∀y2 : set, y2 ∈ SNoS_ (SNoLev x) → (∀z2 : set, z2 ∈ SNoS_ (SNoLev y) → y2 * z2 * z = (y2 * z2) * z))
L544
Hypothesis H7 : (∀y2 : set, y2 ∈ SNoS_ (SNoLev x) → (∀z2 : set, z2 ∈ SNoS_ (SNoLev z) → y2 * y * z2 = (y2 * y) * z2))
L545
Hypothesis H8 : (∀y2 : set, y2 ∈ SNoS_ (SNoLev y) → (∀z2 : set, z2 ∈ SNoS_ (SNoLev z) → x * y2 * z2 = (x * y2) * z2))
L546
Hypothesis H9 : (∀y2 : set, y2 ∈ SNoS_ (SNoLev x) → (∀z2 : set, z2 ∈ SNoS_ (SNoLev y) → (∀w2 : set, w2 ∈ SNoS_ (SNoLev z) → y2 * z2 * w2 = (y2 * z2) * w2)))
L547
Hypothesis H10 : SNo (x * y)
L548
Hypothesis H11 : SNoCutP w u
L549
Hypothesis H12 : (∀y2 : set, y2 ∈ w → (∀P : prop, (∀z2 : set, z2 ∈ SNoL x → (∀w2 : set, w2 ∈ SNoL (y * z) → y2 = z2 * y * z + x * w2 + - (z2 * w2) → P)) → (∀z2 : set, z2 ∈ SNoR x → (∀w2 : set, w2 ∈ SNoR (y * z) → y2 = z2 * y * z + x * w2 + - (z2 * w2) → P)) → P))
L550
Hypothesis H14 : x * y * z = SNoCut w u
L551
Hypothesis H15 : SNoCutP v x2
L552
Hypothesis H16 : (∀y2 : set, y2 ∈ v → (∀P : prop, (∀z2 : set, z2 ∈ SNoL (x * y) → (∀w2 : set, w2 ∈ SNoL z → y2 = z2 * z + (x * y) * w2 + - (z2 * w2) → P)) → (∀z2 : set, z2 ∈ SNoR (x * y) → (∀w2 : set, w2 ∈ SNoR z → y2 = z2 * z + (x * y) * w2 + - (z2 * w2) → P)) → P))
L553
Hypothesis H17 : (∀y2 : set, y2 ∈ x2 → (∀P : prop, (∀z2 : set, z2 ∈ SNoL (x * y) → (∀w2 : set, w2 ∈ SNoR z → y2 = z2 * z + (x * y) * w2 + - (z2 * w2) → P)) → (∀z2 : set, z2 ∈ SNoR (x * y) → (∀w2 : set, w2 ∈ SNoL z → y2 = z2 * z + (x * y) * w2 + - (z2 * w2) → P)) → P))
L554
Hypothesis H18 : (x * y) * z = SNoCut v x2
L555
Hypothesis H19 : (∀y2 : set, ∀z2 : set, SNo y2 → SNo z2 → (∀w2 : set, w2 ∈ SNoL (z2 * y2) → (∀P : prop, (∀u2 : set, u2 ∈ SNoL y2 → (∀v2 : set, v2 ∈ SNoL z2 → (w2 + v2 * u2) ≤ z2 * u2 + v2 * y2 → P)) → (∀u2 : set, u2 ∈ SNoR y2 → (∀v2 : set, v2 ∈ SNoR z2 → (w2 + v2 * u2) ≤ z2 * u2 + v2 * y2 → P)) → P)))
L556
Theorem. (Conj_mul_SNo_assoc__10__13)
(∀y2 : set, ∀z2 : set, SNo y2 → SNo z2 → (∀w2 : set, w2 ∈ SNoR (z2 * y2) → (∀P : prop, (∀u2 : set, u2 ∈ SNoL y2 → (∀v2 : set, v2 ∈ SNoR z2 → (z2 * u2 + v2 * y2) ≤ w2 + v2 * u2 → P)) → (∀u2 : set, u2 ∈ SNoR y2 → (∀v2 : set, v2 ∈ SNoL z2 → (z2 * u2 + v2 * y2) ≤ w2 + v2 * u2 → P)) → P))) → SNoCut w u = SNoCut v x2
In Proofgold the corresponding term root is 2cc89a... and proposition id is f2c203...
Proof:
Proof not loaded.
End of Section Conj_mul_SNo_assoc__10__13
Beginning of Section Conj_mul_SNo_assoc__11__11
L562
Variable x : set
(*** Conj_mul_SNo_assoc__11__11 TMRopZCqC2UzTt1mCdi3YVQyXXXjLVBzuPP bounty of about 25 bars ***)
L563
Variable y : set
L564
Variable z : set
L565
Variable w : set
L566
Variable u : set
L567
Variable v : set
L568
Variable x2 : set
L569
Hypothesis H0 : SNo x
L570
Hypothesis H1 : SNo y
L571
Hypothesis H2 : SNo z
L572
Hypothesis H3 : (∀y2 : set, y2 ∈ SNoS_ (SNoLev x) → y2 * y * z = (y2 * y) * z)
L573
Hypothesis H4 : (∀y2 : set, y2 ∈ SNoS_ (SNoLev y) → x * y2 * z = (x * y2) * z)
L574
Hypothesis H5 : (∀y2 : set, y2 ∈ SNoS_ (SNoLev z) → x * y * y2 = (x * y) * y2)
L575
Hypothesis H6 : (∀y2 : set, y2 ∈ SNoS_ (SNoLev x) → (∀z2 : set, z2 ∈ SNoS_ (SNoLev y) → y2 * z2 * z = (y2 * z2) * z))
L576
Hypothesis H7 : (∀y2 : set, y2 ∈ SNoS_ (SNoLev x) → (∀z2 : set, z2 ∈ SNoS_ (SNoLev z) → y2 * y * z2 = (y2 * y) * z2))
L577
Hypothesis H8 : (∀y2 : set, y2 ∈ SNoS_ (SNoLev y) → (∀z2 : set, z2 ∈ SNoS_ (SNoLev z) → x * y2 * z2 = (x * y2) * z2))
L578
Hypothesis H9 : (∀y2 : set, y2 ∈ SNoS_ (SNoLev x) → (∀z2 : set, z2 ∈ SNoS_ (SNoLev y) → (∀w2 : set, w2 ∈ SNoS_ (SNoLev z) → y2 * z2 * w2 = (y2 * z2) * w2)))
L579
Hypothesis H10 : SNo (x * y)
L580
Hypothesis H12 : (∀y2 : set, y2 ∈ w → (∀P : prop, (∀z2 : set, z2 ∈ SNoL x → (∀w2 : set, w2 ∈ SNoL (y * z) → y2 = z2 * y * z + x * w2 + - (z2 * w2) → P)) → (∀z2 : set, z2 ∈ SNoR x → (∀w2 : set, w2 ∈ SNoR (y * z) → y2 = z2 * y * z + x * w2 + - (z2 * w2) → P)) → P))
L581
Hypothesis H13 : (∀y2 : set, y2 ∈ u → (∀P : prop, (∀z2 : set, z2 ∈ SNoL x → (∀w2 : set, w2 ∈ SNoR (y * z) → y2 = z2 * y * z + x * w2 + - (z2 * w2) → P)) → (∀z2 : set, z2 ∈ SNoR x → (∀w2 : set, w2 ∈ SNoL (y * z) → y2 = z2 * y * z + x * w2 + - (z2 * w2) → P)) → P))
L582
Hypothesis H14 : x * y * z = SNoCut w u
L583
Hypothesis H15 : SNoCutP v x2
L584
Hypothesis H16 : (∀y2 : set, y2 ∈ v → (∀P : prop, (∀z2 : set, z2 ∈ SNoL (x * y) → (∀w2 : set, w2 ∈ SNoL z → y2 = z2 * z + (x * y) * w2 + - (z2 * w2) → P)) → (∀z2 : set, z2 ∈ SNoR (x * y) → (∀w2 : set, w2 ∈ SNoR z → y2 = z2 * z + (x * y) * w2 + - (z2 * w2) → P)) → P))
L585
Hypothesis H17 : (∀y2 : set, y2 ∈ x2 → (∀P : prop, (∀z2 : set, z2 ∈ SNoL (x * y) → (∀w2 : set, w2 ∈ SNoR z → y2 = z2 * z + (x * y) * w2 + - (z2 * w2) → P)) → (∀z2 : set, z2 ∈ SNoR (x * y) → (∀w2 : set, w2 ∈ SNoL z → y2 = z2 * z + (x * y) * w2 + - (z2 * w2) → P)) → P))
L586
Hypothesis H18 : (x * y) * z = SNoCut v x2
L587
Theorem. (Conj_mul_SNo_assoc__11__11)
(∀y2 : set, ∀z2 : set, SNo y2 → SNo z2 → (∀w2 : set, w2 ∈ SNoL (z2 * y2) → (∀P : prop, (∀u2 : set, u2 ∈ SNoL y2 → (∀v2 : set, v2 ∈ SNoL z2 → (w2 + v2 * u2) ≤ z2 * u2 + v2 * y2 → P)) → (∀u2 : set, u2 ∈ SNoR y2 → (∀v2 : set, v2 ∈ SNoR z2 → (w2 + v2 * u2) ≤ z2 * u2 + v2 * y2 → P)) → P))) → SNoCut w u = SNoCut v x2
In Proofgold the corresponding term root is 0d541f... and proposition id is 81c4bb...
Proof:
Proof not loaded.
End of Section Conj_mul_SNo_assoc__11__11
Beginning of Section Conj_nonneg_mul_SNo_Le__1__2
L593
Variable x : set
(*** Conj_nonneg_mul_SNo_Le__1__2 TMGjsorauqZGxDnLx5NCzWFx3XxQR3FrwLZ bounty of about 25 bars ***)
L594
Variable y : set
L595
Variable z : set
L596
Hypothesis H0 : SNo x
L597
Hypothesis H1 : Empty ≤ x
L598
Hypothesis H3 : SNo z
L599
Hypothesis H4 : y ≤ z
L600
Hypothesis H5 : Empty * z + x * y = x * y
L601
Theorem. (Conj_nonneg_mul_SNo_Le__1__2)
x * z + Empty * y = x * z → x * y ≤ x * z
In Proofgold the corresponding term root is 56603a... and proposition id is ee4568...
Proof:
Proof not loaded.
End of Section Conj_nonneg_mul_SNo_Le__1__2
Beginning of Section Conj_neg_mul_SNo_Lt__1__3
L607
Variable x : set
(*** Conj_neg_mul_SNo_Lt__1__3 TMQYrmVJstqALQKER1bHkUVBt6XMTuEMYVk bounty of about 25 bars ***)
L608
Variable y : set
L609
Variable z : set
L610
Hypothesis H0 : SNo x
L611
Hypothesis H1 : x < Empty
L612
Hypothesis H2 : SNo y
L613
Hypothesis H4 : z < y
L614
Hypothesis H5 : x * y + Empty * z = x * y
L615
Theorem. (Conj_neg_mul_SNo_Lt__1__3)
Empty * y + x * z = x * z → x * y < x * z
In Proofgold the corresponding term root is 14ee30... and proposition id is 9ba346...
Proof:
Proof not loaded.
End of Section Conj_neg_mul_SNo_Lt__1__3
Beginning of Section Conj_neg_mul_SNo_Lt__2__0
L621
Variable x : set
(*** Conj_neg_mul_SNo_Lt__2__0 TMQuAriBkzHzjXJUFnKvjvGhYrnszCs6Wf1 bounty of about 25 bars ***)
L622
Variable y : set
L623
Variable z : set
L624
Hypothesis H1 : x < Empty
L625
Hypothesis H2 : SNo y
L626
Hypothesis H3 : SNo z
L627
Hypothesis H4 : z < y
L628
Theorem. (Conj_neg_mul_SNo_Lt__2__0)
x * y + Empty * z = x * y → x * y < x * z
In Proofgold the corresponding term root is 0bddac... and proposition id is b380b2...
Proof:
Proof not loaded.
End of Section Conj_neg_mul_SNo_Lt__2__0
Beginning of Section Conj_SNo_foil_mm__1__1
L634
Variable x : set
(*** Conj_SNo_foil_mm__1__1 TMaW8BBqSXjX9KmK5wAmbVfzkxNknuatzyX bounty of about 25 bars ***)
L635
Variable y : set
L636
Variable z : set
L637
Variable w : set
L638
Hypothesis H0 : SNo x
L639
Hypothesis H2 : SNo z
L640
Hypothesis H3 : SNo w
L641
Hypothesis H4 : SNo (- y)
L642
Theorem. (Conj_SNo_foil_mm__1__1)
SNo (- w) → (x + - y) * (z + - w) = x * z + - (x * w) + - (y * z) + y * w
In Proofgold the corresponding term root is 1abd68... and proposition id is 08278d...
Proof:
Proof not loaded.
End of Section Conj_SNo_foil_mm__1__1
Beginning of Section Conj_SNo_foil_mm__2__0
L648
Variable x : set
(*** Conj_SNo_foil_mm__2__0 TMaK8SfLjZuPR25H9cpASD1XaMLmrzPeQDc bounty of about 25 bars ***)
L649
Variable y : set
L650
Variable z : set
L651
Variable w : set
L652
Hypothesis H1 : SNo y
L653
Hypothesis H2 : SNo z
L654
Hypothesis H3 : SNo w
L655
Theorem. (Conj_SNo_foil_mm__2__0)
SNo (- y) → (x + - y) * (z + - w) = x * z + - (x * w) + - (y * z) + y * w
In Proofgold the corresponding term root is eea339... and proposition id is 753923...
Proof:
Proof not loaded.
End of Section Conj_SNo_foil_mm__2__0
Beginning of Section Conj_eps_ordsucc_half_add__7__0
L661
Variable x : set
(*** Conj_eps_ordsucc_half_add__7__0 TMJb5N2jfvVFzbeYQ6r75deVvFdQFo1Qypx bounty of about 25 bars ***)
L662
Variable y : set
L663
Hypothesis H1 : x ∈ ω
L664
Hypothesis H2 : SNo (eps_ (ordsucc x))
L665
Hypothesis H3 : SNo y
L666
Hypothesis H4 : SNoLev y ∈ ordsucc (ordsucc x)
L667
Hypothesis H5 : y < eps_ (ordsucc x)
L668
Theorem. (Conj_eps_ordsucc_half_add__7__0)
y ≤ Empty → (y + eps_ (ordsucc x)) < eps_ x ∧ (eps_ (ordsucc x) + y) < eps_ x
In Proofgold the corresponding term root is dc63fe... and proposition id is 47fc97...
Proof:
Proof not loaded.
End of Section Conj_eps_ordsucc_half_add__7__0
Beginning of Section Conj_eps_ordsucc_half_add__7__1
L674
Variable x : set
(*** Conj_eps_ordsucc_half_add__7__1 TMW7rp3z3vveRW15DuHC5HQbXg6EiCzmAAW bounty of about 25 bars ***)
L675
Variable y : set
L676
Hypothesis H0 : nat_p x
L677
Hypothesis H2 : SNo (eps_ (ordsucc x))
L678
Hypothesis H3 : SNo y
L679
Hypothesis H4 : SNoLev y ∈ ordsucc (ordsucc x)
L680
Hypothesis H5 : y < eps_ (ordsucc x)
L681
Theorem. (Conj_eps_ordsucc_half_add__7__1)
y ≤ Empty → (y + eps_ (ordsucc x)) < eps_ x ∧ (eps_ (ordsucc x) + y) < eps_ x
In Proofgold the corresponding term root is 9c6845... and proposition id is c7d134...
Proof:
Proof not loaded.
End of Section Conj_eps_ordsucc_half_add__7__1
Beginning of Section Conj_eps_ordsucc_half_add__11__1
L687
Variable x : set
(*** Conj_eps_ordsucc_half_add__11__1 TMahqJLBEpHsK9f9TGcEDFnXXjWDWeTQuhJ bounty of about 25 bars ***)
L688
Hypothesis H0 : nat_p x
L689
Theorem. (Conj_eps_ordsucc_half_add__11__1)
x ∈ ω → eps_ (ordsucc x) + eps_ (ordsucc x) = eps_ x
In Proofgold the corresponding term root is c75204... and proposition id is ca45cf...
Proof:
Proof not loaded.
End of Section Conj_eps_ordsucc_half_add__11__1
Beginning of Section Conj_double_eps_1__1__1
L695
Variable x : set
(*** Conj_double_eps_1__1__1 TMVZyUnQoZ91cmJhoJwV56N6UVUMW2JWgPE bounty of about 25 bars ***)
L696
Variable y : set
L697
Variable z : set
L698
Hypothesis H0 : SNo x
L699
Hypothesis H2 : SNo z
L700
Hypothesis H3 : x + x = y + z
L701
Theorem. (Conj_double_eps_1__1__1)
SNo (y + z) → x = eps_ (ordsucc Empty) * (y + z)
In Proofgold the corresponding term root is 4c5418... and proposition id is 7bd1dc...
Proof:
Proof not loaded.
End of Section Conj_double_eps_1__1__1
Beginning of Section Conj_exp_SNo_1_bd__1__1
L707
Variable x : set
(*** Conj_exp_SNo_1_bd__1__1 TMN8o59sZ5Twx4D5Kz4PctF7qpkiRee7boh bounty of about 25 bars ***)
L708
Variable y : set
L709
Hypothesis H0 : SNo x
L710
Hypothesis H2 : nat_p y
L711
Hypothesis H3 : ordsucc Empty ≤ exp_SNo_nat x y
L712
Theorem. (Conj_exp_SNo_1_bd__1__1)
SNo (exp_SNo_nat x y) → ordsucc Empty ≤ exp_SNo_nat x (ordsucc y)
In Proofgold the corresponding term root is 1a0306... and proposition id is 9d546f...
Proof:
Proof not loaded.
End of Section Conj_exp_SNo_1_bd__1__1
Beginning of Section Conj_mul_SNo_eps_eps_add_SNo__5__0
L718
Variable x : set
(*** Conj_mul_SNo_eps_eps_add_SNo__5__0 TMKoaaH7VK9hZvbaEJDT5pusMejBDhJJmaS bounty of about 25 bars ***)
L719
Variable y : set
L720
Hypothesis H1 : y ∈ ω
L721
Hypothesis H2 : x + y ∈ ω
L722
Hypothesis H3 : nat_p (x + y)
L723
Hypothesis H4 : SNo (eps_ x)
L724
Theorem. (Conj_mul_SNo_eps_eps_add_SNo__5__0)
SNo (eps_ y) → eps_ x * eps_ y = eps_ (x + y)
In Proofgold the corresponding term root is e456ab... and proposition id is ea47ad...
Proof:
Proof not loaded.
End of Section Conj_mul_SNo_eps_eps_add_SNo__5__0
Beginning of Section Conj_mul_SNo_eps_eps_add_SNo__8__0
L730
Variable x : set
(*** Conj_mul_SNo_eps_eps_add_SNo__8__0 TMFzVzKiacct23f44X1LXmF5ChpVSdiggCF bounty of about 25 bars ***)
L731
Variable y : set
L732
Hypothesis H1 : y ∈ ω
L733
Theorem. (Conj_mul_SNo_eps_eps_add_SNo__8__0)
x + y ∈ ω → eps_ x * eps_ y = eps_ (x + y)
In Proofgold the corresponding term root is ca3632... and proposition id is 756fb4...
Proof:
Proof not loaded.
End of Section Conj_mul_SNo_eps_eps_add_SNo__8__0
Beginning of Section Conj_SNoS_omega_Lev_equip__9__0
L739
Variable x : set
(*** Conj_SNoS_omega_Lev_equip__9__0 TMG1KMpBBNaaizgKcjtixYkiVWrSygoDhCN bounty of about 25 bars ***)
L740
Variable f : (set → set)
L741
Variable f2 : (set → set)
L742
Variable y : set
L743
Variable z : set
L744
Hypothesis H1 : (∀w : set, x ∈ w → f2 w = f (binintersect w (SNoElts_ x)))
L745
Hypothesis H2 : f z = y
L746
Hypothesis H3 : SNoLev z = x
L747
Hypothesis H4 : SNo z
L748
Hypothesis H5 : SNoLev (SNo_extend1 z) = ordsucc x
L749
Theorem. (Conj_SNoS_omega_Lev_equip__9__0)
x ∈ SNo_extend1 z → (∃w : set, w ∈ Sep (SNoS_ ω) (λu : set ⇒ SNoLev u = ordsucc x) ∧ f2 w = y)
In Proofgold the corresponding term root is ed5a3e... and proposition id is 386344...
Proof:
Proof not loaded.
End of Section Conj_SNoS_omega_Lev_equip__9__0
Beginning of Section Conj_SNoS_omega_Lev_equip__9__1
L755
Variable x : set
(*** Conj_SNoS_omega_Lev_equip__9__1 TMZ43Wjbdf6opM763pZyvVW4PMWiQ9pdxQQ bounty of about 25 bars ***)
L756
Variable f : (set → set)
L757
Variable f2 : (set → set)
L758
Variable y : set
L759
Variable z : set
L760
Hypothesis H0 : nat_p x
L761
Hypothesis H2 : f z = y
L762
Hypothesis H3 : SNoLev z = x
L763
Hypothesis H4 : SNo z
L764
Hypothesis H5 : SNoLev (SNo_extend1 z) = ordsucc x
L765
Theorem. (Conj_SNoS_omega_Lev_equip__9__1)
x ∈ SNo_extend1 z → (∃w : set, w ∈ Sep (SNoS_ ω) (λu : set ⇒ SNoLev u = ordsucc x) ∧ f2 w = y)
In Proofgold the corresponding term root is bbdd8d... and proposition id is 6e0662...
Proof:
Proof not loaded.
End of Section Conj_SNoS_omega_Lev_equip__9__1
Beginning of Section Conj_SNoS_omega_Lev_equip__13__7
L771
Variable x : set
(*** Conj_SNoS_omega_Lev_equip__13__7 TMWUqrVPWYxKfk53cR1PDhFuwKVQETiy8Aq bounty of about 25 bars ***)
L772
Variable f : (set → set)
L773
Variable f2 : (set → set)
L774
Variable y : set
L775
Hypothesis H0 : nat_p x
L776
Hypothesis H1 : nat_p (exp_SNo_nat (ordsucc (ordsucc Empty)) x)
L777
Hypothesis H2 : SNo (exp_SNo_nat (ordsucc (ordsucc Empty)) x)
L778
Hypothesis H3 : nat_p (exp_SNo_nat (ordsucc (ordsucc Empty)) x + exp_SNo_nat (ordsucc (ordsucc Empty)) x)
L779
Hypothesis H4 : (∀z : set, z ∈ exp_SNo_nat (ordsucc (ordsucc Empty)) x → (∃w : set, w ∈ Sep (SNoS_ ω) (λu : set ⇒ SNoLev u = x) ∧ f w = z))
L780
Hypothesis H5 : (∀z : set, x ∈ z → f2 z = f (binintersect z (SNoElts_ x)))
L781
Hypothesis H6 : (∀z : set, nIn x z → f2 z = exp_SNo_nat (ordsucc (ordsucc Empty)) x + f (binintersect z (SNoElts_ x)))
L782
Theorem. (Conj_SNoS_omega_Lev_equip__13__7)
nat_p y → (∃z : set, z ∈ Sep (SNoS_ ω) (λw : set ⇒ SNoLev w = ordsucc x) ∧ f2 z = y)
In Proofgold the corresponding term root is 8c36e5... and proposition id is 705159...
Proof:
Proof not loaded.
End of Section Conj_SNoS_omega_Lev_equip__13__7
Beginning of Section Conj_SNoS_omega_Lev_equip__16__1
L788
Variable x : set
(*** Conj_SNoS_omega_Lev_equip__16__1 TMHRRgFXkDjLdVTJV7gdBgWYzLCBrp6Ehyj bounty of about 25 bars ***)
L789
Variable f : (set → set)
L790
Variable f2 : (set → set)
L791
Variable y : set
L792
Variable z : set
L793
Hypothesis H0 : SNo (exp_SNo_nat (ordsucc (ordsucc Empty)) x)
L794
Hypothesis H2 : (∀w : set, x ∈ w → f2 w = f (binintersect w (SNoElts_ x)))
L795
Hypothesis H3 : (∀w : set, nIn x w → f2 w = exp_SNo_nat (ordsucc (ordsucc Empty)) x + f (binintersect w (SNoElts_ x)))
L796
Hypothesis H4 : SNo y
L797
Hypothesis H5 : SNoLev y = ordsucc x
L798
Hypothesis H6 : binintersect y (SNoElts_ x) ∈ Sep (SNoS_ ω) (λw : set ⇒ SNoLev w = x)
L799
Hypothesis H7 : nat_p (f (binintersect y (SNoElts_ x)))
L800
Hypothesis H8 : SNo (f (binintersect y (SNoElts_ x)))
L801
Hypothesis H9 : f (binintersect y (SNoElts_ x)) < exp_SNo_nat (ordsucc (ordsucc Empty)) x
L802
Hypothesis H10 : SNo z
L803
Hypothesis H11 : SNoLev z = ordsucc x
L804
Hypothesis H12 : binintersect z (SNoElts_ x) ∈ Sep (SNoS_ ω) (λw : set ⇒ SNoLev w = x)
L805
Hypothesis H13 : nat_p (f (binintersect z (SNoElts_ x)))
L806
Hypothesis H14 : SNo (f (binintersect z (SNoElts_ x)))
L807
Hypothesis H15 : f (binintersect z (SNoElts_ x)) < exp_SNo_nat (ordsucc (ordsucc Empty)) x
L808
Hypothesis H16 : x ∈ SNoLev y
L809
Theorem. (Conj_SNoS_omega_Lev_equip__16__1)
x ∈ SNoLev z → f2 y = f2 z → y = z
In Proofgold the corresponding term root is 3c8bdb... and proposition id is 3903c0...
Proof:
Proof not loaded.
End of Section Conj_SNoS_omega_Lev_equip__16__1
Beginning of Section Conj_SNoS_omega_Lev_equip__16__4
L815
Variable x : set
(*** Conj_SNoS_omega_Lev_equip__16__4 TMK9XmRHcc5LWNQNgymtMPERAYJ6SLKNqLB bounty of about 25 bars ***)
L816
Variable f : (set → set)
L817
Variable f2 : (set → set)
L818
Variable y : set
L819
Variable z : set
L820
Hypothesis H0 : SNo (exp_SNo_nat (ordsucc (ordsucc Empty)) x)
L821
Hypothesis H1 : (∀w : set, w ∈ Sep (SNoS_ ω) (λu : set ⇒ SNoLev u = x) → (∀u : set, u ∈ Sep (SNoS_ ω) (λv : set ⇒ SNoLev v = x) → f w = f u → w = u))
L822
Hypothesis H2 : (∀w : set, x ∈ w → f2 w = f (binintersect w (SNoElts_ x)))
L823
Hypothesis H3 : (∀w : set, nIn x w → f2 w = exp_SNo_nat (ordsucc (ordsucc Empty)) x + f (binintersect w (SNoElts_ x)))
L824
Hypothesis H5 : SNoLev y = ordsucc x
L825
Hypothesis H6 : binintersect y (SNoElts_ x) ∈ Sep (SNoS_ ω) (λw : set ⇒ SNoLev w = x)
L826
Hypothesis H7 : nat_p (f (binintersect y (SNoElts_ x)))
L827
Hypothesis H8 : SNo (f (binintersect y (SNoElts_ x)))
L828
Hypothesis H9 : f (binintersect y (SNoElts_ x)) < exp_SNo_nat (ordsucc (ordsucc Empty)) x
L829
Hypothesis H10 : SNo z
L830
Hypothesis H11 : SNoLev z = ordsucc x
L831
Hypothesis H12 : binintersect z (SNoElts_ x) ∈ Sep (SNoS_ ω) (λw : set ⇒ SNoLev w = x)
L832
Hypothesis H13 : nat_p (f (binintersect z (SNoElts_ x)))
L833
Hypothesis H14 : SNo (f (binintersect z (SNoElts_ x)))
L834
Hypothesis H15 : f (binintersect z (SNoElts_ x)) < exp_SNo_nat (ordsucc (ordsucc Empty)) x
L835
Hypothesis H16 : x ∈ SNoLev y
L836
Theorem. (Conj_SNoS_omega_Lev_equip__16__4)
x ∈ SNoLev z → f2 y = f2 z → y = z
In Proofgold the corresponding term root is c3c206... and proposition id is bef6b1...
Proof:
Proof not loaded.
End of Section Conj_SNoS_omega_Lev_equip__16__4
Beginning of Section Conj_SNoS_omega_Lev_equip__16__6
L842
Variable x : set
(*** Conj_SNoS_omega_Lev_equip__16__6 TMKmR8LzxLFHc2vi3R6hxPCwXhBiKHXquGw bounty of about 25 bars ***)
L843
Variable f : (set → set)
L844
Variable f2 : (set → set)
L845
Variable y : set
L846
Variable z : set
L847
Hypothesis H0 : SNo (exp_SNo_nat (ordsucc (ordsucc Empty)) x)
L848
Hypothesis H1 : (∀w : set, w ∈ Sep (SNoS_ ω) (λu : set ⇒ SNoLev u = x) → (∀u : set, u ∈ Sep (SNoS_ ω) (λv : set ⇒ SNoLev v = x) → f w = f u → w = u))
L849
Hypothesis H2 : (∀w : set, x ∈ w → f2 w = f (binintersect w (SNoElts_ x)))
L850
Hypothesis H3 : (∀w : set, nIn x w → f2 w = exp_SNo_nat (ordsucc (ordsucc Empty)) x + f (binintersect w (SNoElts_ x)))
L851
Hypothesis H4 : SNo y
L852
Hypothesis H5 : SNoLev y = ordsucc x
L853
Hypothesis H7 : nat_p (f (binintersect y (SNoElts_ x)))
L854
Hypothesis H8 : SNo (f (binintersect y (SNoElts_ x)))
L855
Hypothesis H9 : f (binintersect y (SNoElts_ x)) < exp_SNo_nat (ordsucc (ordsucc Empty)) x
L856
Hypothesis H10 : SNo z
L857
Hypothesis H11 : SNoLev z = ordsucc x
L858
Hypothesis H12 : binintersect z (SNoElts_ x) ∈ Sep (SNoS_ ω) (λw : set ⇒ SNoLev w = x)
L859
Hypothesis H13 : nat_p (f (binintersect z (SNoElts_ x)))
L860
Hypothesis H14 : SNo (f (binintersect z (SNoElts_ x)))
L861
Hypothesis H15 : f (binintersect z (SNoElts_ x)) < exp_SNo_nat (ordsucc (ordsucc Empty)) x
L862
Hypothesis H16 : x ∈ SNoLev y
L863
Theorem. (Conj_SNoS_omega_Lev_equip__16__6)
x ∈ SNoLev z → f2 y = f2 z → y = z
In Proofgold the corresponding term root is 01607e... and proposition id is ac9f35...
Proof:
Proof not loaded.
End of Section Conj_SNoS_omega_Lev_equip__16__6
Beginning of Section Conj_SNoS_omega_Lev_equip__17__14
L869
Variable x : set
(*** Conj_SNoS_omega_Lev_equip__17__14 TMFXYCzERqRqqzAiHctBqYKnhwPEiCRfkRV bounty of about 25 bars ***)
L870
Variable f : (set → set)
L871
Variable f2 : (set → set)
L872
Variable y : set
L873
Variable z : set
L874
Hypothesis H0 : SNo (exp_SNo_nat (ordsucc (ordsucc Empty)) x)
L875
Hypothesis H1 : (∀w : set, w ∈ Sep (SNoS_ ω) (λu : set ⇒ SNoLev u = x) → (∀u : set, u ∈ Sep (SNoS_ ω) (λv : set ⇒ SNoLev v = x) → f w = f u → w = u))
L876
Hypothesis H2 : (∀w : set, x ∈ w → f2 w = f (binintersect w (SNoElts_ x)))
L877
Hypothesis H3 : (∀w : set, nIn x w → f2 w = exp_SNo_nat (ordsucc (ordsucc Empty)) x + f (binintersect w (SNoElts_ x)))
L878
Hypothesis H4 : SNo y
L879
Hypothesis H5 : SNoLev y = ordsucc x
L880
Hypothesis H6 : binintersect y (SNoElts_ x) ∈ Sep (SNoS_ ω) (λw : set ⇒ SNoLev w = x)
L881
Hypothesis H7 : nat_p (f (binintersect y (SNoElts_ x)))
L882
Hypothesis H8 : SNo (f (binintersect y (SNoElts_ x)))
L883
Hypothesis H9 : f (binintersect y (SNoElts_ x)) < exp_SNo_nat (ordsucc (ordsucc Empty)) x
L884
Hypothesis H10 : SNo z
L885
Hypothesis H11 : SNoLev z = ordsucc x
L886
Hypothesis H12 : binintersect z (SNoElts_ x) ∈ Sep (SNoS_ ω) (λw : set ⇒ SNoLev w = x)
L887
Hypothesis H13 : nat_p (f (binintersect z (SNoElts_ x)))
L888
Hypothesis H15 : f (binintersect z (SNoElts_ x)) < exp_SNo_nat (ordsucc (ordsucc Empty)) x
L889
Theorem. (Conj_SNoS_omega_Lev_equip__17__14)
x ∈ SNoLev y → f2 y = f2 z → y = z
In Proofgold the corresponding term root is c05dec... and proposition id is 1cc56d...
Proof:
Proof not loaded.
End of Section Conj_SNoS_omega_Lev_equip__17__14
Beginning of Section Conj_SNoS_omega_Lev_equip__18__0
L895
Variable x : set
(*** Conj_SNoS_omega_Lev_equip__18__0 TMTqn1mJ5us4t33BrhY11rQJ9Ufr6SiRkwf bounty of about 25 bars ***)
L896
Variable f : (set → set)
L897
Hypothesis H1 : nat_p (exp_SNo_nat (ordsucc (ordsucc Empty)) x)
L898
Hypothesis H2 : ordinal (exp_SNo_nat (ordsucc (ordsucc Empty)) x)
L899
Hypothesis H3 : SNo (exp_SNo_nat (ordsucc (ordsucc Empty)) x)
L900
Hypothesis H4 : nat_p (exp_SNo_nat (ordsucc (ordsucc Empty)) x + exp_SNo_nat (ordsucc (ordsucc Empty)) x)
L901
Hypothesis H5 : ordinal (exp_SNo_nat (ordsucc (ordsucc Empty)) x + exp_SNo_nat (ordsucc (ordsucc Empty)) x)
L902
Hypothesis H6 : SNo (exp_SNo_nat (ordsucc (ordsucc Empty)) x + exp_SNo_nat (ordsucc (ordsucc Empty)) x)
L903
Hypothesis H7 : (∀y : set, SNo y → y < exp_SNo_nat (ordsucc (ordsucc Empty)) x → (exp_SNo_nat (ordsucc (ordsucc Empty)) x + y) < exp_SNo_nat (ordsucc (ordsucc Empty)) x + exp_SNo_nat (ordsucc (ordsucc Empty)) x)
L904
Hypothesis H8 : exp_SNo_nat (ordsucc (ordsucc Empty)) x < exp_SNo_nat (ordsucc (ordsucc Empty)) x + exp_SNo_nat (ordsucc (ordsucc Empty)) x
L905
Hypothesis H9 : (∀y : set, y ∈ Sep (SNoS_ ω) (λz : set ⇒ SNoLev z = x) → (∀z : set, z ∈ Sep (SNoS_ ω) (λw : set ⇒ SNoLev w = x) → f y = f z → y = z))
L906
Hypothesis H10 : (∀y : set, y ∈ exp_SNo_nat (ordsucc (ordsucc Empty)) x → (∃z : set, z ∈ Sep (SNoS_ ω) (λw : set ⇒ SNoLev w = x) ∧ f z = y))
L907
Hypothesis H11 : (∀y : set, y ∈ Sep (SNoS_ ω) (λz : set ⇒ SNoLev z = ordsucc x) → (∀P : prop, (SNo y → SNoLev y = ordsucc x → binintersect y (SNoElts_ x) ∈ Sep (SNoS_ ω) (λz : set ⇒ SNoLev z = x) → SNo (binintersect y (SNoElts_ x)) → SNoLev (binintersect y (SNoElts_ x)) = x → P) → P))
L908
Hypothesis H12 : (∀y : set, y ∈ Sep (SNoS_ ω) (λz : set ⇒ SNoLev z = ordsucc x) → (∀P : prop, (nat_p (f (binintersect y (SNoElts_ x))) → ordinal (f (binintersect y (SNoElts_ x))) → SNo (f (binintersect y (SNoElts_ x))) → f (binintersect y (SNoElts_ x)) < exp_SNo_nat (ordsucc (ordsucc Empty)) x → P) → P))
L909
Theorem. (Conj_SNoS_omega_Lev_equip__18__0)
(∃f2 : set → set, (∀y : set, x ∈ y → f2 y = f (binintersect y (SNoElts_ x))) ∧ (∀y : set, nIn x y → f2 y = exp_SNo_nat (ordsucc (ordsucc Empty)) x + f (binintersect y (SNoElts_ x)))) → (∃f2 : set → set, bij (Sep (SNoS_ ω) (λy : set ⇒ SNoLev y = ordsucc x)) (exp_SNo_nat (ordsucc (ordsucc Empty)) x + exp_SNo_nat (ordsucc (ordsucc Empty)) x) f2)
In Proofgold the corresponding term root is e6f7d9... and proposition id is 74cbfb...
Proof:
Proof not loaded.
End of Section Conj_SNoS_omega_Lev_equip__18__0
Beginning of Section Conj_SNoS_omega_Lev_equip__18__3
L915
Variable x : set
(*** Conj_SNoS_omega_Lev_equip__18__3 TMFuqnSfwmRA9e6CivrUJRN2ZiuA1zNDFAL bounty of about 25 bars ***)
L916
Variable f : (set → set)
L917
Hypothesis H0 : nat_p x
L918
Hypothesis H1 : nat_p (exp_SNo_nat (ordsucc (ordsucc Empty)) x)
L919
Hypothesis H2 : ordinal (exp_SNo_nat (ordsucc (ordsucc Empty)) x)
L920
Hypothesis H4 : nat_p (exp_SNo_nat (ordsucc (ordsucc Empty)) x + exp_SNo_nat (ordsucc (ordsucc Empty)) x)
L921
Hypothesis H5 : ordinal (exp_SNo_nat (ordsucc (ordsucc Empty)) x + exp_SNo_nat (ordsucc (ordsucc Empty)) x)
L922
Hypothesis H6 : SNo (exp_SNo_nat (ordsucc (ordsucc Empty)) x + exp_SNo_nat (ordsucc (ordsucc Empty)) x)
L923
Hypothesis H7 : (∀y : set, SNo y → y < exp_SNo_nat (ordsucc (ordsucc Empty)) x → (exp_SNo_nat (ordsucc (ordsucc Empty)) x + y) < exp_SNo_nat (ordsucc (ordsucc Empty)) x + exp_SNo_nat (ordsucc (ordsucc Empty)) x)
L924
Hypothesis H8 : exp_SNo_nat (ordsucc (ordsucc Empty)) x < exp_SNo_nat (ordsucc (ordsucc Empty)) x + exp_SNo_nat (ordsucc (ordsucc Empty)) x
L925
Hypothesis H9 : (∀y : set, y ∈ Sep (SNoS_ ω) (λz : set ⇒ SNoLev z = x) → (∀z : set, z ∈ Sep (SNoS_ ω) (λw : set ⇒ SNoLev w = x) → f y = f z → y = z))
L926
Hypothesis H10 : (∀y : set, y ∈ exp_SNo_nat (ordsucc (ordsucc Empty)) x → (∃z : set, z ∈ Sep (SNoS_ ω) (λw : set ⇒ SNoLev w = x) ∧ f z = y))
L927
Hypothesis H11 : (∀y : set, y ∈ Sep (SNoS_ ω) (λz : set ⇒ SNoLev z = ordsucc x) → (∀P : prop, (SNo y → SNoLev y = ordsucc x → binintersect y (SNoElts_ x) ∈ Sep (SNoS_ ω) (λz : set ⇒ SNoLev z = x) → SNo (binintersect y (SNoElts_ x)) → SNoLev (binintersect y (SNoElts_ x)) = x → P) → P))
L928
Hypothesis H12 : (∀y : set, y ∈ Sep (SNoS_ ω) (λz : set ⇒ SNoLev z = ordsucc x) → (∀P : prop, (nat_p (f (binintersect y (SNoElts_ x))) → ordinal (f (binintersect y (SNoElts_ x))) → SNo (f (binintersect y (SNoElts_ x))) → f (binintersect y (SNoElts_ x)) < exp_SNo_nat (ordsucc (ordsucc Empty)) x → P) → P))
L929
Theorem. (Conj_SNoS_omega_Lev_equip__18__3)
(∃f2 : set → set, (∀y : set, x ∈ y → f2 y = f (binintersect y (SNoElts_ x))) ∧ (∀y : set, nIn x y → f2 y = exp_SNo_nat (ordsucc (ordsucc Empty)) x + f (binintersect y (SNoElts_ x)))) → (∃f2 : set → set, bij (Sep (SNoS_ ω) (λy : set ⇒ SNoLev y = ordsucc x)) (exp_SNo_nat (ordsucc (ordsucc Empty)) x + exp_SNo_nat (ordsucc (ordsucc Empty)) x) f2)
In Proofgold the corresponding term root is c92c97... and proposition id is 9a24e9...
Proof:
Proof not loaded.
End of Section Conj_SNoS_omega_Lev_equip__18__3
Beginning of Section Conj_SNoS_omega_Lev_equip__18__6
L935
Variable x : set
(*** Conj_SNoS_omega_Lev_equip__18__6 TMb7s34n5EcAcXeNvjtjLjFMEDPVoMYjU13 bounty of about 25 bars ***)
L936
Variable f : (set → set)
L937
Hypothesis H0 : nat_p x
L938
Hypothesis H1 : nat_p (exp_SNo_nat (ordsucc (ordsucc Empty)) x)
L939
Hypothesis H2 : ordinal (exp_SNo_nat (ordsucc (ordsucc Empty)) x)
L940
Hypothesis H3 : SNo (exp_SNo_nat (ordsucc (ordsucc Empty)) x)
L941
Hypothesis H4 : nat_p (exp_SNo_nat (ordsucc (ordsucc Empty)) x + exp_SNo_nat (ordsucc (ordsucc Empty)) x)
L942
Hypothesis H5 : ordinal (exp_SNo_nat (ordsucc (ordsucc Empty)) x + exp_SNo_nat (ordsucc (ordsucc Empty)) x)
L943
Hypothesis H7 : (∀y : set, SNo y → y < exp_SNo_nat (ordsucc (ordsucc Empty)) x → (exp_SNo_nat (ordsucc (ordsucc Empty)) x + y) < exp_SNo_nat (ordsucc (ordsucc Empty)) x + exp_SNo_nat (ordsucc (ordsucc Empty)) x)
L944
Hypothesis H8 : exp_SNo_nat (ordsucc (ordsucc Empty)) x < exp_SNo_nat (ordsucc (ordsucc Empty)) x + exp_SNo_nat (ordsucc (ordsucc Empty)) x
L945
Hypothesis H9 : (∀y : set, y ∈ Sep (SNoS_ ω) (λz : set ⇒ SNoLev z = x) → (∀z : set, z ∈ Sep (SNoS_ ω) (λw : set ⇒ SNoLev w = x) → f y = f z → y = z))
L946
Hypothesis H10 : (∀y : set, y ∈ exp_SNo_nat (ordsucc (ordsucc Empty)) x → (∃z : set, z ∈ Sep (SNoS_ ω) (λw : set ⇒ SNoLev w = x) ∧ f z = y))
L947
Hypothesis H11 : (∀y : set, y ∈ Sep (SNoS_ ω) (λz : set ⇒ SNoLev z = ordsucc x) → (∀P : prop, (SNo y → SNoLev y = ordsucc x → binintersect y (SNoElts_ x) ∈ Sep (SNoS_ ω) (λz : set ⇒ SNoLev z = x) → SNo (binintersect y (SNoElts_ x)) → SNoLev (binintersect y (SNoElts_ x)) = x → P) → P))
L948
Hypothesis H12 : (∀y : set, y ∈ Sep (SNoS_ ω) (λz : set ⇒ SNoLev z = ordsucc x) → (∀P : prop, (nat_p (f (binintersect y (SNoElts_ x))) → ordinal (f (binintersect y (SNoElts_ x))) → SNo (f (binintersect y (SNoElts_ x))) → f (binintersect y (SNoElts_ x)) < exp_SNo_nat (ordsucc (ordsucc Empty)) x → P) → P))
L949
Theorem. (Conj_SNoS_omega_Lev_equip__18__6)
(∃f2 : set → set, (∀y : set, x ∈ y → f2 y = f (binintersect y (SNoElts_ x))) ∧ (∀y : set, nIn x y → f2 y = exp_SNo_nat (ordsucc (ordsucc Empty)) x + f (binintersect y (SNoElts_ x)))) → (∃f2 : set → set, bij (Sep (SNoS_ ω) (λy : set ⇒ SNoLev y = ordsucc x)) (exp_SNo_nat (ordsucc (ordsucc Empty)) x + exp_SNo_nat (ordsucc (ordsucc Empty)) x) f2)
In Proofgold the corresponding term root is e4e991... and proposition id is e04fb4...
Proof:
Proof not loaded.
End of Section Conj_SNoS_omega_Lev_equip__18__6
Beginning of Section Conj_SNoS_omega_Lev_equip__22__1
L955
Variable x : set
(*** Conj_SNoS_omega_Lev_equip__22__1 TMbCvbPwp2FHpwPLDi6PGgLkykMDv6mpwkf bounty of about 25 bars ***)
L956
Hypothesis H0 : nat_p x
L957
Hypothesis H2 : nat_p (exp_SNo_nat (ordsucc (ordsucc Empty)) x)
L958
Hypothesis H3 : ordinal (exp_SNo_nat (ordsucc (ordsucc Empty)) x)
L959
Hypothesis H4 : SNo (exp_SNo_nat (ordsucc (ordsucc Empty)) x)
L960
Hypothesis H5 : nat_p (exp_SNo_nat (ordsucc (ordsucc Empty)) x + exp_SNo_nat (ordsucc (ordsucc Empty)) x)
L961
Hypothesis H6 : ordinal (exp_SNo_nat (ordsucc (ordsucc Empty)) x + exp_SNo_nat (ordsucc (ordsucc Empty)) x)
L962
Hypothesis H7 : SNo (exp_SNo_nat (ordsucc (ordsucc Empty)) x + exp_SNo_nat (ordsucc (ordsucc Empty)) x)
L963
Theorem. (Conj_SNoS_omega_Lev_equip__22__1)
(∀y : set, SNo y → y < exp_SNo_nat (ordsucc (ordsucc Empty)) x → (exp_SNo_nat (ordsucc (ordsucc Empty)) x + y) < exp_SNo_nat (ordsucc (ordsucc Empty)) x + exp_SNo_nat (ordsucc (ordsucc Empty)) x) → equip (Sep (SNoS_ ω) (λy : set ⇒ SNoLev y = ordsucc x)) (exp_SNo_nat (ordsucc (ordsucc Empty)) x + exp_SNo_nat (ordsucc (ordsucc Empty)) x)
In Proofgold the corresponding term root is e53d15... and proposition id is 784414...
Proof:
Proof not loaded.
End of Section Conj_SNoS_omega_Lev_equip__22__1
Beginning of Section Conj_SNoS_omega_Lev_equip__22__4
L969
Variable x : set
(*** Conj_SNoS_omega_Lev_equip__22__4 TMX9cVQboUEkonQDqDTa96BnZyBLSe2y3hr bounty of about 25 bars ***)
L970
Hypothesis H0 : nat_p x
L971
Hypothesis H1 : equip (Sep (SNoS_ ω) (λy : set ⇒ SNoLev y = x)) (exp_SNo_nat (ordsucc (ordsucc Empty)) x)
L972
Hypothesis H2 : nat_p (exp_SNo_nat (ordsucc (ordsucc Empty)) x)
L973
Hypothesis H3 : ordinal (exp_SNo_nat (ordsucc (ordsucc Empty)) x)
L974
Hypothesis H5 : nat_p (exp_SNo_nat (ordsucc (ordsucc Empty)) x + exp_SNo_nat (ordsucc (ordsucc Empty)) x)
L975
Hypothesis H6 : ordinal (exp_SNo_nat (ordsucc (ordsucc Empty)) x + exp_SNo_nat (ordsucc (ordsucc Empty)) x)
L976
Hypothesis H7 : SNo (exp_SNo_nat (ordsucc (ordsucc Empty)) x + exp_SNo_nat (ordsucc (ordsucc Empty)) x)
L977
Theorem. (Conj_SNoS_omega_Lev_equip__22__4)
(∀y : set, SNo y → y < exp_SNo_nat (ordsucc (ordsucc Empty)) x → (exp_SNo_nat (ordsucc (ordsucc Empty)) x + y) < exp_SNo_nat (ordsucc (ordsucc Empty)) x + exp_SNo_nat (ordsucc (ordsucc Empty)) x) → equip (Sep (SNoS_ ω) (λy : set ⇒ SNoLev y = ordsucc x)) (exp_SNo_nat (ordsucc (ordsucc Empty)) x + exp_SNo_nat (ordsucc (ordsucc Empty)) x)
In Proofgold the corresponding term root is c352f3... and proposition id is 2a6a0a...
Proof:
Proof not loaded.
End of Section Conj_SNoS_omega_Lev_equip__22__4
Beginning of Section Conj_SNoS_omega_Lev_equip__24__5
L983
Variable x : set
(*** Conj_SNoS_omega_Lev_equip__24__5 TMGzrd2L5pFtAh1bvZmqPG45HaLvWYfdbMi bounty of about 25 bars ***)
L984
Hypothesis H0 : nat_p x
L985
Hypothesis H1 : equip (Sep (SNoS_ ω) (λy : set ⇒ SNoLev y = x)) (exp_SNo_nat (ordsucc (ordsucc Empty)) x)
L986
Hypothesis H2 : nat_p (exp_SNo_nat (ordsucc (ordsucc Empty)) x)
L987
Hypothesis H3 : ordinal (exp_SNo_nat (ordsucc (ordsucc Empty)) x)
L988
Hypothesis H4 : SNo (exp_SNo_nat (ordsucc (ordsucc Empty)) x)
L989
Theorem. (Conj_SNoS_omega_Lev_equip__24__5)
ordinal (exp_SNo_nat (ordsucc (ordsucc Empty)) x + exp_SNo_nat (ordsucc (ordsucc Empty)) x) → equip (Sep (SNoS_ ω) (λy : set ⇒ SNoLev y = ordsucc x)) (exp_SNo_nat (ordsucc (ordsucc Empty)) x + exp_SNo_nat (ordsucc (ordsucc Empty)) x)
In Proofgold the corresponding term root is ee5e0e... and proposition id is a9f787...
Proof:
Proof not loaded.
End of Section Conj_SNoS_omega_Lev_equip__24__5
Beginning of Section Conj_int_add_SNo__1__1
L995
Variable x : set
(*** Conj_int_add_SNo__1__1 TMKXdnndWzphwh27gBanBBbCiKsrkD3NNJS bounty of about 25 bars ***)
L996
Variable y : set
L997
Hypothesis H0 : x ∈ ω
L998
Hypothesis H2 : SNo x
L999
Theorem. (Conj_int_add_SNo__1__1)
SNo y → - x + - y ∈ int
In Proofgold the corresponding term root is 1b05cb... and proposition id is 0b6fee...
Proof:
Proof not loaded.
End of Section Conj_int_add_SNo__1__1
Beginning of Section Conj_int_mul_SNo__3__2
L1005
Variable x : set
(*** Conj_int_mul_SNo__3__2 TMKLzQZv7tetX9ynJbWSocrRgmXDh9JCEBo bounty of about 25 bars ***)
L1006
Variable y : set
L1007
Hypothesis H0 : x ∈ ω
L1008
Hypothesis H1 : SNo x
L1009
Theorem. (Conj_int_mul_SNo__3__2)
nat_p y → (- x) * (- y) ∈ int
In Proofgold the corresponding term root is 118cbc... and proposition id is 164f11...
Proof:
Proof not loaded.
End of Section Conj_int_mul_SNo__3__2
Beginning of Section Conj_int_mul_SNo__10__2
L1015
Variable x : set
(*** Conj_int_mul_SNo__10__2 TMT5DAmPEb4avysJVsgfnV7iERVYNbhZHmz bounty of about 25 bars ***)
L1016
Variable y : set
L1017
Hypothesis H0 : x ∈ ω
L1018
Hypothesis H1 : SNo x
L1019
Hypothesis H3 : ordinal y
L1020
Theorem. (Conj_int_mul_SNo__10__2)
SNo y → x * (- y) ∈ int
In Proofgold the corresponding term root is e29249... and proposition id is 581271...
Proof:
Proof not loaded.
End of Section Conj_int_mul_SNo__10__2
Beginning of Section Conj_SNo_triangle2__2__0
L1026
Variable x : set
(*** Conj_SNo_triangle2__2__0 TMVycYNJ7jNg6EL94FLXgfKSZ4XDHoiNVvn bounty of about 25 bars ***)
L1027
Variable y : set
L1028
Variable z : set
L1029
Hypothesis H1 : SNo y
L1030
Hypothesis H2 : SNo z
L1031
Hypothesis H3 : SNo (- y)
L1032
Theorem. (Conj_SNo_triangle2__2__0)
SNo (- z) → abs_SNo (x + - z) ≤ abs_SNo (x + - y) + abs_SNo (y + - z)
In Proofgold the corresponding term root is e956d2... and proposition id is be2756...
Proof:
Proof not loaded.
End of Section Conj_SNo_triangle2__2__0
Beginning of Section Conj_double_SNo_max_1__1__6
L1038
Variable x : set
(*** Conj_double_SNo_max_1__1__6 TMZPPjfSoNYFqfCdsNA9tXEqcv7W6qY9vUm bounty of about 25 bars ***)
L1039
Variable y : set
L1040
Variable z : set
L1041
Hypothesis H0 : SNo x
L1042
Hypothesis H1 : SNo y
L1043
Hypothesis H2 : (∀w : set, w ∈ SNoL x → SNo w → w ≤ y)
L1044
Hypothesis H3 : SNoLev y ∈ SNoLev x
L1045
Hypothesis H4 : SNo z
L1046
Hypothesis H5 : SNoLev z ∈ SNoLev y
L1047
Hypothesis H7 : z < x
L1048
Theorem. (Conj_double_SNo_max_1__1__6)
z ∈ SNoL x → x ≤ z
In Proofgold the corresponding term root is e157c8... and proposition id is 879a40...
Proof:
Proof not loaded.
End of Section Conj_double_SNo_max_1__1__6
Beginning of Section Conj_double_SNo_max_1__2__2
L1054
Variable x : set
(*** Conj_double_SNo_max_1__2__2 TMVXxdnLVVAoJGMcYJ7Jzr1zVbL8oRqhwLw bounty of about 25 bars ***)
L1055
Variable y : set
L1056
Variable z : set
L1057
Variable w : set
L1058
Hypothesis H0 : SNo x
L1059
Hypothesis H1 : SNo z
L1060
Hypothesis H3 : x < z
L1061
Hypothesis H4 : w ∈ SNoR z
L1062
Hypothesis H5 : (y + w) < x + x
L1063
Hypothesis H6 : SNo w
L1064
Hypothesis H7 : SNoLev w ∈ SNoLev z
L1065
Hypothesis H8 : z < w
L1066
Theorem. (Conj_double_SNo_max_1__2__2)
(∃u : set, u ∈ SNoR w ∧ y + u = x + x) → (∃u : set, u ∈ SNoR z ∧ y + u = x + x)
In Proofgold the corresponding term root is 7ccdd6... and proposition id is f75cab...
Proof:
Proof not loaded.
End of Section Conj_double_SNo_max_1__2__2
Beginning of Section Conj_double_SNo_min_1__5__1
L1072
Variable x : set
(*** Conj_double_SNo_min_1__5__1 TMSyUoeoMqqKNLxkEMukT3vhRJnwTb8aeAP bounty of about 25 bars ***)
L1073
Variable y : set
L1074
Variable z : set
L1075
Hypothesis H0 : SNo x
L1076
Hypothesis H2 : SNo y
L1077
Hypothesis H3 : SNo z
L1078
Hypothesis H4 : z < x
L1079
Hypothesis H5 : (x + x) < y + z
L1080
Hypothesis H6 : SNo (- x)
L1081
Hypothesis H7 : SNo (- y)
L1082
Hypothesis H8 : SNo (- z)
L1083
Hypothesis H9 : SNo (x + x)
L1084
Theorem. (Conj_double_SNo_min_1__5__1)
SNo (y + z) → (∃w : set, w ∈ SNoL z ∧ y + w = x + x)
In Proofgold the corresponding term root is 081aea... and proposition id is d12cee...
Proof:
Proof not loaded.
End of Section Conj_double_SNo_min_1__5__1
Beginning of Section Conj_double_SNo_min_1__5__5
L1090
Variable x : set
(*** Conj_double_SNo_min_1__5__5 TMdAnCS2pcYxFsvXwBUiWkWREPawhPn472r bounty of about 25 bars ***)
L1091
Variable y : set
L1092
Variable z : set
L1093
Hypothesis H0 : SNo x
L1094
Hypothesis H1 : SNo_min_of (SNoR x) y
L1095
Hypothesis H2 : SNo y
L1096
Hypothesis H3 : SNo z
L1097
Hypothesis H4 : z < x
L1098
Hypothesis H6 : SNo (- x)
L1099
Hypothesis H7 : SNo (- y)
L1100
Hypothesis H8 : SNo (- z)
L1101
Hypothesis H9 : SNo (x + x)
L1102
Theorem. (Conj_double_SNo_min_1__5__5)
SNo (y + z) → (∃w : set, w ∈ SNoL z ∧ y + w = x + x)
In Proofgold the corresponding term root is 4fe4b4... and proposition id is 3cb5f3...
Proof:
Proof not loaded.
End of Section Conj_double_SNo_min_1__5__5
Beginning of Section Conj_double_SNo_min_1__5__6
L1108
Variable x : set
(*** Conj_double_SNo_min_1__5__6 TMXudoMGdEhm5rdPuCb6JxQ3V1RvK6b3y58 bounty of about 25 bars ***)
L1109
Variable y : set
L1110
Variable z : set
L1111
Hypothesis H0 : SNo x
L1112
Hypothesis H1 : SNo_min_of (SNoR x) y
L1113
Hypothesis H2 : SNo y
L1114
Hypothesis H3 : SNo z
L1115
Hypothesis H4 : z < x
L1116
Hypothesis H5 : (x + x) < y + z
L1117
Hypothesis H7 : SNo (- y)
L1118
Hypothesis H8 : SNo (- z)
L1119
Hypothesis H9 : SNo (x + x)
L1120
Theorem. (Conj_double_SNo_min_1__5__6)
SNo (y + z) → (∃w : set, w ∈ SNoL z ∧ y + w = x + x)
In Proofgold the corresponding term root is 88fd6d... and proposition id is e1aaae...
Proof:
Proof not loaded.
End of Section Conj_double_SNo_min_1__5__6
Beginning of Section Conj_double_SNo_min_1__5__9
L1126
Variable x : set
(*** Conj_double_SNo_min_1__5__9 TMbkDCJzS7Y61htx9xAZn22zkqtvqwNqRhh bounty of about 25 bars ***)
L1127
Variable y : set
L1128
Variable z : set
L1129
Hypothesis H0 : SNo x
L1130
Hypothesis H1 : SNo_min_of (SNoR x) y
L1131
Hypothesis H2 : SNo y
L1132
Hypothesis H3 : SNo z
L1133
Hypothesis H4 : z < x
L1134
Hypothesis H5 : (x + x) < y + z
L1135
Hypothesis H6 : SNo (- x)
L1136
Hypothesis H7 : SNo (- y)
L1137
Hypothesis H8 : SNo (- z)
L1138
Theorem. (Conj_double_SNo_min_1__5__9)
SNo (y + z) → (∃w : set, w ∈ SNoL z ∧ y + w = x + x)
In Proofgold the corresponding term root is a42ead... and proposition id is b81828...
Proof:
Proof not loaded.
End of Section Conj_double_SNo_min_1__5__9
Beginning of Section Conj_double_SNo_min_1__7__1
L1144
Variable x : set
(*** Conj_double_SNo_min_1__7__1 TMPqQUwKPm1Pn9YnUN5fkbwQBCGdRbzAk3Y bounty of about 25 bars ***)
L1145
Variable y : set
L1146
Variable z : set
L1147
Hypothesis H0 : SNo x
L1148
Hypothesis H2 : SNo y
L1149
Hypothesis H3 : SNo z
L1150
Hypothesis H4 : z < x
L1151
Hypothesis H5 : (x + x) < y + z
L1152
Hypothesis H6 : SNo (- x)
L1153
Hypothesis H7 : SNo (- y)
L1154
Theorem. (Conj_double_SNo_min_1__7__1)
SNo (- z) → (∃w : set, w ∈ SNoL z ∧ y + w = x + x)
In Proofgold the corresponding term root is 697571... and proposition id is 110d2a...
Proof:
Proof not loaded.
End of Section Conj_double_SNo_min_1__7__1
Beginning of Section Conj_finite_max_exists__3__1
L1160
Variable x : set
(*** Conj_finite_max_exists__3__1 TMKFqbw3kax2YeVnPNwjr7GebiYjLjbwXJT bounty of about 25 bars ***)
L1161
Variable y : set
L1162
Variable f : (set → set)
L1163
Hypothesis H0 : (∀z : set, (∀w : set, w ∈ z → SNo w) → equip z (ordsucc x) → (∃w : set, SNo_max_of z w))
L1164
Hypothesis H2 : (∀z : set, z ∈ ordsucc (ordsucc x) → f z ∈ y)
L1165
Hypothesis H3 : (∀z : set, z ∈ ordsucc (ordsucc x) → (∀w : set, w ∈ ordsucc (ordsucc x) → f z = f w → z = w))
L1166
Hypothesis H4 : (∀z : set, z ∈ y → (∃w : set, w ∈ ordsucc (ordsucc x) ∧ f w = z))
L1167
Hypothesis H5 : Subq (Repl (ordsucc x) f) y
L1168
Theorem. (Conj_finite_max_exists__3__1)
equip (Repl (ordsucc x) f) (ordsucc x) → (∃z : set, SNo_max_of y z)
In Proofgold the corresponding term root is 4f3ed1... and proposition id is f9b93e...
Proof:
Proof not loaded.
End of Section Conj_finite_max_exists__3__1
Beginning of Section Conj_SNoS_omega_SNoL_max_exists__1__0
L1174
Variable x : set
(*** Conj_SNoS_omega_SNoL_max_exists__1__0 TMMwZtwZzq24ogbQh2HpJPu3BXWecQciwYA bounty of about 25 bars ***)
L1175
Hypothesis H1 : SNoL x ≠ Empty
L1176
Theorem. (Conj_SNoS_omega_SNoL_max_exists__1__0)
(∀y : set, y ∈ SNoL x → SNo y) → (∃y : set, SNo_max_of (SNoL x) y)
In Proofgold the corresponding term root is 752ac0... and proposition id is 2232e2...
Proof:
Proof not loaded.
End of Section Conj_SNoS_omega_SNoL_max_exists__1__0
Beginning of Section Conj_SNoS_omega_SNoR_min_exists__1__0
L1182
Variable x : set
(*** Conj_SNoS_omega_SNoR_min_exists__1__0 TMUvJaTLXjppUiwg1ff7hJFuuqfrnBGEuAo bounty of about 25 bars ***)
L1183
Hypothesis H1 : SNoR x ≠ Empty
L1184
Theorem. (Conj_SNoS_omega_SNoR_min_exists__1__0)
(∀y : set, y ∈ SNoR x → SNo y) → (∃y : set, SNo_min_of (SNoR x) y)
In Proofgold the corresponding term root is 0fb500... and proposition id is e0ac64...
Proof:
Proof not loaded.
End of Section Conj_SNoS_omega_SNoR_min_exists__1__0
Beginning of Section Conj_minus_SNo_diadic_rational_p__3__0
L1190
Variable x : set
(*** Conj_minus_SNo_diadic_rational_p__3__0 TMWSVpsoFVQdr3wEzGzqQmPprFstpFeKZRs bounty of about 25 bars ***)
L1191
Variable y : set
L1192
Theorem. (Conj_minus_SNo_diadic_rational_p__3__0)
SNo (eps_ y) → (∃z : set, z ∈ int ∧ x = eps_ y * z) → diadic_rational_p (- x)
In Proofgold the corresponding term root is 09351f... and proposition id is 92ab71...
Proof:
Proof not loaded.
End of Section Conj_minus_SNo_diadic_rational_p__3__0
Beginning of Section Conj_mul_SNo_diadic_rational_p__1__7
L1198
Variable x : set
(*** Conj_mul_SNo_diadic_rational_p__1__7 TMZPdyy2iYeBRUmj813vgMVaruTJNRNNUai bounty of about 25 bars ***)
L1199
Variable y : set
L1200
Variable z : set
L1201
Variable w : set
L1202
Variable u : set
L1203
Variable v : set
L1204
Hypothesis H0 : z ∈ ω
L1205
Hypothesis H1 : SNo (eps_ z)
L1206
Hypothesis H2 : w ∈ int
L1207
Hypothesis H3 : x = eps_ z * w
L1208
Hypothesis H4 : SNo w
L1209
Hypothesis H5 : u ∈ ω
L1210
Hypothesis H6 : SNo (eps_ u)
L1211
Hypothesis H8 : SNo v
L1212
Theorem. (Conj_mul_SNo_diadic_rational_p__1__7)
SNo (eps_ u * v) → y = eps_ u * v → (∃x2 : set, x2 ∈ ω ∧ (∃y2 : set, y2 ∈ int ∧ x * y = eps_ x2 * y2))
In Proofgold the corresponding term root is d9b1f0... and proposition id is 0fb620...
Proof:
Proof not loaded.
End of Section Conj_mul_SNo_diadic_rational_p__1__7
Beginning of Section Conj_mul_SNo_diadic_rational_p__3__2
L1218
Variable x : set
(*** Conj_mul_SNo_diadic_rational_p__3__2 TMV7CPWxjJBTbrsyZrRPh15spXZb46NKRQ8 bounty of about 25 bars ***)
L1219
Variable y : set
L1220
Variable z : set
L1221
Variable w : set
L1222
Variable u : set
L1223
Hypothesis H0 : z ∈ ω
L1224
Hypothesis H1 : SNo (eps_ z)
L1225
Hypothesis H3 : x = eps_ z * w
L1226
Hypothesis H4 : SNo w
L1227
Hypothesis H5 : u ∈ ω
L1228
Theorem. (Conj_mul_SNo_diadic_rational_p__3__2)
SNo (eps_ u) → (∃v : set, v ∈ int ∧ y = eps_ u * v) → diadic_rational_p (x * y)
In Proofgold the corresponding term root is 825e7e... and proposition id is 238c1f...
Proof:
Proof not loaded.
End of Section Conj_mul_SNo_diadic_rational_p__3__2
Beginning of Section Conj_add_SNo_diadic_rational_p__1__8
L1234
Variable x : set
(*** Conj_add_SNo_diadic_rational_p__1__8 TMV7au4o5UWBPWULwLsgJgoqFNa7fsVK2PP bounty of about 25 bars ***)
L1235
Variable y : set
L1236
Variable z : set
L1237
Variable w : set
L1238
Variable u : set
L1239
Variable v : set
L1240
Hypothesis H0 : z ∈ ω
L1241
Hypothesis H1 : SNo (eps_ z)
L1242
Hypothesis H2 : w ∈ int
L1243
Hypothesis H3 : x = eps_ z * w
L1244
Hypothesis H4 : u ∈ ω
L1245
Hypothesis H5 : SNo (eps_ u)
L1246
Hypothesis H6 : v ∈ int
L1247
Hypothesis H7 : y = eps_ u * v
L1248
Hypothesis H9 : exp_SNo_nat (ordsucc (ordsucc Empty)) u ∈ int
L1249
Hypothesis H10 : exp_SNo_nat (ordsucc (ordsucc Empty)) u * w ∈ int
L1250
Hypothesis H11 : exp_SNo_nat (ordsucc (ordsucc Empty)) z ∈ int
L1251
Theorem. (Conj_add_SNo_diadic_rational_p__1__8)
exp_SNo_nat (ordsucc (ordsucc Empty)) z * v ∈ int → exp_SNo_nat (ordsucc (ordsucc Empty)) u * w + exp_SNo_nat (ordsucc (ordsucc Empty)) z * v ∈ int ∧ x + y = eps_ (z + u) * (exp_SNo_nat (ordsucc (ordsucc Empty)) u * w + exp_SNo_nat (ordsucc (ordsucc Empty)) z * v)
In Proofgold the corresponding term root is a1ce09... and proposition id is 867cbc...
Proof:
Proof not loaded.
End of Section Conj_add_SNo_diadic_rational_p__1__8
Beginning of Section Conj_add_SNo_diadic_rational_p__1__11
L1257
Variable x : set
(*** Conj_add_SNo_diadic_rational_p__1__11 TMF7t8L5f1mERyj3VtKZmbW1k8aSteUSi4c bounty of about 25 bars ***)
L1258
Variable y : set
L1259
Variable z : set
L1260
Variable w : set
L1261
Variable u : set
L1262
Variable v : set
L1263
Hypothesis H0 : z ∈ ω
L1264
Hypothesis H1 : SNo (eps_ z)
L1265
Hypothesis H2 : w ∈ int
L1266
Hypothesis H3 : x = eps_ z * w
L1267
Hypothesis H4 : u ∈ ω
L1268
Hypothesis H5 : SNo (eps_ u)
L1269
Hypothesis H6 : v ∈ int
L1270
Hypothesis H7 : y = eps_ u * v
L1271
Hypothesis H8 : SNo (eps_ u * v)
L1272
Hypothesis H9 : exp_SNo_nat (ordsucc (ordsucc Empty)) u ∈ int
L1273
Hypothesis H10 : exp_SNo_nat (ordsucc (ordsucc Empty)) u * w ∈ int
L1274
Theorem. (Conj_add_SNo_diadic_rational_p__1__11)
exp_SNo_nat (ordsucc (ordsucc Empty)) z * v ∈ int → exp_SNo_nat (ordsucc (ordsucc Empty)) u * w + exp_SNo_nat (ordsucc (ordsucc Empty)) z * v ∈ int ∧ x + y = eps_ (z + u) * (exp_SNo_nat (ordsucc (ordsucc Empty)) u * w + exp_SNo_nat (ordsucc (ordsucc Empty)) z * v)
In Proofgold the corresponding term root is 4cbb3d... and proposition id is 71a20e...
Proof:
Proof not loaded.
End of Section Conj_add_SNo_diadic_rational_p__1__11
Beginning of Section Conj_add_SNo_diadic_rational_p__4__0
L1280
Variable x : set
(*** Conj_add_SNo_diadic_rational_p__4__0 TMVQZGxMcdDK2HdqW6srTE6PnvWE7RutNY8 bounty of about 25 bars ***)
L1281
Variable y : set
L1282
Variable z : set
L1283
Variable w : set
L1284
Variable u : set
L1285
Variable v : set
L1286
Hypothesis H1 : SNo (eps_ z)
L1287
Hypothesis H2 : w ∈ int
L1288
Hypothesis H3 : x = eps_ z * w
L1289
Hypothesis H4 : u ∈ ω
L1290
Hypothesis H5 : SNo (eps_ u)
L1291
Hypothesis H6 : v ∈ int
L1292
Hypothesis H7 : y = eps_ u * v
L1293
Hypothesis H8 : SNo (eps_ u * v)
L1294
Theorem. (Conj_add_SNo_diadic_rational_p__4__0)
exp_SNo_nat (ordsucc (ordsucc Empty)) u ∈ int → exp_SNo_nat (ordsucc (ordsucc Empty)) u * w + exp_SNo_nat (ordsucc (ordsucc Empty)) z * v ∈ int ∧ x + y = eps_ (z + u) * (exp_SNo_nat (ordsucc (ordsucc Empty)) u * w + exp_SNo_nat (ordsucc (ordsucc Empty)) z * v)
In Proofgold the corresponding term root is b83185... and proposition id is be5fe2...
Proof:
Proof not loaded.
End of Section Conj_add_SNo_diadic_rational_p__4__0
Beginning of Section Conj_add_SNo_diadic_rational_p__4__7
L1300
Variable x : set
(*** Conj_add_SNo_diadic_rational_p__4__7 TMJQSBjAjm7vxyCNHT8UK7C3xm6F6X2ZViw bounty of about 25 bars ***)
L1301
Variable y : set
L1302
Variable z : set
L1303
Variable w : set
L1304
Variable u : set
L1305
Variable v : set
L1306
Hypothesis H0 : z ∈ ω
L1307
Hypothesis H1 : SNo (eps_ z)
L1308
Hypothesis H2 : w ∈ int
L1309
Hypothesis H3 : x = eps_ z * w
L1310
Hypothesis H4 : u ∈ ω
L1311
Hypothesis H5 : SNo (eps_ u)
L1312
Hypothesis H6 : v ∈ int
L1313
Hypothesis H8 : SNo (eps_ u * v)
L1314
Theorem. (Conj_add_SNo_diadic_rational_p__4__7)
exp_SNo_nat (ordsucc (ordsucc Empty)) u ∈ int → exp_SNo_nat (ordsucc (ordsucc Empty)) u * w + exp_SNo_nat (ordsucc (ordsucc Empty)) z * v ∈ int ∧ x + y = eps_ (z + u) * (exp_SNo_nat (ordsucc (ordsucc Empty)) u * w + exp_SNo_nat (ordsucc (ordsucc Empty)) z * v)
In Proofgold the corresponding term root is 191436... and proposition id is f7b3e9...
Proof:
Proof not loaded.
End of Section Conj_add_SNo_diadic_rational_p__4__7
Beginning of Section Conj_add_SNo_diadic_rational_p__5__6
L1320
Variable x : set
(*** Conj_add_SNo_diadic_rational_p__5__6 TMcSyLcDxLNE2aXkeSJV1cFt1WMD3zE8VRq bounty of about 25 bars ***)
L1321
Variable y : set
L1322
Variable z : set
L1323
Variable w : set
L1324
Variable u : set
L1325
Variable v : set
L1326
Hypothesis H0 : z ∈ ω
L1327
Hypothesis H1 : SNo (eps_ z)
L1328
Hypothesis H2 : w ∈ int
L1329
Hypothesis H3 : x = eps_ z * w
L1330
Hypothesis H4 : u ∈ ω
L1331
Hypothesis H5 : SNo (eps_ u)
L1332
Hypothesis H7 : y = eps_ u * v
L1333
Hypothesis H8 : SNo v
L1334
Theorem. (Conj_add_SNo_diadic_rational_p__5__6)
SNo (eps_ u * v) → (∃x2 : set, x2 ∈ ω ∧ (∃y2 : set, y2 ∈ int ∧ x + y = eps_ x2 * y2))
In Proofgold the corresponding term root is ab8348... and proposition id is 1c1f1f...
Proof:
Proof not loaded.
End of Section Conj_add_SNo_diadic_rational_p__5__6
Beginning of Section Conj_add_SNo_diadic_rational_p__5__7
L1340
Variable x : set
(*** Conj_add_SNo_diadic_rational_p__5__7 TMLqfYjB4T25CFT6n1vBfEqM7TVqVLGRipj bounty of about 25 bars ***)
L1341
Variable y : set
L1342
Variable z : set
L1343
Variable w : set
L1344
Variable u : set
L1345
Variable v : set
L1346
Hypothesis H0 : z ∈ ω
L1347
Hypothesis H1 : SNo (eps_ z)
L1348
Hypothesis H2 : w ∈ int
L1349
Hypothesis H3 : x = eps_ z * w
L1350
Hypothesis H4 : u ∈ ω
L1351
Hypothesis H5 : SNo (eps_ u)
L1352
Hypothesis H6 : v ∈ int
L1353
Hypothesis H8 : SNo v
L1354
Theorem. (Conj_add_SNo_diadic_rational_p__5__7)
SNo (eps_ u * v) → (∃x2 : set, x2 ∈ ω ∧ (∃y2 : set, y2 ∈ int ∧ x + y = eps_ x2 * y2))
In Proofgold the corresponding term root is 3b4ecf... and proposition id is 7ddba9...
Proof:
Proof not loaded.
End of Section Conj_add_SNo_diadic_rational_p__5__7
Beginning of Section Conj_add_SNo_diadic_rational_p__7__1
L1360
Variable x : set
(*** Conj_add_SNo_diadic_rational_p__7__1 TMH2zbFVvL5Fn9rePTs2NsVfjAQYqPkcdeT bounty of about 25 bars ***)
L1361
Variable y : set
L1362
Variable z : set
L1363
Variable w : set
L1364
Variable u : set
L1365
Hypothesis H0 : z ∈ ω
L1366
Hypothesis H2 : w ∈ int
L1367
Hypothesis H3 : x = eps_ z * w
L1368
Hypothesis H4 : u ∈ ω
L1369
Theorem. (Conj_add_SNo_diadic_rational_p__7__1)
SNo (eps_ u) → (∃v : set, v ∈ int ∧ y = eps_ u * v) → diadic_rational_p (x + y)
In Proofgold the corresponding term root is 404f40... and proposition id is 88c5df...
Proof:
Proof not loaded.
End of Section Conj_add_SNo_diadic_rational_p__7__1
Beginning of Section Conj_SNoS_omega_diadic_rational_p_lem__3__1
L1375
Variable x : set
(*** Conj_SNoS_omega_diadic_rational_p_lem__3__1 TMLBMcYi4nnEoz3iCtCGCnSDkvRmMhSJP8W bounty of about 25 bars ***)
L1376
Variable y : set
L1377
Hypothesis H0 : nat_p x
L1378
Hypothesis H2 : ¬ diadic_rational_p y
L1379
Hypothesis H3 : ordinal y
L1380
Theorem. (Conj_SNoS_omega_diadic_rational_p_lem__3__1)
y = x → (∃z : set, SNo_min_of (SNoR y) z)
In Proofgold the corresponding term root is b7d4e2... and proposition id is e3fbe2...
Proof:
Proof not loaded.
End of Section Conj_SNoS_omega_diadic_rational_p_lem__3__1
Beginning of Section Conj_SNoS_omega_diadic_rational_p_lem__9__2
L1386
Variable x : set
(*** Conj_SNoS_omega_diadic_rational_p_lem__9__2 TMXzCynKgAxiP9828r6dDAMEsNa733ewRCT bounty of about 25 bars ***)
L1387
Variable y : set
L1388
Variable z : set
L1389
Variable w : set
L1390
Variable u : set
L1391
Hypothesis H0 : nat_p x
L1392
Hypothesis H1 : (∀v : set, v ∈ x → (∀x2 : set, SNo x2 → SNoLev x2 = v → diadic_rational_p x2))
L1393
Hypothesis H3 : SNoLev y = x
L1394
Hypothesis H4 : ¬ diadic_rational_p y
L1395
Hypothesis H5 : SNoLev z ∈ SNoLev y
L1396
Hypothesis H6 : SNo w
L1397
Hypothesis H7 : diadic_rational_p w
L1398
Hypothesis H8 : w + u = y + y
L1399
Hypothesis H9 : SNo u
L1400
Hypothesis H10 : SNoLev u ∈ SNoLev z
L1401
Theorem. (Conj_SNoS_omega_diadic_rational_p_lem__9__2)
¬ diadic_rational_p u
In Proofgold the corresponding term root is 651c43... and proposition id is dbf4e7...
Proof:
Proof not loaded.
End of Section Conj_SNoS_omega_diadic_rational_p_lem__9__2
Beginning of Section Conj_SNoS_omega_diadic_rational_p_lem__10__10
L1407
Variable x : set
(*** Conj_SNoS_omega_diadic_rational_p_lem__10__10 TMYEZA12F1rvpuV3jvJVpv7awmgwB2XXuHt bounty of about 25 bars ***)
L1408
Variable y : set
L1409
Variable z : set
L1410
Variable w : set
L1411
Hypothesis H0 : nat_p x
L1412
Hypothesis H1 : (∀u : set, u ∈ x → (∀v : set, SNo v → SNoLev v = u → diadic_rational_p v))
L1413
Hypothesis H2 : SNo y
L1414
Hypothesis H3 : SNoLev y = x
L1415
Hypothesis H4 : ¬ diadic_rational_p y
L1416
Hypothesis H5 : SNo_max_of (SNoL y) z
L1417
Hypothesis H6 : SNo z
L1418
Hypothesis H7 : SNoLev z ∈ SNoLev y
L1419
Hypothesis H8 : z < y
L1420
Hypothesis H9 : SNo_min_of (SNoR y) w
L1421
Hypothesis H11 : SNoLev w ∈ SNoLev y
L1422
Hypothesis H12 : y < w
L1423
Hypothesis H13 : SNo (y + y)
L1424
Hypothesis H14 : SNo (z + w)
L1425
Hypothesis H15 : diadic_rational_p z
L1426
Theorem. (Conj_SNoS_omega_diadic_rational_p_lem__10__10)
¬ diadic_rational_p w
In Proofgold the corresponding term root is 9d29bd... and proposition id is 2167e9...
Proof:
Proof not loaded.
End of Section Conj_SNoS_omega_diadic_rational_p_lem__10__10
Beginning of Section Conj_SNoS_omega_diadic_rational_p_lem__10__12
L1432
Variable x : set
(*** Conj_SNoS_omega_diadic_rational_p_lem__10__12 TMUiBGSafg3P5vutrDH7n1TJihAtNuHMe3y bounty of about 25 bars ***)
L1433
Variable y : set
L1434
Variable z : set
L1435
Variable w : set
L1436
Hypothesis H0 : nat_p x
L1437
Hypothesis H1 : (∀u : set, u ∈ x → (∀v : set, SNo v → SNoLev v = u → diadic_rational_p v))
L1438
Hypothesis H2 : SNo y
L1439
Hypothesis H3 : SNoLev y = x
L1440
Hypothesis H4 : ¬ diadic_rational_p y
L1441
Hypothesis H5 : SNo_max_of (SNoL y) z
L1442
Hypothesis H6 : SNo z
L1443
Hypothesis H7 : SNoLev z ∈ SNoLev y
L1444
Hypothesis H8 : z < y
L1445
Hypothesis H9 : SNo_min_of (SNoR y) w
L1446
Hypothesis H10 : SNo w
L1447
Hypothesis H11 : SNoLev w ∈ SNoLev y
L1448
Hypothesis H13 : SNo (y + y)
L1449
Hypothesis H14 : SNo (z + w)
L1450
Hypothesis H15 : diadic_rational_p z
L1451
Theorem. (Conj_SNoS_omega_diadic_rational_p_lem__10__12)
¬ diadic_rational_p w
In Proofgold the corresponding term root is ce518a... and proposition id is 9e6ca7...
Proof:
Proof not loaded.
End of Section Conj_SNoS_omega_diadic_rational_p_lem__10__12
Beginning of Section Conj_SNoS_omega_diadic_rational_p_lem__11__3
L1457
Variable x : set
(*** Conj_SNoS_omega_diadic_rational_p_lem__11__3 TMWXKUzSL25jh9dJYXM8nsfLs7ZCr8E7kLY bounty of about 25 bars ***)
L1458
Variable y : set
L1459
Variable z : set
L1460
Variable w : set
L1461
Hypothesis H0 : nat_p x
L1462
Hypothesis H1 : (∀u : set, u ∈ x → (∀v : set, SNo v → SNoLev v = u → diadic_rational_p v))
L1463
Hypothesis H2 : SNo y
L1464
Hypothesis H4 : ¬ diadic_rational_p y
L1465
Hypothesis H5 : SNo_max_of (SNoL y) z
L1466
Hypothesis H6 : SNo z
L1467
Hypothesis H7 : SNoLev z ∈ SNoLev y
L1468
Hypothesis H8 : z < y
L1469
Hypothesis H9 : SNo_min_of (SNoR y) w
L1470
Hypothesis H10 : SNo w
L1471
Hypothesis H11 : SNoLev w ∈ SNoLev y
L1472
Hypothesis H12 : y < w
L1473
Hypothesis H13 : SNo (y + y)
L1474
Hypothesis H14 : SNo (z + w)
L1475
Theorem. (Conj_SNoS_omega_diadic_rational_p_lem__11__3)
¬ diadic_rational_p z
In Proofgold the corresponding term root is bf55aa... and proposition id is 07a99d...
Proof:
Proof not loaded.
End of Section Conj_SNoS_omega_diadic_rational_p_lem__11__3
Beginning of Section Conj_SNoS_omega_diadic_rational_p_lem__11__5
L1481
Variable x : set
(*** Conj_SNoS_omega_diadic_rational_p_lem__11__5 TMYDTWoPKKhTi2KKbDQjUw4AAX1HXJFFktq bounty of about 25 bars ***)
L1482
Variable y : set
L1483
Variable z : set
L1484
Variable w : set
L1485
Hypothesis H0 : nat_p x
L1486
Hypothesis H1 : (∀u : set, u ∈ x → (∀v : set, SNo v → SNoLev v = u → diadic_rational_p v))
L1487
Hypothesis H2 : SNo y
L1488
Hypothesis H3 : SNoLev y = x
L1489
Hypothesis H4 : ¬ diadic_rational_p y
L1490
Hypothesis H6 : SNo z
L1491
Hypothesis H7 : SNoLev z ∈ SNoLev y
L1492
Hypothesis H8 : z < y
L1493
Hypothesis H9 : SNo_min_of (SNoR y) w
L1494
Hypothesis H10 : SNo w
L1495
Hypothesis H11 : SNoLev w ∈ SNoLev y
L1496
Hypothesis H12 : y < w
L1497
Hypothesis H13 : SNo (y + y)
L1498
Hypothesis H14 : SNo (z + w)
L1499
Theorem. (Conj_SNoS_omega_diadic_rational_p_lem__11__5)
¬ diadic_rational_p z
In Proofgold the corresponding term root is 808680... and proposition id is ff2111...
Proof:
Proof not loaded.
End of Section Conj_SNoS_omega_diadic_rational_p_lem__11__5
Beginning of Section Conj_SNoS_ordsucc_omega_bdd_above__4__0
L1505
Variable x : set
(*** Conj_SNoS_ordsucc_omega_bdd_above__4__0 TMNgxC5YVMUhEsNkAtGvVTQ22pyheQnVo9p bounty of about 25 bars ***)
L1506
Variable y : set
L1507
Hypothesis H1 : SNo y
L1508
Hypothesis H2 : x < y
L1509
Hypothesis H3 : SNoLev y ∈ ω
L1510
Theorem. (Conj_SNoS_ordsucc_omega_bdd_above__4__0)
ordinal (SNoLev y) → (∃z : set, z ∈ ω ∧ x < z)
In Proofgold the corresponding term root is cf2710... and proposition id is 5303d1...
Proof:
Proof not loaded.
End of Section Conj_SNoS_ordsucc_omega_bdd_above__4__0
Beginning of Section Conj_SNoS_ordsucc_omega_bdd_drat_intvl__3__0
L1516
Variable x : set
(*** Conj_SNoS_ordsucc_omega_bdd_drat_intvl__3__0 TMaa94vJggKcEJFWqbFYFKdcrHSdHcZG8uE bounty of about 25 bars ***)
L1517
Variable y : set
L1518
Hypothesis H1 : nIn x (SNoS_ ω)
L1519
Hypothesis H2 : nat_p y
L1520
Hypothesis H3 : - y < x → x < y → (∃z : set, z ∈ SNoS_ ω ∧ (z < x ∧ x < z + ordsucc Empty))
L1521
Hypothesis H4 : - (ordsucc y) < x
L1522
Hypothesis H5 : x < ordsucc y
L1523
Theorem. (Conj_SNoS_ordsucc_omega_bdd_drat_intvl__3__0)
SNo y → (∃z : set, z ∈ SNoS_ ω ∧ (z < x ∧ x < z + ordsucc Empty))
In Proofgold the corresponding term root is 5a7cc1... and proposition id is 236f7c...
Proof:
Proof not loaded.
End of Section Conj_SNoS_ordsucc_omega_bdd_drat_intvl__3__0
Beginning of Section Conj_SNoS_ordsucc_omega_bdd_drat_intvl__5__2
L1529
Variable x : set
(*** Conj_SNoS_ordsucc_omega_bdd_drat_intvl__5__2 TMFZ1ipbbA3ajjDAfQeAJQr3peiEPMuJTxR bounty of about 25 bars ***)
L1530
Hypothesis H0 : x ∈ SNoS_ (ordsucc ω)
L1531
Hypothesis H1 : - ω < x
L1532
Hypothesis H3 : SNo x
L1533
Hypothesis H4 : ¬ (∀y : set, y ∈ ω → (∃z : set, z ∈ SNoS_ ω ∧ (z < x ∧ x < z + eps_ y)))
L1534
Hypothesis H5 : nIn x (SNoS_ ω)
L1535
Theorem. (Conj_SNoS_ordsucc_omega_bdd_drat_intvl__5__2)
¬ (∀y : set, nat_p y → (∃z : set, z ∈ SNoS_ ω ∧ (z < x ∧ x < z + eps_ y)))
In Proofgold the corresponding term root is 901492... and proposition id is e3ee47...
Proof:
Proof not loaded.
End of Section Conj_SNoS_ordsucc_omega_bdd_drat_intvl__5__2
Beginning of Section Conj_real_E__3__6
L1541
Variable x : set
(*** Conj_real_E__3__6 TMe18YcuNkFzjiGrF2jdbxviCg1KLnzATQm bounty of about 25 bars ***)
L1542
Variable P : prop
L1543
Hypothesis H0 : SNo x → SNoLev x ∈ ordsucc ω → x ∈ SNoS_ (ordsucc ω) → - ω < x → x < ω → (∀y : set, y ∈ SNoS_ ω → (∀z : set, z ∈ ω → abs_SNo (y + - x) < eps_ z) → y = x) → (∀y : set, y ∈ ω → (∃z : set, z ∈ SNoS_ ω ∧ (z < x ∧ x < z + eps_ y))) → P
L1544
Hypothesis H1 : x ∈ SNoS_ (ordsucc ω)
L1545
Hypothesis H2 : (∀y : set, y ∈ SNoS_ ω → (∀z : set, z ∈ ω → abs_SNo (y + - x) < eps_ z) → y = x)
L1546
Hypothesis H3 : SNoLev x ∈ ordsucc ω
L1547
Hypothesis H4 : SNo x
L1548
Hypothesis H5 : x < ω
L1549
Theorem. (Conj_real_E__3__6)
(∀y : set, y ∈ ω → (∃z : set, z ∈ SNoS_ ω ∧ (z < x ∧ x < z + eps_ y))) → P
In Proofgold the corresponding term root is 7043e0... and proposition id is b49ded...
Proof:
Proof not loaded.
End of Section Conj_real_E__3__6
Beginning of Section Conj_real_E__4__0
L1555
Variable x : set
(*** Conj_real_E__4__0 TMYoeUL4MnkVKSXDUUvwyAQhi28mUYMvQNA bounty of about 25 bars ***)
L1556
Variable P : prop
L1557
Hypothesis H1 : x ∈ SNoS_ (ordsucc ω)
L1558
Hypothesis H2 : x ≠ - ω
L1559
Hypothesis H3 : (∀y : set, y ∈ SNoS_ ω → (∀z : set, z ∈ ω → abs_SNo (y + - x) < eps_ z) → y = x)
L1560
Hypothesis H4 : SNoLev x ∈ ordsucc ω
L1561
Hypothesis H5 : SNo x
L1562
Hypothesis H6 : x < ω
L1563
Theorem. (Conj_real_E__4__0)
- ω < x → P
In Proofgold the corresponding term root is a792d2... and proposition id is c82708...
Proof:
Proof not loaded.
End of Section Conj_real_E__4__0
Beginning of Section Conj_real_E__4__6
L1569
Variable x : set
(*** Conj_real_E__4__6 TMMBHs78zrRNyKW3cd7bJkyzvCZ4xFze9N9 bounty of about 25 bars ***)
L1570
Variable P : prop
L1571
Hypothesis H0 : SNo x → SNoLev x ∈ ordsucc ω → x ∈ SNoS_ (ordsucc ω) → - ω < x → x < ω → (∀y : set, y ∈ SNoS_ ω → (∀z : set, z ∈ ω → abs_SNo (y + - x) < eps_ z) → y = x) → (∀y : set, y ∈ ω → (∃z : set, z ∈ SNoS_ ω ∧ (z < x ∧ x < z + eps_ y))) → P
L1572
Hypothesis H1 : x ∈ SNoS_ (ordsucc ω)
L1573
Hypothesis H2 : x ≠ - ω
L1574
Hypothesis H3 : (∀y : set, y ∈ SNoS_ ω → (∀z : set, z ∈ ω → abs_SNo (y + - x) < eps_ z) → y = x)
L1575
Hypothesis H4 : SNoLev x ∈ ordsucc ω
L1576
Hypothesis H5 : SNo x
L1577
Theorem. (Conj_real_E__4__6)
- ω < x → P
In Proofgold the corresponding term root is 9d32dc... and proposition id is 799804...
Proof:
Proof not loaded.
End of Section Conj_real_E__4__6
Beginning of Section Conj_SNoS_omega_real__2__0
L1583
Variable x : set
(*** Conj_SNoS_omega_real__2__0 TMXFqoGGxtT83bBhWxAzWzHtocSTXsvwuk9 bounty of about 25 bars ***)
L1584
Variable y : set
L1585
Hypothesis H1 : y ∈ SNoS_ ω
L1586
Hypothesis H2 : (∀z : set, z ∈ ω → abs_SNo (y + - x) < eps_ z)
L1587
Hypothesis H3 : Empty < y + - x
L1588
Theorem. (Conj_SNoS_omega_real__2__0)
¬ y + - x ∈ SNoS_ ω
In Proofgold the corresponding term root is 3c410d... and proposition id is 3dcf84...
Proof:
Proof not loaded.
End of Section Conj_SNoS_omega_real__2__0
Beginning of Section Conj_SNoS_omega_real__5__1
L1594
Variable x : set
(*** Conj_SNoS_omega_real__5__1 TMLbwz9KJVwZGYYgtEDwM3M7xaUNdgpV5Z6 bounty of about 25 bars ***)
L1595
Variable y : set
L1596
Hypothesis H0 : x ∈ SNoS_ ω
L1597
Hypothesis H2 : y ∈ SNoS_ ω
L1598
Hypothesis H3 : (∀z : set, z ∈ ω → abs_SNo (y + - x) < eps_ z)
L1599
Hypothesis H4 : SNo y
L1600
Hypothesis H5 : Empty < x + - y
L1601
Theorem. (Conj_SNoS_omega_real__5__1)
¬ x + - y ∈ SNoS_ ω
In Proofgold the corresponding term root is a37630... and proposition id is beddcd...
Proof:
Proof not loaded.
End of Section Conj_SNoS_omega_real__5__1
Beginning of Section Conj_SNo_prereal_incr_lower_pos__4__6
L1607
Variable x : set
(*** Conj_SNo_prereal_incr_lower_pos__4__6 TMJdQt7Y9beuSxZr6aRpknYi2Z1jHnGSPzL bounty of about 25 bars ***)
L1608
Variable y : set
L1609
Variable P : prop
L1610
Variable z : set
L1611
Variable w : set
L1612
Hypothesis H0 : SNo x
L1613
Hypothesis H1 : y ∈ ω
L1614
Hypothesis H2 : (∀u : set, u ∈ SNoS_ ω → Empty < u → u < x → x < u + eps_ y → P)
L1615
Hypothesis H3 : x < z + eps_ y
L1616
Hypothesis H4 : SNo z
L1617
Hypothesis H5 : z ≤ Empty
L1618
Hypothesis H7 : w ∈ ω
L1619
Hypothesis H8 : eps_ w ≤ x
L1620
Theorem. (Conj_SNo_prereal_incr_lower_pos__4__6)
SNo (eps_ w) → x < eps_ w
In Proofgold the corresponding term root is 6245e9... and proposition id is 55931d...
Proof:
Proof not loaded.
End of Section Conj_SNo_prereal_incr_lower_pos__4__6
Beginning of Section Conj_SNoCutP_SNoCut_lim__4__4
L1626
Variable x : set
(*** Conj_SNoCutP_SNoCut_lim__4__4 TMbtyipTzmyeFPY82KXogBudj14dNt3eXNQ bounty of about 25 bars ***)
L1627
Variable y : set
L1628
Variable z : set
L1629
Hypothesis H0 : ordinal x
L1630
Hypothesis H1 : (∀w : set, w ∈ x → ordsucc w ∈ x)
L1631
Hypothesis H2 : Subq y (SNoS_ x)
L1632
Hypothesis H3 : Subq z (SNoS_ x)
L1633
Hypothesis H5 : (∀w : set, w ∈ y → SNoLev w ∈ x)
L1634
Theorem. (Conj_SNoCutP_SNoCut_lim__4__4)
(∀w : set, w ∈ z → SNoLev w ∈ x) → SNoLev (SNoCut y z) ∈ ordsucc x
In Proofgold the corresponding term root is 5b6e65... and proposition id is c14ca3...
Proof:
Proof not loaded.
End of Section Conj_SNoCutP_SNoCut_lim__4__4
Beginning of Section Conj_SNo_approx_real__4__12
L1640
Variable x : set
(*** Conj_SNo_approx_real__4__12 TMPqS7JkbD2JqeTFSqysXfXdQ46YXcWzDnw bounty of about 25 bars ***)
L1641
Variable y : set
L1642
Variable z : set
L1643
Variable w : set
L1644
Hypothesis H0 : SNo x
L1645
Hypothesis H1 : (∀u : set, u ∈ ω → SNo (ap y u))
L1646
Hypothesis H2 : z ∈ SNoS_ ω
L1647
Hypothesis H3 : (∀u : set, u ∈ ω → abs_SNo (z + - x) < eps_ u)
L1648
Hypothesis H4 : SNo z
L1649
Hypothesis H5 : SNo (- z)
L1650
Hypothesis H6 : SNo (x + - z)
L1651
Hypothesis H7 : w ∈ ω
L1652
Hypothesis H8 : (∀u : set, u ∈ SNoS_ ω → (∀v : set, v ∈ ω → abs_SNo (u + - (ap y (ordsucc w))) < eps_ v) → u = ap y (ordsucc w))
L1653
Hypothesis H9 : SNo (ap y (ordsucc w))
L1654
Hypothesis H10 : z < ap y (ordsucc w)
L1655
Hypothesis H11 : Empty < ap y (ordsucc w) + - z
L1656
Hypothesis H13 : ap y (ordsucc w) < x
L1657
Theorem. (Conj_SNo_approx_real__4__12)
z < x → ap y w < z
In Proofgold the corresponding term root is 58b1de... and proposition id is 6c1a6c...
Proof:
Proof not loaded.
End of Section Conj_SNo_approx_real__4__12
Beginning of Section Conj_SNo_approx_real__9__0
L1663
Variable x : set
(*** Conj_SNo_approx_real__9__0 TMbXApEiiZa9o5aMfiQaEauJP71gJVrdqj2 bounty of about 25 bars ***)
L1664
Variable y : set
L1665
Variable z : set
L1666
Variable w : set
L1667
Hypothesis H1 : (∀u : set, u ∈ ω → ap y u < x)
L1668
Hypothesis H2 : (∀u : set, u ∈ ω → (∀v : set, v ∈ u → ap y v < ap y u))
L1669
Hypothesis H3 : (∀u : set, u ∈ ω → SNo (ap y u))
L1670
Hypothesis H4 : z ∈ SNoS_ ω
L1671
Hypothesis H5 : (∀u : set, u ∈ ω → abs_SNo (z + - x) < eps_ u)
L1672
Hypothesis H6 : SNo z
L1673
Hypothesis H7 : SNo (- z)
L1674
Hypothesis H8 : SNo (x + - z)
L1675
Hypothesis H9 : w ∈ ω
L1676
Hypothesis H10 : z ≤ ap y w
L1677
Hypothesis H11 : (∀u : set, u ∈ SNoS_ ω → (∀v : set, v ∈ ω → abs_SNo (u + - (ap y (ordsucc w))) < eps_ v) → u = ap y (ordsucc w))
L1678
Theorem. (Conj_SNo_approx_real__9__0)
SNo (ap y (ordsucc w)) → ap y w < z
In Proofgold the corresponding term root is aa0ea7... and proposition id is 74c90f...
Proof:
Proof not loaded.
End of Section Conj_SNo_approx_real__9__0
Beginning of Section Conj_SNo_approx_real__9__9
L1684
Variable x : set
(*** Conj_SNo_approx_real__9__9 TMWJEMzAVTiFGKkg68ZoAaYLYCDa23XkVQG bounty of about 25 bars ***)
L1685
Variable y : set
L1686
Variable z : set
L1687
Variable w : set
L1688
Hypothesis H0 : SNo x
L1689
Hypothesis H1 : (∀u : set, u ∈ ω → ap y u < x)
L1690
Hypothesis H2 : (∀u : set, u ∈ ω → (∀v : set, v ∈ u → ap y v < ap y u))
L1691
Hypothesis H3 : (∀u : set, u ∈ ω → SNo (ap y u))
L1692
Hypothesis H4 : z ∈ SNoS_ ω
L1693
Hypothesis H5 : (∀u : set, u ∈ ω → abs_SNo (z + - x) < eps_ u)
L1694
Hypothesis H6 : SNo z
L1695
Hypothesis H7 : SNo (- z)
L1696
Hypothesis H8 : SNo (x + - z)
L1697
Hypothesis H10 : z ≤ ap y w
L1698
Hypothesis H11 : (∀u : set, u ∈ SNoS_ ω → (∀v : set, v ∈ ω → abs_SNo (u + - (ap y (ordsucc w))) < eps_ v) → u = ap y (ordsucc w))
L1699
Theorem. (Conj_SNo_approx_real__9__9)
SNo (ap y (ordsucc w)) → ap y w < z
In Proofgold the corresponding term root is 58e29d... and proposition id is 22a5e5...
Proof:
Proof not loaded.
End of Section Conj_SNo_approx_real__9__9
Beginning of Section Conj_SNo_approx_real__10__10
L1705
Variable x : set
(*** Conj_SNo_approx_real__10__10 TMcRdM5CBJVSRpd581R1cLb2Q3dopbZ7TnZ bounty of about 25 bars ***)
L1706
Variable y : set
L1707
Variable z : set
L1708
Variable w : set
L1709
Hypothesis H0 : SNo x
L1710
Hypothesis H1 : SNo (- x)
L1711
Hypothesis H2 : z ∈ SNoS_ ω
L1712
Hypothesis H3 : (∀u : set, u ∈ ω → abs_SNo (z + - x) < eps_ u)
L1713
Hypothesis H4 : SNo z
L1714
Hypothesis H5 : SNo (z + - x)
L1715
Hypothesis H6 : (∀u : set, u ∈ SNoS_ ω → (∀v : set, v ∈ ω → abs_SNo (u + - (ap y (ordsucc w))) < eps_ v) → u = ap y (ordsucc w))
L1716
Hypothesis H7 : SNo (ap y (ordsucc w))
L1717
Hypothesis H8 : ap y (ordsucc w) < z
L1718
Hypothesis H9 : Empty < z + - (ap y (ordsucc w))
L1719
Hypothesis H11 : x < ap y (ordsucc w)
L1720
Hypothesis H12 : abs_SNo (z + - x) = z + - x
L1721
Theorem. (Conj_SNo_approx_real__10__10)
z = ap y (ordsucc w) → z < ap y w
In Proofgold the corresponding term root is 500c65... and proposition id is 694cf5...
Proof:
Proof not loaded.
End of Section Conj_SNo_approx_real__10__10
Beginning of Section Conj_SNo_approx_real__12__7
L1727
Variable x : set
(*** Conj_SNo_approx_real__12__7 TMK6gp8homc4N8ArrWNNBbXtYzvk4KH5ePC bounty of about 25 bars ***)
L1728
Variable y : set
L1729
Variable z : set
L1730
Variable w : set
L1731
Hypothesis H0 : SNo x
L1732
Hypothesis H1 : SNo (- x)
L1733
Hypothesis H2 : z ∈ SNoS_ ω
L1734
Hypothesis H3 : (∀u : set, u ∈ ω → abs_SNo (z + - x) < eps_ u)
L1735
Hypothesis H4 : SNo z
L1736
Hypothesis H5 : SNo (z + - x)
L1737
Hypothesis H6 : (∀u : set, u ∈ SNoS_ ω → (∀v : set, v ∈ ω → abs_SNo (u + - (ap y (ordsucc w))) < eps_ v) → u = ap y (ordsucc w))
L1738
Hypothesis H8 : ap y (ordsucc w) < z
L1739
Hypothesis H9 : Empty < z + - (ap y (ordsucc w))
L1740
Hypothesis H10 : SNo (z + - (ap y (ordsucc w)))
L1741
Hypothesis H11 : x < ap y (ordsucc w)
L1742
Hypothesis H12 : x < z
L1743
Theorem. (Conj_SNo_approx_real__12__7)
Empty < z + - x → z < ap y w
In Proofgold the corresponding term root is a15a58... and proposition id is 4bd124...
Proof:
Proof not loaded.
End of Section Conj_SNo_approx_real__12__7
Beginning of Section Conj_SNo_approx_real__12__9
L1749
Variable x : set
(*** Conj_SNo_approx_real__12__9 TMUL1QBL724QNP1qronWW1bmpYBBAxu6p1T bounty of about 25 bars ***)
L1750
Variable y : set
L1751
Variable z : set
L1752
Variable w : set
L1753
Hypothesis H0 : SNo x
L1754
Hypothesis H1 : SNo (- x)
L1755
Hypothesis H2 : z ∈ SNoS_ ω
L1756
Hypothesis H3 : (∀u : set, u ∈ ω → abs_SNo (z + - x) < eps_ u)
L1757
Hypothesis H4 : SNo z
L1758
Hypothesis H5 : SNo (z + - x)
L1759
Hypothesis H6 : (∀u : set, u ∈ SNoS_ ω → (∀v : set, v ∈ ω → abs_SNo (u + - (ap y (ordsucc w))) < eps_ v) → u = ap y (ordsucc w))
L1760
Hypothesis H7 : SNo (ap y (ordsucc w))
L1761
Hypothesis H8 : ap y (ordsucc w) < z
L1762
Hypothesis H10 : SNo (z + - (ap y (ordsucc w)))
L1763
Hypothesis H11 : x < ap y (ordsucc w)
L1764
Hypothesis H12 : x < z
L1765
Theorem. (Conj_SNo_approx_real__12__9)
Empty < z + - x → z < ap y w
In Proofgold the corresponding term root is 2608bc... and proposition id is d49c69...
Proof:
Proof not loaded.
End of Section Conj_SNo_approx_real__12__9
Beginning of Section Conj_SNo_approx_real__14__8
L1771
Variable x : set
(*** Conj_SNo_approx_real__14__8 TMa7acqvLaZq3E4YiBxxEDY4jDZfPPv2q1B bounty of about 25 bars ***)
L1772
Variable y : set
L1773
Variable z : set
L1774
Variable w : set
L1775
Hypothesis H0 : SNo x
L1776
Hypothesis H1 : (∀u : set, u ∈ ω → x < ap y u)
L1777
Hypothesis H2 : SNo (- x)
L1778
Hypothesis H3 : z ∈ SNoS_ ω
L1779
Hypothesis H4 : (∀u : set, u ∈ ω → abs_SNo (z + - x) < eps_ u)
L1780
Hypothesis H5 : SNo z
L1781
Hypothesis H6 : SNo (z + - x)
L1782
Hypothesis H7 : w ∈ ω
L1783
Hypothesis H9 : SNo (ap y (ordsucc w))
L1784
Hypothesis H10 : ap y (ordsucc w) < z
L1785
Hypothesis H11 : Empty < z + - (ap y (ordsucc w))
L1786
Hypothesis H12 : SNo (z + - (ap y (ordsucc w)))
L1787
Theorem. (Conj_SNo_approx_real__14__8)
x < ap y (ordsucc w) → z < ap y w
In Proofgold the corresponding term root is 606262... and proposition id is c71e9d...
Proof:
Proof not loaded.
End of Section Conj_SNo_approx_real__14__8
Beginning of Section Conj_SNo_approx_real__14__10
L1793
Variable x : set
(*** Conj_SNo_approx_real__14__10 TMcBEa1ubBhnXLmDbp3krvHuEu7fQmftYge bounty of about 25 bars ***)
L1794
Variable y : set
L1795
Variable z : set
L1796
Variable w : set
L1797
Hypothesis H0 : SNo x
L1798
Hypothesis H1 : (∀u : set, u ∈ ω → x < ap y u)
L1799
Hypothesis H2 : SNo (- x)
L1800
Hypothesis H3 : z ∈ SNoS_ ω
L1801
Hypothesis H4 : (∀u : set, u ∈ ω → abs_SNo (z + - x) < eps_ u)
L1802
Hypothesis H5 : SNo z
L1803
Hypothesis H6 : SNo (z + - x)
L1804
Hypothesis H7 : w ∈ ω
L1805
Hypothesis H8 : (∀u : set, u ∈ SNoS_ ω → (∀v : set, v ∈ ω → abs_SNo (u + - (ap y (ordsucc w))) < eps_ v) → u = ap y (ordsucc w))
L1806
Hypothesis H9 : SNo (ap y (ordsucc w))
L1807
Hypothesis H11 : Empty < z + - (ap y (ordsucc w))
L1808
Hypothesis H12 : SNo (z + - (ap y (ordsucc w)))
L1809
Theorem. (Conj_SNo_approx_real__14__10)
x < ap y (ordsucc w) → z < ap y w
In Proofgold the corresponding term root is 38b583... and proposition id is dccc1e...
Proof:
Proof not loaded.
End of Section Conj_SNo_approx_real__14__10
Beginning of Section Conj_SNo_approx_real__18__0
L1815
Variable x : set
(*** Conj_SNo_approx_real__18__0 TMJV1dcJwndTioAauemfR7EZFgLbLEQgvJn bounty of about 25 bars ***)
L1816
Variable y : set
L1817
Variable z : set
L1818
Variable w : set
L1819
Hypothesis H1 : (∀u : set, u ∈ ω → x < ap y u)
L1820
Hypothesis H2 : (∀u : set, u ∈ ω → (∀v : set, v ∈ u → ap y u < ap y v))
L1821
Hypothesis H3 : SNo (- x)
L1822
Hypothesis H4 : (∀u : set, u ∈ ω → SNo (ap y u))
L1823
Hypothesis H5 : z ∈ SNoS_ ω
L1824
Hypothesis H6 : (∀u : set, u ∈ ω → abs_SNo (z + - x) < eps_ u)
L1825
Hypothesis H7 : SNo z
L1826
Hypothesis H8 : SNo (z + - x)
L1827
Hypothesis H9 : w ∈ ω
L1828
Hypothesis H10 : ap y w ≤ z
L1829
Hypothesis H11 : (∀u : set, u ∈ SNoS_ ω → (∀v : set, v ∈ ω → abs_SNo (u + - (ap y (ordsucc w))) < eps_ v) → u = ap y (ordsucc w))
L1830
Theorem. (Conj_SNo_approx_real__18__0)
SNo (ap y (ordsucc w)) → z < ap y w
In Proofgold the corresponding term root is 1657da... and proposition id is 4b64d7...
Proof:
Proof not loaded.
End of Section Conj_SNo_approx_real__18__0
Beginning of Section Conj_SNo_approx_real__18__8
L1836
Variable x : set
(*** Conj_SNo_approx_real__18__8 TMVMMwG9jvsqDv6d9Zzj9y9rNAiwckvtarc bounty of about 25 bars ***)
L1837
Variable y : set
L1838
Variable z : set
L1839
Variable w : set
L1840
Hypothesis H0 : SNo x
L1841
Hypothesis H1 : (∀u : set, u ∈ ω → x < ap y u)
L1842
Hypothesis H2 : (∀u : set, u ∈ ω → (∀v : set, v ∈ u → ap y u < ap y v))
L1843
Hypothesis H3 : SNo (- x)
L1844
Hypothesis H4 : (∀u : set, u ∈ ω → SNo (ap y u))
L1845
Hypothesis H5 : z ∈ SNoS_ ω
L1846
Hypothesis H6 : (∀u : set, u ∈ ω → abs_SNo (z + - x) < eps_ u)
L1847
Hypothesis H7 : SNo z
L1848
Hypothesis H9 : w ∈ ω
L1849
Hypothesis H10 : ap y w ≤ z
L1850
Hypothesis H11 : (∀u : set, u ∈ SNoS_ ω → (∀v : set, v ∈ ω → abs_SNo (u + - (ap y (ordsucc w))) < eps_ v) → u = ap y (ordsucc w))
L1851
Theorem. (Conj_SNo_approx_real__18__8)
SNo (ap y (ordsucc w)) → z < ap y w
In Proofgold the corresponding term root is 6222bd... and proposition id is 66c812...
Proof:
Proof not loaded.
End of Section Conj_SNo_approx_real__18__8
Beginning of Section Conj_SNo_approx_real__19__13
L1857
Variable x : set
(*** Conj_SNo_approx_real__19__13 TMajuAh8BJqhtZB8ApXe4wjvzZsfA6EMj72 bounty of about 25 bars ***)
L1858
Variable y : set
L1859
Variable z : set
L1860
Variable w : set
L1861
Hypothesis H0 : SNo x
L1862
Hypothesis H1 : z ∈ setexp (SNoS_ ω) ω
L1863
Hypothesis H2 : (∀u : set, u ∈ ω → x < ap z u)
L1864
Hypothesis H3 : (∀u : set, u ∈ ω → (∀v : set, v ∈ u → ap z u < ap z v))
L1865
Hypothesis H4 : x = SNoCut (Repl ω (ap y)) (Repl ω (ap z))
L1866
Hypothesis H5 : SNo (- x)
L1867
Hypothesis H6 : (∀u : set, u ∈ ω → SNo (ap z u))
L1868
Hypothesis H7 : (∀u : set, SNo u → (∀v : set, v ∈ Repl ω (ap y) → v < u) → (∀v : set, v ∈ Repl ω (ap z) → u < v) → Subq (SNoLev (SNoCut (Repl ω (ap y)) (Repl ω (ap z)))) (SNoLev u) ∧ SNoEq_ (SNoLev (SNoCut (Repl ω (ap y)) (Repl ω (ap z)))) (SNoCut (Repl ω (ap y)) (Repl ω (ap z))) u)
L1869
Hypothesis H8 : SNoLev x = ω
L1870
Hypothesis H9 : w ∈ SNoS_ ω
L1871
Hypothesis H10 : (∀u : set, u ∈ ω → abs_SNo (w + - x) < eps_ u)
L1872
Hypothesis H11 : SNoLev w ∈ ω
L1873
Hypothesis H12 : SNo w
L1874
Hypothesis H14 : (∀u : set, u ∈ Repl ω (ap y) → u < w)
L1875
Theorem. (Conj_SNo_approx_real__19__13)
(∀u : set, u ∈ Repl ω (ap z) → w < u) → w = x
In Proofgold the corresponding term root is 527c9a... and proposition id is 95fb88...
Proof:
Proof not loaded.
End of Section Conj_SNo_approx_real__19__13
Beginning of Section Conj_SNo_approx_real__20__16
L1881
Variable x : set
(*** Conj_SNo_approx_real__20__16 TMZkg4T1RLwosEcFjgCUMMZNKwFZjbyAaVK bounty of about 25 bars ***)
L1882
Variable y : set
L1883
Variable z : set
L1884
Variable w : set
L1885
Hypothesis H0 : SNo x
L1886
Hypothesis H1 : y ∈ setexp (SNoS_ ω) ω
L1887
Hypothesis H2 : z ∈ setexp (SNoS_ ω) ω
L1888
Hypothesis H3 : (∀u : set, u ∈ ω → ap y u < x)
L1889
Hypothesis H4 : (∀u : set, u ∈ ω → (∀v : set, v ∈ u → ap y v < ap y u))
L1890
Hypothesis H5 : (∀u : set, u ∈ ω → x < ap z u)
L1891
Hypothesis H6 : (∀u : set, u ∈ ω → (∀v : set, v ∈ u → ap z u < ap z v))
L1892
Hypothesis H7 : x = SNoCut (Repl ω (ap y)) (Repl ω (ap z))
L1893
Hypothesis H8 : SNo (- x)
L1894
Hypothesis H9 : (∀u : set, u ∈ ω → SNo (ap y u))
L1895
Hypothesis H10 : (∀u : set, u ∈ ω → SNo (ap z u))
L1896
Hypothesis H11 : (∀u : set, SNo u → (∀v : set, v ∈ Repl ω (ap y) → v < u) → (∀v : set, v ∈ Repl ω (ap z) → u < v) → Subq (SNoLev (SNoCut (Repl ω (ap y)) (Repl ω (ap z)))) (SNoLev u) ∧ SNoEq_ (SNoLev (SNoCut (Repl ω (ap y)) (Repl ω (ap z)))) (SNoCut (Repl ω (ap y)) (Repl ω (ap z))) u)
L1897
Hypothesis H12 : SNoLev x = ω
L1898
Hypothesis H13 : w ∈ SNoS_ ω
L1899
Hypothesis H14 : (∀u : set, u ∈ ω → abs_SNo (w + - x) < eps_ u)
L1900
Hypothesis H15 : SNoLev w ∈ ω
L1901
Hypothesis H17 : SNo (- w)
L1902
Hypothesis H18 : SNo (x + - w)
L1903
Hypothesis H19 : SNo (w + - x)
L1904
Theorem. (Conj_SNo_approx_real__20__16)
(∀u : set, u ∈ Repl ω (ap y) → u < w) → w = x
In Proofgold the corresponding term root is 096659... and proposition id is 9fca98...
Proof:
Proof not loaded.
End of Section Conj_SNo_approx_real__20__16
Beginning of Section Conj_SNo_approx_real__21__1
L1910
Variable x : set
(*** Conj_SNo_approx_real__21__1 TMdvyEdxk1oh6DGGmCKEHbfX8bkafMLPXw9 bounty of about 25 bars ***)
L1911
Variable y : set
L1912
Variable z : set
L1913
Variable w : set
L1914
Hypothesis H0 : SNo x
L1915
Hypothesis H2 : z ∈ setexp (SNoS_ ω) ω
L1916
Hypothesis H3 : (∀u : set, u ∈ ω → ap y u < x)
L1917
Hypothesis H4 : (∀u : set, u ∈ ω → (∀v : set, v ∈ u → ap y v < ap y u))
L1918
Hypothesis H5 : (∀u : set, u ∈ ω → x < ap z u)
L1919
Hypothesis H6 : (∀u : set, u ∈ ω → (∀v : set, v ∈ u → ap z u < ap z v))
L1920
Hypothesis H7 : x = SNoCut (Repl ω (ap y)) (Repl ω (ap z))
L1921
Hypothesis H8 : SNo (- x)
L1922
Hypothesis H9 : (∀u : set, u ∈ ω → SNo (ap y u))
L1923
Hypothesis H10 : (∀u : set, u ∈ ω → SNo (ap z u))
L1924
Hypothesis H11 : (∀u : set, SNo u → (∀v : set, v ∈ Repl ω (ap y) → v < u) → (∀v : set, v ∈ Repl ω (ap z) → u < v) → Subq (SNoLev (SNoCut (Repl ω (ap y)) (Repl ω (ap z)))) (SNoLev u) ∧ SNoEq_ (SNoLev (SNoCut (Repl ω (ap y)) (Repl ω (ap z)))) (SNoCut (Repl ω (ap y)) (Repl ω (ap z))) u)
L1925
Hypothesis H12 : SNoLev x = ω
L1926
Hypothesis H13 : w ∈ SNoS_ ω
L1927
Hypothesis H14 : (∀u : set, u ∈ ω → abs_SNo (w + - x) < eps_ u)
L1928
Hypothesis H15 : SNoLev w ∈ ω
L1929
Hypothesis H16 : SNo w
L1930
Hypothesis H17 : SNo (- w)
L1931
Hypothesis H18 : SNo (x + - w)
L1932
Theorem. (Conj_SNo_approx_real__21__1)
SNo (w + - x) → w = x
In Proofgold the corresponding term root is 960735... and proposition id is 40825c...
Proof:
Proof not loaded.
End of Section Conj_SNo_approx_real__21__1
Beginning of Section Conj_SNo_approx_real__21__2
L1938
Variable x : set
(*** Conj_SNo_approx_real__21__2 TMZW1pgKP5r9GC4dH6YZLPkFRVoi8KjKsJP bounty of about 25 bars ***)
L1939
Variable y : set
L1940
Variable z : set
L1941
Variable w : set
L1942
Hypothesis H0 : SNo x
L1943
Hypothesis H1 : y ∈ setexp (SNoS_ ω) ω
L1944
Hypothesis H3 : (∀u : set, u ∈ ω → ap y u < x)
L1945
Hypothesis H4 : (∀u : set, u ∈ ω → (∀v : set, v ∈ u → ap y v < ap y u))
L1946
Hypothesis H5 : (∀u : set, u ∈ ω → x < ap z u)
L1947
Hypothesis H6 : (∀u : set, u ∈ ω → (∀v : set, v ∈ u → ap z u < ap z v))
L1948
Hypothesis H7 : x = SNoCut (Repl ω (ap y)) (Repl ω (ap z))
L1949
Hypothesis H8 : SNo (- x)
L1950
Hypothesis H9 : (∀u : set, u ∈ ω → SNo (ap y u))
L1951
Hypothesis H10 : (∀u : set, u ∈ ω → SNo (ap z u))
L1952
Hypothesis H11 : (∀u : set, SNo u → (∀v : set, v ∈ Repl ω (ap y) → v < u) → (∀v : set, v ∈ Repl ω (ap z) → u < v) → Subq (SNoLev (SNoCut (Repl ω (ap y)) (Repl ω (ap z)))) (SNoLev u) ∧ SNoEq_ (SNoLev (SNoCut (Repl ω (ap y)) (Repl ω (ap z)))) (SNoCut (Repl ω (ap y)) (Repl ω (ap z))) u)
L1953
Hypothesis H12 : SNoLev x = ω
L1954
Hypothesis H13 : w ∈ SNoS_ ω
L1955
Hypothesis H14 : (∀u : set, u ∈ ω → abs_SNo (w + - x) < eps_ u)
L1956
Hypothesis H15 : SNoLev w ∈ ω
L1957
Hypothesis H16 : SNo w
L1958
Hypothesis H17 : SNo (- w)
L1959
Hypothesis H18 : SNo (x + - w)
L1960
Theorem. (Conj_SNo_approx_real__21__2)
SNo (w + - x) → w = x
In Proofgold the corresponding term root is 6697fb... and proposition id is 16df87...
Proof:
Proof not loaded.
End of Section Conj_SNo_approx_real__21__2
Beginning of Section Conj_SNo_approx_real__22__3
L1966
Variable x : set
(*** Conj_SNo_approx_real__22__3 TMSJhNkxnEy5fwwk5nfNKVWd18nEJFwsVBm bounty of about 25 bars ***)
L1967
Variable y : set
L1968
Variable z : set
L1969
Variable w : set
L1970
Hypothesis H0 : SNo x
L1971
Hypothesis H1 : y ∈ setexp (SNoS_ ω) ω
L1972
Hypothesis H2 : z ∈ setexp (SNoS_ ω) ω
L1973
Hypothesis H4 : (∀u : set, u ∈ ω → (∀v : set, v ∈ u → ap y v < ap y u))
L1974
Hypothesis H5 : (∀u : set, u ∈ ω → x < ap z u)
L1975
Hypothesis H6 : (∀u : set, u ∈ ω → (∀v : set, v ∈ u → ap z u < ap z v))
L1976
Hypothesis H7 : x = SNoCut (Repl ω (ap y)) (Repl ω (ap z))
L1977
Hypothesis H8 : SNo (- x)
L1978
Hypothesis H9 : (∀u : set, u ∈ ω → SNo (ap y u))
L1979
Hypothesis H10 : (∀u : set, u ∈ ω → SNo (ap z u))
L1980
Hypothesis H11 : (∀u : set, SNo u → (∀v : set, v ∈ Repl ω (ap y) → v < u) → (∀v : set, v ∈ Repl ω (ap z) → u < v) → Subq (SNoLev (SNoCut (Repl ω (ap y)) (Repl ω (ap z)))) (SNoLev u) ∧ SNoEq_ (SNoLev (SNoCut (Repl ω (ap y)) (Repl ω (ap z)))) (SNoCut (Repl ω (ap y)) (Repl ω (ap z))) u)
L1981
Hypothesis H12 : SNoLev x = ω
L1982
Hypothesis H13 : w ∈ SNoS_ ω
L1983
Hypothesis H14 : (∀u : set, u ∈ ω → abs_SNo (w + - x) < eps_ u)
L1984
Hypothesis H15 : SNoLev w ∈ ω
L1985
Hypothesis H16 : SNo w
L1986
Hypothesis H17 : SNo (- w)
L1987
Theorem. (Conj_SNo_approx_real__22__3)
SNo (x + - w) → w = x
In Proofgold the corresponding term root is 07b04b... and proposition id is 8ed05f...
Proof:
Proof not loaded.
End of Section Conj_SNo_approx_real__22__3
Beginning of Section Conj_SNo_approx_real__22__9
L1993
Variable x : set
(*** Conj_SNo_approx_real__22__9 TMKHHLPgAtoYdmZD13wKmjxUsMvC8dGCCVp bounty of about 25 bars ***)
L1994
Variable y : set
L1995
Variable z : set
L1996
Variable w : set
L1997
Hypothesis H0 : SNo x
L1998
Hypothesis H1 : y ∈ setexp (SNoS_ ω) ω
L1999
Hypothesis H2 : z ∈ setexp (SNoS_ ω) ω
L2000
Hypothesis H3 : (∀u : set, u ∈ ω → ap y u < x)
L2001
Hypothesis H4 : (∀u : set, u ∈ ω → (∀v : set, v ∈ u → ap y v < ap y u))
L2002
Hypothesis H5 : (∀u : set, u ∈ ω → x < ap z u)
L2003
Hypothesis H6 : (∀u : set, u ∈ ω → (∀v : set, v ∈ u → ap z u < ap z v))
L2004
Hypothesis H7 : x = SNoCut (Repl ω (ap y)) (Repl ω (ap z))
L2005
Hypothesis H8 : SNo (- x)
L2006
Hypothesis H10 : (∀u : set, u ∈ ω → SNo (ap z u))
L2007
Hypothesis H11 : (∀u : set, SNo u → (∀v : set, v ∈ Repl ω (ap y) → v < u) → (∀v : set, v ∈ Repl ω (ap z) → u < v) → Subq (SNoLev (SNoCut (Repl ω (ap y)) (Repl ω (ap z)))) (SNoLev u) ∧ SNoEq_ (SNoLev (SNoCut (Repl ω (ap y)) (Repl ω (ap z)))) (SNoCut (Repl ω (ap y)) (Repl ω (ap z))) u)
L2008
Hypothesis H12 : SNoLev x = ω
L2009
Hypothesis H13 : w ∈ SNoS_ ω
L2010
Hypothesis H14 : (∀u : set, u ∈ ω → abs_SNo (w + - x) < eps_ u)
L2011
Hypothesis H15 : SNoLev w ∈ ω
L2012
Hypothesis H16 : SNo w
L2013
Hypothesis H17 : SNo (- w)
L2014
Theorem. (Conj_SNo_approx_real__22__9)
SNo (x + - w) → w = x
In Proofgold the corresponding term root is 5062cf... and proposition id is b22985...
Proof:
Proof not loaded.
End of Section Conj_SNo_approx_real__22__9
Beginning of Section Conj_SNo_approx_real__23__2
L2020
Variable x : set
(*** Conj_SNo_approx_real__23__2 TMcco6JvwnvX2cxjjrXGPkhnk5j2tpwFAuX bounty of about 25 bars ***)
L2021
Variable y : set
L2022
Variable z : set
L2023
Variable w : set
L2024
Hypothesis H0 : SNo x
L2025
Hypothesis H1 : y ∈ setexp (SNoS_ ω) ω
L2026
Hypothesis H3 : (∀u : set, u ∈ ω → ap y u < x)
L2027
Hypothesis H4 : (∀u : set, u ∈ ω → (∀v : set, v ∈ u → ap y v < ap y u))
L2028
Hypothesis H5 : (∀u : set, u ∈ ω → x < ap z u)
L2029
Hypothesis H6 : (∀u : set, u ∈ ω → (∀v : set, v ∈ u → ap z u < ap z v))
L2030
Hypothesis H7 : x = SNoCut (Repl ω (ap y)) (Repl ω (ap z))
L2031
Hypothesis H8 : SNo (- x)
L2032
Hypothesis H9 : (∀u : set, u ∈ ω → SNo (ap y u))
L2033
Hypothesis H10 : (∀u : set, u ∈ ω → SNo (ap z u))
L2034
Hypothesis H11 : (∀u : set, SNo u → (∀v : set, v ∈ Repl ω (ap y) → v < u) → (∀v : set, v ∈ Repl ω (ap z) → u < v) → Subq (SNoLev (SNoCut (Repl ω (ap y)) (Repl ω (ap z)))) (SNoLev u) ∧ SNoEq_ (SNoLev (SNoCut (Repl ω (ap y)) (Repl ω (ap z)))) (SNoCut (Repl ω (ap y)) (Repl ω (ap z))) u)
L2035
Hypothesis H12 : SNoLev x = ω
L2036
Hypothesis H13 : w ∈ SNoS_ ω
L2037
Hypothesis H14 : (∀u : set, u ∈ ω → abs_SNo (w + - x) < eps_ u)
L2038
Hypothesis H15 : SNoLev w ∈ ω
L2039
Hypothesis H16 : SNo w
L2040
Theorem. (Conj_SNo_approx_real__23__2)
SNo (- w) → w = x
In Proofgold the corresponding term root is a96760... and proposition id is f7cead...
Proof:
Proof not loaded.
End of Section Conj_SNo_approx_real__23__2
Beginning of Section Conj_SNo_approx_real__26__1
L2046
Variable x : set
(*** Conj_SNo_approx_real__26__1 TMKyprpZaE3jjjLHjcTEzpwXghbSiYXeWT9 bounty of about 25 bars ***)
L2047
Variable y : set
L2048
Variable z : set
L2049
Hypothesis H0 : SNo x
L2050
Hypothesis H2 : z ∈ setexp (SNoS_ ω) ω
L2051
Hypothesis H3 : (∀w : set, w ∈ ω → ap y w < x)
L2052
Hypothesis H4 : (∀w : set, w ∈ ω → (∀u : set, u ∈ w → ap y u < ap y w))
L2053
Hypothesis H5 : (∀w : set, w ∈ ω → x < ap z w)
L2054
Hypothesis H6 : (∀w : set, w ∈ ω → (∀u : set, u ∈ w → ap z w < ap z u))
L2055
Hypothesis H7 : x = SNoCut (Repl ω (ap y)) (Repl ω (ap z))
L2056
Hypothesis H8 : SNo (- x)
L2057
Hypothesis H9 : (∀w : set, w ∈ ω → SNo (ap y w))
L2058
Hypothesis H10 : (∀w : set, w ∈ ω → SNo (ap z w))
L2059
Theorem. (Conj_SNo_approx_real__26__1)
(∀w : set, w ∈ ω → (∀u : set, u ∈ ω → ap y w < ap z u)) → x ∈ real
In Proofgold the corresponding term root is b1c1c5... and proposition id is c27b0f...
Proof:
Proof not loaded.
End of Section Conj_SNo_approx_real__26__1
Beginning of Section Conj_SNo_approx_real__28__6
L2065
Variable x : set
(*** Conj_SNo_approx_real__28__6 TMHbiUhmam383wkNS2DVV7TFtkWVoKasHp4 bounty of about 25 bars ***)
L2066
Variable y : set
L2067
Variable z : set
L2068
Hypothesis H0 : SNo x
L2069
Hypothesis H1 : y ∈ setexp (SNoS_ ω) ω
L2070
Hypothesis H2 : z ∈ setexp (SNoS_ ω) ω
L2071
Hypothesis H3 : (∀w : set, w ∈ ω → ap y w < x)
L2072
Hypothesis H4 : (∀w : set, w ∈ ω → (∀u : set, u ∈ w → ap y u < ap y w))
L2073
Hypothesis H5 : (∀w : set, w ∈ ω → x < ap z w)
L2074
Hypothesis H7 : x = SNoCut (Repl ω (ap y)) (Repl ω (ap z))
L2075
Hypothesis H8 : SNo (- x)
L2076
Theorem. (Conj_SNo_approx_real__28__6)
(∀w : set, w ∈ ω → SNo (ap y w)) → x ∈ real
In Proofgold the corresponding term root is 51f1f6... and proposition id is 3e8cc0...
Proof:
Proof not loaded.
End of Section Conj_SNo_approx_real__28__6
Beginning of Section Conj_SNo_approx_real_rep__1__0
L2082
Variable x : set
(*** Conj_SNo_approx_real_rep__1__0 TMPswjThdqLyamR76SpcC6kdeKHkjMeCG8q bounty of about 25 bars ***)
L2083
Variable y : set
L2084
Hypothesis H1 : (∀z : set, z ∈ SNoS_ ω → (∀w : set, w ∈ ω → abs_SNo (z + - x) < eps_ w) → z = x)
L2085
Hypothesis H2 : SNo y
L2086
Hypothesis H3 : x < y
L2087
Hypothesis H4 : y ∈ SNoS_ ω
L2088
Hypothesis H5 : Empty < y + - x
L2089
Hypothesis H6 : ¬ (∃z : set, z ∈ ω ∧ (x + eps_ z) ≤ y)
L2090
Theorem. (Conj_SNo_approx_real_rep__1__0)
y ≠ x
In Proofgold the corresponding term root is 6971a8... and proposition id is fb18d3...
Proof:
Proof not loaded.
End of Section Conj_SNo_approx_real_rep__1__0
Beginning of Section Conj_SNo_approx_real_rep__1__1
L2096
Variable x : set
(*** Conj_SNo_approx_real_rep__1__1 TMLUKQ6pYN7VGx5kXhsLmCsEDg8Fppg9mKf bounty of about 25 bars ***)
L2097
Variable y : set
L2098
Hypothesis H0 : SNo x
L2099
Hypothesis H2 : SNo y
L2100
Hypothesis H3 : x < y
L2101
Hypothesis H4 : y ∈ SNoS_ ω
L2102
Hypothesis H5 : Empty < y + - x
L2103
Hypothesis H6 : ¬ (∃z : set, z ∈ ω ∧ (x + eps_ z) ≤ y)
L2104
Theorem. (Conj_SNo_approx_real_rep__1__1)
y ≠ x
In Proofgold the corresponding term root is e9b82c... and proposition id is 3aec6f...
Proof:
Proof not loaded.
End of Section Conj_SNo_approx_real_rep__1__1
Beginning of Section Conj_SNo_approx_real_rep__3__0
L2110
Variable x : set
(*** Conj_SNo_approx_real_rep__3__0 TMHCAkDxZzUi1rkKiKnR2P4yMtAcv3NDmES bounty of about 25 bars ***)
L2111
Variable y : set
L2112
Variable z : set
L2113
Variable w : set
L2114
Hypothesis H1 : (∀u : set, u ∈ SNoS_ ω → (∀v : set, v ∈ ω → abs_SNo (u + - x) < eps_ v) → u = x)
L2115
Hypothesis H2 : (∀u : set, u ∈ ω → SNo (ap z u))
L2116
Hypothesis H3 : (∀u : set, u ∈ ω → (ap z u + - (eps_ u)) < x)
L2117
Hypothesis H4 : SNo (SNoCut (Repl ω (ap y)) (Repl ω (ap z)))
L2118
Hypothesis H5 : (∀u : set, u ∈ ω → SNoCut (Repl ω (ap y)) (Repl ω (ap z)) < ap z u)
L2119
Hypothesis H6 : SNo w
L2120
Hypothesis H7 : x < w
L2121
Hypothesis H8 : w ∈ SNoS_ ω
L2122
Theorem. (Conj_SNo_approx_real_rep__3__0)
Empty < w + - x → SNoCut (Repl ω (ap y)) (Repl ω (ap z)) < w
In Proofgold the corresponding term root is 955f33... and proposition id is 94cd09...
Proof:
Proof not loaded.
End of Section Conj_SNo_approx_real_rep__3__0
Beginning of Section Conj_SNo_approx_real_rep__6__2
L2128
Variable x : set
(*** Conj_SNo_approx_real_rep__6__2 TMZmsUvnGDWMjz4sAcPdhLK7UaGtpZJgpDn bounty of about 25 bars ***)
L2129
Variable y : set
L2130
Variable z : set
L2131
Variable w : set
L2132
Hypothesis H0 : SNo x
L2133
Hypothesis H1 : (∀u : set, u ∈ SNoS_ ω → (∀v : set, v ∈ ω → abs_SNo (u + - x) < eps_ v) → u = x)
L2134
Hypothesis H3 : (∀u : set, u ∈ ω → SNo (ap y u))
L2135
Hypothesis H4 : SNo (SNoCut (Repl ω (ap y)) (Repl ω (ap z)))
L2136
Hypothesis H5 : (∀u : set, u ∈ ω → ap y u < SNoCut (Repl ω (ap y)) (Repl ω (ap z)))
L2137
Hypothesis H6 : SNo w
L2138
Hypothesis H7 : w < x
L2139
Hypothesis H8 : w ∈ SNoS_ ω
L2140
Hypothesis H9 : Empty < x + - w
L2141
Theorem. (Conj_SNo_approx_real_rep__6__2)
(∃u : set, u ∈ ω ∧ (w + eps_ u) ≤ x) → w < SNoCut (Repl ω (ap y)) (Repl ω (ap z))
In Proofgold the corresponding term root is a417e1... and proposition id is 57a124...
Proof:
Proof not loaded.
End of Section Conj_SNo_approx_real_rep__6__2
Beginning of Section Conj_SNo_approx_real_rep__6__4
L2147
Variable x : set
(*** Conj_SNo_approx_real_rep__6__4 TMPpGQHycudyQ8ukFfoNh1GKVQkBKjKpz4m bounty of about 25 bars ***)
L2148
Variable y : set
L2149
Variable z : set
L2150
Variable w : set
L2151
Hypothesis H0 : SNo x
L2152
Hypothesis H1 : (∀u : set, u ∈ SNoS_ ω → (∀v : set, v ∈ ω → abs_SNo (u + - x) < eps_ v) → u = x)
L2153
Hypothesis H2 : (∀u : set, u ∈ ω → ap y u < x ∧ x < ap y u + eps_ u ∧ (∀v : set, v ∈ u → ap y v < ap y u))
L2154
Hypothesis H3 : (∀u : set, u ∈ ω → SNo (ap y u))
L2155
Hypothesis H5 : (∀u : set, u ∈ ω → ap y u < SNoCut (Repl ω (ap y)) (Repl ω (ap z)))
L2156
Hypothesis H6 : SNo w
L2157
Hypothesis H7 : w < x
L2158
Hypothesis H8 : w ∈ SNoS_ ω
L2159
Hypothesis H9 : Empty < x + - w
L2160
Theorem. (Conj_SNo_approx_real_rep__6__4)
(∃u : set, u ∈ ω ∧ (w + eps_ u) ≤ x) → w < SNoCut (Repl ω (ap y)) (Repl ω (ap z))
In Proofgold the corresponding term root is feadc2... and proposition id is f66efe...
Proof:
Proof not loaded.
End of Section Conj_SNo_approx_real_rep__6__4
Beginning of Section Conj_SNo_approx_real_rep__7__2
L2166
Variable x : set
(*** Conj_SNo_approx_real_rep__7__2 TMHffEj8MiT2BxNpdHqm3u3cD5Ns6RkGGRw bounty of about 25 bars ***)
L2167
Variable y : set
L2168
Variable z : set
L2169
Variable w : set
L2170
Hypothesis H0 : SNo x
L2171
Hypothesis H1 : (∀u : set, u ∈ SNoS_ ω → (∀v : set, v ∈ ω → abs_SNo (u + - x) < eps_ v) → u = x)
L2172
Hypothesis H3 : (∀u : set, u ∈ ω → SNo (ap y u))
L2173
Hypothesis H4 : SNo (SNoCut (Repl ω (ap y)) (Repl ω (ap z)))
L2174
Hypothesis H5 : (∀u : set, u ∈ ω → ap y u < SNoCut (Repl ω (ap y)) (Repl ω (ap z)))
L2175
Hypothesis H6 : SNo w
L2176
Hypothesis H7 : w < x
L2177
Hypothesis H8 : w ∈ SNoS_ ω
L2178
Theorem. (Conj_SNo_approx_real_rep__7__2)
Empty < x + - w → w < SNoCut (Repl ω (ap y)) (Repl ω (ap z))
In Proofgold the corresponding term root is 585f09... and proposition id is 8a039c...
Proof:
Proof not loaded.
End of Section Conj_SNo_approx_real_rep__7__2
Beginning of Section Conj_SNo_approx_real_rep__9__9
L2184
Variable x : set
(*** Conj_SNo_approx_real_rep__9__9 TMP6JXqsMz2cuVZqivvisGMJYMZ16muRMnS bounty of about 25 bars ***)
L2185
Variable P : prop
L2186
Variable y : set
L2187
Variable z : set
L2188
Hypothesis H0 : x ∈ real
L2189
Hypothesis H1 : (∀w : set, w ∈ setexp (SNoS_ ω) ω → (∀u : set, u ∈ setexp (SNoS_ ω) ω → (∀v : set, v ∈ ω → ap w v < x) → (∀v : set, v ∈ ω → x < ap w v + eps_ v) → (∀v : set, v ∈ ω → (∀x2 : set, x2 ∈ v → ap w x2 < ap w v)) → (∀v : set, v ∈ ω → (ap u v + - (eps_ v)) < x) → (∀v : set, v ∈ ω → x < ap u v) → (∀v : set, v ∈ ω → (∀x2 : set, x2 ∈ v → ap u v < ap u x2)) → SNoCutP (Repl ω (ap w)) (Repl ω (ap u)) → x = SNoCut (Repl ω (ap w)) (Repl ω (ap u)) → P))
L2190
Hypothesis H2 : SNo x
L2191
Hypothesis H3 : (∀w : set, w ∈ SNoS_ ω → (∀u : set, u ∈ ω → abs_SNo (w + - x) < eps_ u) → w = x)
L2192
Hypothesis H4 : y ∈ setexp (SNoS_ ω) ω
L2193
Hypothesis H5 : (∀w : set, w ∈ ω → ap y w < x ∧ x < ap y w + eps_ w ∧ (∀u : set, u ∈ w → ap y u < ap y w))
L2194
Hypothesis H6 : z ∈ setexp (SNoS_ ω) ω
L2195
Hypothesis H7 : (∀w : set, w ∈ ω → SNo (ap y w))
L2196
Hypothesis H8 : (∀w : set, w ∈ ω → SNo (ap z w))
L2197
Hypothesis H10 : (∀w : set, w ∈ ω → x < ap y w + eps_ w)
L2198
Hypothesis H11 : (∀w : set, w ∈ ω → (∀u : set, u ∈ w → ap y u < ap y w))
L2199
Hypothesis H12 : (∀w : set, w ∈ ω → (ap z w + - (eps_ w)) < x)
L2200
Hypothesis H13 : (∀w : set, w ∈ ω → x < ap z w)
L2201
Hypothesis H14 : (∀w : set, w ∈ ω → (∀u : set, u ∈ w → ap z w < ap z u))
L2202
Hypothesis H15 : SNoCutP (Repl ω (ap y)) (Repl ω (ap z))
L2203
Hypothesis H16 : SNo (SNoCut (Repl ω (ap y)) (Repl ω (ap z)))
L2204
Hypothesis H17 : (∀w : set, w ∈ ω → ap y w < SNoCut (Repl ω (ap y)) (Repl ω (ap z)))
L2205
Hypothesis H18 : (∀w : set, w ∈ ω → SNoCut (Repl ω (ap y)) (Repl ω (ap z)) < ap z w)
L2206
Theorem. (Conj_SNo_approx_real_rep__9__9)
x = SNoCut (Repl ω (ap y)) (Repl ω (ap z)) → P
In Proofgold the corresponding term root is dfc77a... and proposition id is 8d0a88...
Proof:
Proof not loaded.
End of Section Conj_SNo_approx_real_rep__9__9
Beginning of Section Conj_SNo_approx_real_rep__9__10
L2212
Variable x : set
(*** Conj_SNo_approx_real_rep__9__10 TMSyd6rEBKTrjHxdqN31rPgSF19hZApgB4m bounty of about 25 bars ***)
L2213
Variable P : prop
L2214
Variable y : set
L2215
Variable z : set
L2216
Hypothesis H0 : x ∈ real
L2217
Hypothesis H1 : (∀w : set, w ∈ setexp (SNoS_ ω) ω → (∀u : set, u ∈ setexp (SNoS_ ω) ω → (∀v : set, v ∈ ω → ap w v < x) → (∀v : set, v ∈ ω → x < ap w v + eps_ v) → (∀v : set, v ∈ ω → (∀x2 : set, x2 ∈ v → ap w x2 < ap w v)) → (∀v : set, v ∈ ω → (ap u v + - (eps_ v)) < x) → (∀v : set, v ∈ ω → x < ap u v) → (∀v : set, v ∈ ω → (∀x2 : set, x2 ∈ v → ap u v < ap u x2)) → SNoCutP (Repl ω (ap w)) (Repl ω (ap u)) → x = SNoCut (Repl ω (ap w)) (Repl ω (ap u)) → P))
L2218
Hypothesis H2 : SNo x
L2219
Hypothesis H3 : (∀w : set, w ∈ SNoS_ ω → (∀u : set, u ∈ ω → abs_SNo (w + - x) < eps_ u) → w = x)
L2220
Hypothesis H4 : y ∈ setexp (SNoS_ ω) ω
L2221
Hypothesis H5 : (∀w : set, w ∈ ω → ap y w < x ∧ x < ap y w + eps_ w ∧ (∀u : set, u ∈ w → ap y u < ap y w))
L2222
Hypothesis H6 : z ∈ setexp (SNoS_ ω) ω
L2223
Hypothesis H7 : (∀w : set, w ∈ ω → SNo (ap y w))
L2224
Hypothesis H8 : (∀w : set, w ∈ ω → SNo (ap z w))
L2225
Hypothesis H9 : (∀w : set, w ∈ ω → ap y w < x)
L2226
Hypothesis H11 : (∀w : set, w ∈ ω → (∀u : set, u ∈ w → ap y u < ap y w))
L2227
Hypothesis H12 : (∀w : set, w ∈ ω → (ap z w + - (eps_ w)) < x)
L2228
Hypothesis H13 : (∀w : set, w ∈ ω → x < ap z w)
L2229
Hypothesis H14 : (∀w : set, w ∈ ω → (∀u : set, u ∈ w → ap z w < ap z u))
L2230
Hypothesis H15 : SNoCutP (Repl ω (ap y)) (Repl ω (ap z))
L2231
Hypothesis H16 : SNo (SNoCut (Repl ω (ap y)) (Repl ω (ap z)))
L2232
Hypothesis H17 : (∀w : set, w ∈ ω → ap y w < SNoCut (Repl ω (ap y)) (Repl ω (ap z)))
L2233
Hypothesis H18 : (∀w : set, w ∈ ω → SNoCut (Repl ω (ap y)) (Repl ω (ap z)) < ap z w)
L2234
Theorem. (Conj_SNo_approx_real_rep__9__10)
x = SNoCut (Repl ω (ap y)) (Repl ω (ap z)) → P
In Proofgold the corresponding term root is a9cbbc... and proposition id is f0619a...
Proof:
Proof not loaded.
End of Section Conj_SNo_approx_real_rep__9__10
Beginning of Section Conj_SNo_approx_real_rep__11__2
L2240
Variable x : set
(*** Conj_SNo_approx_real_rep__11__2 TMGwyfBiq15RqrJxuD71iMYMRnxgyC1dfU1 bounty of about 25 bars ***)
L2241
Variable P : prop
L2242
Variable y : set
L2243
Variable z : set
L2244
Hypothesis H0 : x ∈ real
L2245
Hypothesis H1 : (∀w : set, w ∈ setexp (SNoS_ ω) ω → (∀u : set, u ∈ setexp (SNoS_ ω) ω → (∀v : set, v ∈ ω → ap w v < x) → (∀v : set, v ∈ ω → x < ap w v + eps_ v) → (∀v : set, v ∈ ω → (∀x2 : set, x2 ∈ v → ap w x2 < ap w v)) → (∀v : set, v ∈ ω → (ap u v + - (eps_ v)) < x) → (∀v : set, v ∈ ω → x < ap u v) → (∀v : set, v ∈ ω → (∀x2 : set, x2 ∈ v → ap u v < ap u x2)) → SNoCutP (Repl ω (ap w)) (Repl ω (ap u)) → x = SNoCut (Repl ω (ap w)) (Repl ω (ap u)) → P))
L2246
Hypothesis H3 : (∀w : set, w ∈ SNoS_ ω → (∀u : set, u ∈ ω → abs_SNo (w + - x) < eps_ u) → w = x)
L2247
Hypothesis H4 : y ∈ setexp (SNoS_ ω) ω
L2248
Hypothesis H5 : (∀w : set, w ∈ ω → ap y w < x ∧ x < ap y w + eps_ w ∧ (∀u : set, u ∈ w → ap y u < ap y w))
L2249
Hypothesis H6 : z ∈ setexp (SNoS_ ω) ω
L2250
Hypothesis H7 : (∀w : set, w ∈ ω → (ap z w + - (eps_ w)) < x ∧ x < ap z w ∧ (∀u : set, u ∈ w → ap z w < ap z u))
L2251
Hypothesis H8 : (∀w : set, w ∈ ω → SNo (ap y w))
L2252
Hypothesis H9 : (∀w : set, w ∈ ω → SNo (ap z w))
L2253
Hypothesis H10 : (∀w : set, w ∈ ω → ap y w < x)
L2254
Hypothesis H11 : (∀w : set, w ∈ ω → x < ap y w + eps_ w)
L2255
Hypothesis H12 : (∀w : set, w ∈ ω → (∀u : set, u ∈ w → ap y u < ap y w))
L2256
Hypothesis H13 : (∀w : set, w ∈ ω → (ap z w + - (eps_ w)) < x)
L2257
Hypothesis H14 : (∀w : set, w ∈ ω → x < ap z w)
L2258
Theorem. (Conj_SNo_approx_real_rep__11__2)
(∀w : set, w ∈ ω → (∀u : set, u ∈ w → ap z w < ap z u)) → P
In Proofgold the corresponding term root is ee67f9... and proposition id is 603954...
Proof:
Proof not loaded.
End of Section Conj_SNo_approx_real_rep__11__2
Beginning of Section Conj_SNo_approx_real_rep__11__3
L2264
Variable x : set
(*** Conj_SNo_approx_real_rep__11__3 TMRS23j1xnPpdb9o7FQ5E2S2uQyZEJGFDxX bounty of about 25 bars ***)
L2265
Variable P : prop
L2266
Variable y : set
L2267
Variable z : set
L2268
Hypothesis H0 : x ∈ real
L2269
Hypothesis H1 : (∀w : set, w ∈ setexp (SNoS_ ω) ω → (∀u : set, u ∈ setexp (SNoS_ ω) ω → (∀v : set, v ∈ ω → ap w v < x) → (∀v : set, v ∈ ω → x < ap w v + eps_ v) → (∀v : set, v ∈ ω → (∀x2 : set, x2 ∈ v → ap w x2 < ap w v)) → (∀v : set, v ∈ ω → (ap u v + - (eps_ v)) < x) → (∀v : set, v ∈ ω → x < ap u v) → (∀v : set, v ∈ ω → (∀x2 : set, x2 ∈ v → ap u v < ap u x2)) → SNoCutP (Repl ω (ap w)) (Repl ω (ap u)) → x = SNoCut (Repl ω (ap w)) (Repl ω (ap u)) → P))
L2270
Hypothesis H2 : SNo x
L2271
Hypothesis H4 : y ∈ setexp (SNoS_ ω) ω
L2272
Hypothesis H5 : (∀w : set, w ∈ ω → ap y w < x ∧ x < ap y w + eps_ w ∧ (∀u : set, u ∈ w → ap y u < ap y w))
L2273
Hypothesis H6 : z ∈ setexp (SNoS_ ω) ω
L2274
Hypothesis H7 : (∀w : set, w ∈ ω → (ap z w + - (eps_ w)) < x ∧ x < ap z w ∧ (∀u : set, u ∈ w → ap z w < ap z u))
L2275
Hypothesis H8 : (∀w : set, w ∈ ω → SNo (ap y w))
L2276
Hypothesis H9 : (∀w : set, w ∈ ω → SNo (ap z w))
L2277
Hypothesis H10 : (∀w : set, w ∈ ω → ap y w < x)
L2278
Hypothesis H11 : (∀w : set, w ∈ ω → x < ap y w + eps_ w)
L2279
Hypothesis H12 : (∀w : set, w ∈ ω → (∀u : set, u ∈ w → ap y u < ap y w))
L2280
Hypothesis H13 : (∀w : set, w ∈ ω → (ap z w + - (eps_ w)) < x)
L2281
Hypothesis H14 : (∀w : set, w ∈ ω → x < ap z w)
L2282
Theorem. (Conj_SNo_approx_real_rep__11__3)
(∀w : set, w ∈ ω → (∀u : set, u ∈ w → ap z w < ap z u)) → P
In Proofgold the corresponding term root is d27589... and proposition id is 025c43...
Proof:
Proof not loaded.
End of Section Conj_SNo_approx_real_rep__11__3
Beginning of Section Conj_SNo_approx_real_rep__14__0
L2288
Variable x : set
(*** Conj_SNo_approx_real_rep__14__0 TMdLLekUxFyp1BUUub1FKbb59PQyesDGDRG bounty of about 25 bars ***)
L2289
Variable P : prop
L2290
Variable y : set
L2291
Variable z : set
L2292
Hypothesis H1 : (∀w : set, w ∈ setexp (SNoS_ ω) ω → (∀u : set, u ∈ setexp (SNoS_ ω) ω → (∀v : set, v ∈ ω → ap w v < x) → (∀v : set, v ∈ ω → x < ap w v + eps_ v) → (∀v : set, v ∈ ω → (∀x2 : set, x2 ∈ v → ap w x2 < ap w v)) → (∀v : set, v ∈ ω → (ap u v + - (eps_ v)) < x) → (∀v : set, v ∈ ω → x < ap u v) → (∀v : set, v ∈ ω → (∀x2 : set, x2 ∈ v → ap u v < ap u x2)) → SNoCutP (Repl ω (ap w)) (Repl ω (ap u)) → x = SNoCut (Repl ω (ap w)) (Repl ω (ap u)) → P))
L2293
Hypothesis H2 : SNo x
L2294
Hypothesis H3 : (∀w : set, w ∈ SNoS_ ω → (∀u : set, u ∈ ω → abs_SNo (w + - x) < eps_ u) → w = x)
L2295
Hypothesis H4 : y ∈ setexp (SNoS_ ω) ω
L2296
Hypothesis H5 : (∀w : set, w ∈ ω → ap y w < x ∧ x < ap y w + eps_ w ∧ (∀u : set, u ∈ w → ap y u < ap y w))
L2297
Hypothesis H6 : z ∈ setexp (SNoS_ ω) ω
L2298
Hypothesis H7 : (∀w : set, w ∈ ω → (ap z w + - (eps_ w)) < x ∧ x < ap z w ∧ (∀u : set, u ∈ w → ap z w < ap z u))
L2299
Hypothesis H8 : (∀w : set, w ∈ ω → SNo (ap y w))
L2300
Hypothesis H9 : (∀w : set, w ∈ ω → SNo (ap z w))
L2301
Hypothesis H10 : (∀w : set, w ∈ ω → ap y w < x)
L2302
Hypothesis H11 : (∀w : set, w ∈ ω → x < ap y w + eps_ w)
L2303
Theorem. (Conj_SNo_approx_real_rep__14__0)
(∀w : set, w ∈ ω → (∀u : set, u ∈ w → ap y u < ap y w)) → P
In Proofgold the corresponding term root is f66a8c... and proposition id is 59c24b...
Proof:
Proof not loaded.
End of Section Conj_SNo_approx_real_rep__14__0
Beginning of Section Conj_SNo_approx_real_rep__14__4
L2309
Variable x : set
(*** Conj_SNo_approx_real_rep__14__4 TMHd3bk5esoJHwbrBU1dVbhMuvmJNMpvYmh bounty of about 25 bars ***)
L2310
Variable P : prop
L2311
Variable y : set
L2312
Variable z : set
L2313
Hypothesis H0 : x ∈ real
L2314
Hypothesis H1 : (∀w : set, w ∈ setexp (SNoS_ ω) ω → (∀u : set, u ∈ setexp (SNoS_ ω) ω → (∀v : set, v ∈ ω → ap w v < x) → (∀v : set, v ∈ ω → x < ap w v + eps_ v) → (∀v : set, v ∈ ω → (∀x2 : set, x2 ∈ v → ap w x2 < ap w v)) → (∀v : set, v ∈ ω → (ap u v + - (eps_ v)) < x) → (∀v : set, v ∈ ω → x < ap u v) → (∀v : set, v ∈ ω → (∀x2 : set, x2 ∈ v → ap u v < ap u x2)) → SNoCutP (Repl ω (ap w)) (Repl ω (ap u)) → x = SNoCut (Repl ω (ap w)) (Repl ω (ap u)) → P))
L2315
Hypothesis H2 : SNo x
L2316
Hypothesis H3 : (∀w : set, w ∈ SNoS_ ω → (∀u : set, u ∈ ω → abs_SNo (w + - x) < eps_ u) → w = x)
L2317
Hypothesis H5 : (∀w : set, w ∈ ω → ap y w < x ∧ x < ap y w + eps_ w ∧ (∀u : set, u ∈ w → ap y u < ap y w))
L2318
Hypothesis H6 : z ∈ setexp (SNoS_ ω) ω
L2319
Hypothesis H7 : (∀w : set, w ∈ ω → (ap z w + - (eps_ w)) < x ∧ x < ap z w ∧ (∀u : set, u ∈ w → ap z w < ap z u))
L2320
Hypothesis H8 : (∀w : set, w ∈ ω → SNo (ap y w))
L2321
Hypothesis H9 : (∀w : set, w ∈ ω → SNo (ap z w))
L2322
Hypothesis H10 : (∀w : set, w ∈ ω → ap y w < x)
L2323
Hypothesis H11 : (∀w : set, w ∈ ω → x < ap y w + eps_ w)
L2324
Theorem. (Conj_SNo_approx_real_rep__14__4)
(∀w : set, w ∈ ω → (∀u : set, u ∈ w → ap y u < ap y w)) → P
In Proofgold the corresponding term root is 8ad048... and proposition id is b1926b...
Proof:
Proof not loaded.
End of Section Conj_SNo_approx_real_rep__14__4
Beginning of Section Conj_SNo_approx_real_rep__14__10
L2330
Variable x : set
(*** Conj_SNo_approx_real_rep__14__10 TMXG7FzpRXAMVDbi71dZodhfv3HgRvPj2HS bounty of about 25 bars ***)
L2331
Variable P : prop
L2332
Variable y : set
L2333
Variable z : set
L2334
Hypothesis H0 : x ∈ real
L2335
Hypothesis H1 : (∀w : set, w ∈ setexp (SNoS_ ω) ω → (∀u : set, u ∈ setexp (SNoS_ ω) ω → (∀v : set, v ∈ ω → ap w v < x) → (∀v : set, v ∈ ω → x < ap w v + eps_ v) → (∀v : set, v ∈ ω → (∀x2 : set, x2 ∈ v → ap w x2 < ap w v)) → (∀v : set, v ∈ ω → (ap u v + - (eps_ v)) < x) → (∀v : set, v ∈ ω → x < ap u v) → (∀v : set, v ∈ ω → (∀x2 : set, x2 ∈ v → ap u v < ap u x2)) → SNoCutP (Repl ω (ap w)) (Repl ω (ap u)) → x = SNoCut (Repl ω (ap w)) (Repl ω (ap u)) → P))
L2336
Hypothesis H2 : SNo x
L2337
Hypothesis H3 : (∀w : set, w ∈ SNoS_ ω → (∀u : set, u ∈ ω → abs_SNo (w + - x) < eps_ u) → w = x)
L2338
Hypothesis H4 : y ∈ setexp (SNoS_ ω) ω
L2339
Hypothesis H5 : (∀w : set, w ∈ ω → ap y w < x ∧ x < ap y w + eps_ w ∧ (∀u : set, u ∈ w → ap y u < ap y w))
L2340
Hypothesis H6 : z ∈ setexp (SNoS_ ω) ω
L2341
Hypothesis H7 : (∀w : set, w ∈ ω → (ap z w + - (eps_ w)) < x ∧ x < ap z w ∧ (∀u : set, u ∈ w → ap z w < ap z u))
L2342
Hypothesis H8 : (∀w : set, w ∈ ω → SNo (ap y w))
L2343
Hypothesis H9 : (∀w : set, w ∈ ω → SNo (ap z w))
L2344
Hypothesis H11 : (∀w : set, w ∈ ω → x < ap y w + eps_ w)
L2345
Theorem. (Conj_SNo_approx_real_rep__14__10)
(∀w : set, w ∈ ω → (∀u : set, u ∈ w → ap y u < ap y w)) → P
In Proofgold the corresponding term root is 0357c8... and proposition id is c0eaed...
Proof:
Proof not loaded.
End of Section Conj_SNo_approx_real_rep__14__10
Beginning of Section Conj_SNo_approx_real_rep__16__2
L2351
Variable x : set
(*** Conj_SNo_approx_real_rep__16__2 TMZC2m1qkeQPJ61k7mqK9SeXH9A98WYVMoD bounty of about 25 bars ***)
L2352
Variable P : prop
L2353
Variable y : set
L2354
Variable z : set
L2355
Hypothesis H0 : x ∈ real
L2356
Hypothesis H1 : (∀w : set, w ∈ setexp (SNoS_ ω) ω → (∀u : set, u ∈ setexp (SNoS_ ω) ω → (∀v : set, v ∈ ω → ap w v < x) → (∀v : set, v ∈ ω → x < ap w v + eps_ v) → (∀v : set, v ∈ ω → (∀x2 : set, x2 ∈ v → ap w x2 < ap w v)) → (∀v : set, v ∈ ω → (ap u v + - (eps_ v)) < x) → (∀v : set, v ∈ ω → x < ap u v) → (∀v : set, v ∈ ω → (∀x2 : set, x2 ∈ v → ap u v < ap u x2)) → SNoCutP (Repl ω (ap w)) (Repl ω (ap u)) → x = SNoCut (Repl ω (ap w)) (Repl ω (ap u)) → P))
L2357
Hypothesis H3 : (∀w : set, w ∈ SNoS_ ω → (∀u : set, u ∈ ω → abs_SNo (w + - x) < eps_ u) → w = x)
L2358
Hypothesis H4 : y ∈ setexp (SNoS_ ω) ω
L2359
Hypothesis H5 : (∀w : set, w ∈ ω → ap y w < x ∧ x < ap y w + eps_ w ∧ (∀u : set, u ∈ w → ap y u < ap y w))
L2360
Hypothesis H6 : z ∈ setexp (SNoS_ ω) ω
L2361
Hypothesis H7 : (∀w : set, w ∈ ω → (ap z w + - (eps_ w)) < x ∧ x < ap z w ∧ (∀u : set, u ∈ w → ap z w < ap z u))
L2362
Hypothesis H8 : (∀w : set, w ∈ ω → SNo (ap y w))
L2363
Hypothesis H9 : (∀w : set, w ∈ ω → SNo (ap z w))
L2364
Theorem. (Conj_SNo_approx_real_rep__16__2)
(∀w : set, w ∈ ω → ap y w < x) → P
In Proofgold the corresponding term root is 4f42c1... and proposition id is 9084b4...
Proof:
Proof not loaded.
End of Section Conj_SNo_approx_real_rep__16__2
Beginning of Section Conj_SNo_approx_real_rep__17__1
L2370
Variable x : set
(*** Conj_SNo_approx_real_rep__17__1 TMPJPiemdmEqE19WLTZU7Z8pEfz8s6bFjSc bounty of about 25 bars ***)
L2371
Variable P : prop
L2372
Variable y : set
L2373
Variable z : set
L2374
Hypothesis H0 : x ∈ real
L2375
Hypothesis H2 : SNo x
L2376
Hypothesis H3 : (∀w : set, w ∈ SNoS_ ω → (∀u : set, u ∈ ω → abs_SNo (w + - x) < eps_ u) → w = x)
L2377
Hypothesis H4 : y ∈ setexp (SNoS_ ω) ω
L2378
Hypothesis H5 : (∀w : set, w ∈ ω → ap y w < x ∧ x < ap y w + eps_ w ∧ (∀u : set, u ∈ w → ap y u < ap y w))
L2379
Hypothesis H6 : z ∈ setexp (SNoS_ ω) ω
L2380
Hypothesis H7 : (∀w : set, w ∈ ω → (ap z w + - (eps_ w)) < x ∧ x < ap z w ∧ (∀u : set, u ∈ w → ap z w < ap z u))
L2381
Hypothesis H8 : (∀w : set, w ∈ ω → SNo (ap y w))
L2382
Theorem. (Conj_SNo_approx_real_rep__17__1)
(∀w : set, w ∈ ω → SNo (ap z w)) → P
In Proofgold the corresponding term root is 841859... and proposition id is e18823...
Proof:
Proof not loaded.
End of Section Conj_SNo_approx_real_rep__17__1
Beginning of Section Conj_real_add_SNo__1__6
L2388
Variable x : set
(*** Conj_real_add_SNo__1__6 TMY8g4rbgXKt846Bb4n2R14nwgb1kh1G566 bounty of about 25 bars ***)
L2389
Variable y : set
L2390
Variable z : set
L2391
Variable w : set
L2392
Variable u : set
L2393
Hypothesis H0 : (∀v : set, v ∈ ω → x < ap z v + eps_ v)
L2394
Hypothesis H1 : (∀v : set, v ∈ ω → y < ap w v + eps_ v)
L2395
Hypothesis H2 : SNo x
L2396
Hypothesis H3 : SNo y
L2397
Hypothesis H4 : (∀v : set, v ∈ ω → SNo (ap z (ordsucc v)))
L2398
Hypothesis H5 : (∀v : set, v ∈ ω → SNo (ap w (ordsucc v)))
L2399
Theorem. (Conj_real_add_SNo__1__6)
SNo (eps_ (ordsucc u)) → (x + y) < (ap z (ordsucc u) + ap w (ordsucc u)) + eps_ (ordsucc u) + eps_ (ordsucc u)
In Proofgold the corresponding term root is 7d2ba4... and proposition id is 3f513a...
Proof:
Proof not loaded.
End of Section Conj_real_add_SNo__1__6
Beginning of Section Conj_real_add_SNo__2__3
L2405
Variable x : set
(*** Conj_real_add_SNo__2__3 TMFfzj64YGAjMcfvhxia4s9aZiABLUimVYc bounty of about 25 bars ***)
L2406
Variable y : set
L2407
Variable z : set
L2408
Variable w : set
L2409
Hypothesis H0 : (∀u : set, u ∈ ω → (∀v : set, v ∈ u → ap x v < ap x u))
L2410
Hypothesis H1 : (∀u : set, u ∈ ω → (∀v : set, v ∈ u → ap y v < ap y u))
L2411
Hypothesis H2 : (∀u : set, u ∈ ω → SNo (ap x (ordsucc u)))
L2412
Hypothesis H4 : (∀u : set, u ∈ ω → ap (Sigma ω (λv : set ⇒ ap x (ordsucc v) + ap y (ordsucc v))) u = ap x (ordsucc u) + ap y (ordsucc u))
L2413
Hypothesis H5 : z ∈ ω
L2414
Hypothesis H6 : w ∈ z
L2415
Theorem. (Conj_real_add_SNo__2__3)
w ∈ ω → ap (Sigma ω (λu : set ⇒ ap x (ordsucc u) + ap y (ordsucc u))) w < ap x (ordsucc z) + ap y (ordsucc z)
In Proofgold the corresponding term root is 171bff... and proposition id is 3925ff...
Proof:
Proof not loaded.
End of Section Conj_real_add_SNo__2__3
Beginning of Section Conj_real_add_SNo__5__5
L2421
Variable x : set
(*** Conj_real_add_SNo__5__5 TMTp7b2uk8a3K12CPsiXeyQrZy98xCtJ2fc bounty of about 25 bars ***)
L2422
Variable y : set
L2423
Variable z : set
L2424
Variable w : set
L2425
Hypothesis H0 : (∀u : set, u ∈ ω → (∀v : set, v ∈ u → ap x u < ap x v))
L2426
Hypothesis H1 : (∀u : set, u ∈ ω → (∀v : set, v ∈ u → ap y u < ap y v))
L2427
Hypothesis H2 : (∀u : set, u ∈ ω → SNo (ap x (ordsucc u)))
L2428
Hypothesis H3 : (∀u : set, u ∈ ω → SNo (ap y (ordsucc u)))
L2429
Hypothesis H4 : (∀u : set, u ∈ ω → ap (Sigma ω (λv : set ⇒ ap x (ordsucc v) + ap y (ordsucc v))) u = ap x (ordsucc u) + ap y (ordsucc u))
L2430
Hypothesis H6 : w ∈ z
L2431
Theorem. (Conj_real_add_SNo__5__5)
w ∈ ω → (ap x (ordsucc z) + ap y (ordsucc z)) < ap (Sigma ω (λu : set ⇒ ap x (ordsucc u) + ap y (ordsucc u))) w
In Proofgold the corresponding term root is 2149c1... and proposition id is 3d06eb...
Proof:
Proof not loaded.
End of Section Conj_real_add_SNo__5__5
Beginning of Section Conj_real_add_SNo__6__6
L2437
Variable x : set
(*** Conj_real_add_SNo__6__6 TMPYT7AKTm8ZDRenp4RckJ5dHQamX9YVnYF bounty of about 25 bars ***)
L2438
Variable y : set
L2439
Variable z : set
L2440
Variable w : set
L2441
Variable u : set
L2442
Hypothesis H0 : SNo x
L2443
Hypothesis H1 : SNo y
L2444
Hypothesis H2 : SNo (x + y)
L2445
Hypothesis H3 : (∀v : set, v ∈ SNoS_ ω → (∀x2 : set, x2 ∈ ω → abs_SNo (v + - y) < eps_ x2) → v = y)
L2446
Hypothesis H4 : (∀v : set, v ∈ ω → SNo (ap (Sigma ω (λx2 : set ⇒ ap z (ordsucc x2) + ap w (ordsucc x2))) v))
L2447
Hypothesis H5 : (∀v : set, v ∈ ω → (ap (Sigma ω (λx2 : set ⇒ ap z (ordsucc x2) + ap w (ordsucc x2))) v + - (eps_ v)) < x + y)
L2448
Hypothesis H7 : y < u
L2449
Hypothesis H8 : u ∈ SNoS_ ω
L2450
Hypothesis H9 : ¬ (∃v : set, v ∈ ω ∧ ap (Sigma ω (λx2 : set ⇒ ap z (ordsucc x2) + ap w (ordsucc x2))) v ≤ x + u)
L2451
Hypothesis H10 : Empty < u + - y
L2452
Theorem. (Conj_real_add_SNo__6__6)
u ≠ y
In Proofgold the corresponding term root is adb1b7... and proposition id is 064121...
Proof:
Proof not loaded.
End of Section Conj_real_add_SNo__6__6
Beginning of Section Conj_real_add_SNo__6__10
L2458
Variable x : set
(*** Conj_real_add_SNo__6__10 TMcutPj1p7BEhHoMCU1NxbLogVgoyXigBiv bounty of about 25 bars ***)
L2459
Variable y : set
L2460
Variable z : set
L2461
Variable w : set
L2462
Variable u : set
L2463
Hypothesis H0 : SNo x
L2464
Hypothesis H1 : SNo y
L2465
Hypothesis H2 : SNo (x + y)
L2466
Hypothesis H3 : (∀v : set, v ∈ SNoS_ ω → (∀x2 : set, x2 ∈ ω → abs_SNo (v + - y) < eps_ x2) → v = y)
L2467
Hypothesis H4 : (∀v : set, v ∈ ω → SNo (ap (Sigma ω (λx2 : set ⇒ ap z (ordsucc x2) + ap w (ordsucc x2))) v))
L2468
Hypothesis H5 : (∀v : set, v ∈ ω → (ap (Sigma ω (λx2 : set ⇒ ap z (ordsucc x2) + ap w (ordsucc x2))) v + - (eps_ v)) < x + y)
L2469
Hypothesis H6 : SNo u
L2470
Hypothesis H7 : y < u
L2471
Hypothesis H8 : u ∈ SNoS_ ω
L2472
Hypothesis H9 : ¬ (∃v : set, v ∈ ω ∧ ap (Sigma ω (λx2 : set ⇒ ap z (ordsucc x2) + ap w (ordsucc x2))) v ≤ x + u)
L2473
Theorem. (Conj_real_add_SNo__6__10)
u ≠ y
In Proofgold the corresponding term root is 40aeaa... and proposition id is d22e6e...
Proof:
Proof not loaded.
End of Section Conj_real_add_SNo__6__10
Beginning of Section Conj_real_add_SNo__7__0
L2479
Variable x : set
(*** Conj_real_add_SNo__7__0 TMR6jCcdJ8uZFm4oSbAXDxCCk9SDMNQhv7V bounty of about 25 bars ***)
L2480
Variable y : set
L2481
Variable z : set
L2482
Variable w : set
L2483
Variable u : set
L2484
Hypothesis H1 : SNo y
L2485
Hypothesis H2 : SNo (x + y)
L2486
Hypothesis H3 : (∀v : set, v ∈ SNoS_ ω → (∀x2 : set, x2 ∈ ω → abs_SNo (v + - y) < eps_ x2) → v = y)
L2487
Hypothesis H4 : (∀v : set, v ∈ ω → SNo (ap (Sigma ω (λx2 : set ⇒ ap z (ordsucc x2) + ap w (ordsucc x2))) v))
L2488
Hypothesis H5 : (∀v : set, v ∈ ω → (ap (Sigma ω (λx2 : set ⇒ ap z (ordsucc x2) + ap w (ordsucc x2))) v + - (eps_ v)) < x + y)
L2489
Hypothesis H6 : SNo u
L2490
Hypothesis H7 : y < u
L2491
Hypothesis H8 : u ∈ SNoS_ ω
L2492
Hypothesis H9 : ¬ (∃v : set, v ∈ ω ∧ ap (Sigma ω (λx2 : set ⇒ ap z (ordsucc x2) + ap w (ordsucc x2))) v ≤ x + u)
L2493
Theorem. (Conj_real_add_SNo__7__0)
¬ Empty < u + - y
In Proofgold the corresponding term root is a302e4... and proposition id is c466cf...
Proof:
Proof not loaded.
End of Section Conj_real_add_SNo__7__0
Beginning of Section Conj_real_add_SNo__8__0
L2499
Variable x : set
(*** Conj_real_add_SNo__8__0 TMVRVB6LkKJ6LRUedjjCz3VBX9nnB9NmnKQ bounty of about 25 bars ***)
L2500
Variable y : set
L2501
Variable z : set
L2502
Variable w : set
L2503
Variable u : set
L2504
Variable v : set
L2505
Variable x2 : set
L2506
Hypothesis H1 : SNo y
L2507
Hypothesis H2 : SNo (x + y)
L2508
Hypothesis H3 : (∀y2 : set, y2 ∈ SNoS_ ω → (∀z2 : set, z2 ∈ ω → abs_SNo (y2 + - y) < eps_ z2) → y2 = y)
L2509
Hypothesis H4 : (∀y2 : set, y2 ∈ ω → SNo (ap (Sigma ω (λz2 : set ⇒ ap w (ordsucc z2) + ap v (ordsucc z2))) y2))
L2510
Hypothesis H5 : (∀y2 : set, y2 ∈ ω → (ap (Sigma ω (λz2 : set ⇒ ap w (ordsucc z2) + ap v (ordsucc z2))) y2 + - (eps_ y2)) < x + y)
L2511
Hypothesis H6 : SNo (SNoCut (Repl ω (ap (Sigma ω (λy2 : set ⇒ ap z (ordsucc y2) + ap u (ordsucc y2))))) (Repl ω (ap (Sigma ω (λy2 : set ⇒ ap w (ordsucc y2) + ap v (ordsucc y2))))))
L2512
Hypothesis H7 : (∀y2 : set, y2 ∈ Repl ω (ap (Sigma ω (λz2 : set ⇒ ap w (ordsucc z2) + ap v (ordsucc z2)))) → SNoCut (Repl ω (ap (Sigma ω (λz2 : set ⇒ ap z (ordsucc z2) + ap u (ordsucc z2))))) (Repl ω (ap (Sigma ω (λz2 : set ⇒ ap w (ordsucc z2) + ap v (ordsucc z2))))) < y2)
L2513
Hypothesis H8 : SNo x2
L2514
Hypothesis H9 : y < x2
L2515
Hypothesis H10 : x2 ∈ SNoS_ ω
L2516
Theorem. (Conj_real_add_SNo__8__0)
(∃y2 : set, y2 ∈ ω ∧ ap (Sigma ω (λz2 : set ⇒ ap w (ordsucc z2) + ap v (ordsucc z2))) y2 ≤ x + x2) → SNoCut (Repl ω (ap (Sigma ω (λy2 : set ⇒ ap z (ordsucc y2) + ap u (ordsucc y2))))) (Repl ω (ap (Sigma ω (λy2 : set ⇒ ap w (ordsucc y2) + ap v (ordsucc y2))))) < x + x2
In Proofgold the corresponding term root is a93197... and proposition id is 2a3041...
Proof:
Proof not loaded.
End of Section Conj_real_add_SNo__8__0
Beginning of Section Conj_real_add_SNo__8__6
L2522
Variable x : set
(*** Conj_real_add_SNo__8__6 TMcKdvaQqnTQBDhQ2wbEoJDhrQhXzwXbBEv bounty of about 25 bars ***)
L2523
Variable y : set
L2524
Variable z : set
L2525
Variable w : set
L2526
Variable u : set
L2527
Variable v : set
L2528
Variable x2 : set
L2529
Hypothesis H0 : SNo x
L2530
Hypothesis H1 : SNo y
L2531
Hypothesis H2 : SNo (x + y)
L2532
Hypothesis H3 : (∀y2 : set, y2 ∈ SNoS_ ω → (∀z2 : set, z2 ∈ ω → abs_SNo (y2 + - y) < eps_ z2) → y2 = y)
L2533
Hypothesis H4 : (∀y2 : set, y2 ∈ ω → SNo (ap (Sigma ω (λz2 : set ⇒ ap w (ordsucc z2) + ap v (ordsucc z2))) y2))
L2534
Hypothesis H5 : (∀y2 : set, y2 ∈ ω → (ap (Sigma ω (λz2 : set ⇒ ap w (ordsucc z2) + ap v (ordsucc z2))) y2 + - (eps_ y2)) < x + y)
L2535
Hypothesis H7 : (∀y2 : set, y2 ∈ Repl ω (ap (Sigma ω (λz2 : set ⇒ ap w (ordsucc z2) + ap v (ordsucc z2)))) → SNoCut (Repl ω (ap (Sigma ω (λz2 : set ⇒ ap z (ordsucc z2) + ap u (ordsucc z2))))) (Repl ω (ap (Sigma ω (λz2 : set ⇒ ap w (ordsucc z2) + ap v (ordsucc z2))))) < y2)
L2536
Hypothesis H8 : SNo x2
L2537
Hypothesis H9 : y < x2
L2538
Hypothesis H10 : x2 ∈ SNoS_ ω
L2539
Theorem. (Conj_real_add_SNo__8__6)
(∃y2 : set, y2 ∈ ω ∧ ap (Sigma ω (λz2 : set ⇒ ap w (ordsucc z2) + ap v (ordsucc z2))) y2 ≤ x + x2) → SNoCut (Repl ω (ap (Sigma ω (λy2 : set ⇒ ap z (ordsucc y2) + ap u (ordsucc y2))))) (Repl ω (ap (Sigma ω (λy2 : set ⇒ ap w (ordsucc y2) + ap v (ordsucc y2))))) < x + x2
In Proofgold the corresponding term root is 31a316... and proposition id is 36f972...
Proof:
Proof not loaded.
End of Section Conj_real_add_SNo__8__6
Beginning of Section Conj_real_add_SNo__9__2
L2545
Variable x : set
(*** Conj_real_add_SNo__9__2 TMNCzjTkDm9KM6EXsMUe1o6HNF1Pq9HQh5K bounty of about 25 bars ***)
L2546
Variable y : set
L2547
Variable z : set
L2548
Variable w : set
L2549
Variable u : set
L2550
Variable v : set
L2551
Variable x2 : set
L2552
Hypothesis H0 : y ∈ real
L2553
Hypothesis H1 : SNo x
L2554
Hypothesis H3 : SNo (x + y)
L2555
Hypothesis H4 : (∀y2 : set, y2 ∈ SNoS_ ω → (∀z2 : set, z2 ∈ ω → abs_SNo (y2 + - y) < eps_ z2) → y2 = y)
L2556
Hypothesis H5 : (∀y2 : set, y2 ∈ ω → SNo (ap (Sigma ω (λz2 : set ⇒ ap w (ordsucc z2) + ap v (ordsucc z2))) y2))
L2557
Hypothesis H6 : (∀y2 : set, y2 ∈ ω → (ap (Sigma ω (λz2 : set ⇒ ap w (ordsucc z2) + ap v (ordsucc z2))) y2 + - (eps_ y2)) < x + y)
L2558
Hypothesis H7 : SNo (SNoCut (Repl ω (ap (Sigma ω (λy2 : set ⇒ ap z (ordsucc y2) + ap u (ordsucc y2))))) (Repl ω (ap (Sigma ω (λy2 : set ⇒ ap w (ordsucc y2) + ap v (ordsucc y2))))))
L2559
Hypothesis H8 : (∀y2 : set, y2 ∈ Repl ω (ap (Sigma ω (λz2 : set ⇒ ap w (ordsucc z2) + ap v (ordsucc z2)))) → SNoCut (Repl ω (ap (Sigma ω (λz2 : set ⇒ ap z (ordsucc z2) + ap u (ordsucc z2))))) (Repl ω (ap (Sigma ω (λz2 : set ⇒ ap w (ordsucc z2) + ap v (ordsucc z2))))) < y2)
L2560
Hypothesis H9 : SNo x2
L2561
Hypothesis H10 : SNoLev x2 ∈ SNoLev y
L2562
Hypothesis H11 : y < x2
L2563
Theorem. (Conj_real_add_SNo__9__2)
x2 ∈ SNoS_ ω → SNoCut (Repl ω (ap (Sigma ω (λy2 : set ⇒ ap z (ordsucc y2) + ap u (ordsucc y2))))) (Repl ω (ap (Sigma ω (λy2 : set ⇒ ap w (ordsucc y2) + ap v (ordsucc y2))))) < x + x2
In Proofgold the corresponding term root is 533132... and proposition id is 119249...
Proof:
Proof not loaded.
End of Section Conj_real_add_SNo__9__2
Beginning of Section Conj_real_add_SNo__10__7
L2569
Variable x : set
(*** Conj_real_add_SNo__10__7 TMUhQBASZxdMD4Z6pq4y6ukZd9yGPz536pF bounty of about 25 bars ***)
L2570
Variable y : set
L2571
Variable z : set
L2572
Variable w : set
L2573
Variable u : set
L2574
Hypothesis H0 : SNo x
L2575
Hypothesis H1 : SNo y
L2576
Hypothesis H2 : SNo (x + y)
L2577
Hypothesis H3 : (∀v : set, v ∈ SNoS_ ω → (∀x2 : set, x2 ∈ ω → abs_SNo (v + - x) < eps_ x2) → v = x)
L2578
Hypothesis H4 : (∀v : set, v ∈ ω → SNo (ap (Sigma ω (λx2 : set ⇒ ap z (ordsucc x2) + ap w (ordsucc x2))) v))
L2579
Hypothesis H5 : (∀v : set, v ∈ ω → (ap (Sigma ω (λx2 : set ⇒ ap z (ordsucc x2) + ap w (ordsucc x2))) v + - (eps_ v)) < x + y)
L2580
Hypothesis H6 : SNo u
L2581
Hypothesis H8 : u ∈ SNoS_ ω
L2582
Hypothesis H9 : ¬ (∃v : set, v ∈ ω ∧ ap (Sigma ω (λx2 : set ⇒ ap z (ordsucc x2) + ap w (ordsucc x2))) v ≤ u + y)
L2583
Hypothesis H10 : Empty < u + - x
L2584
Theorem. (Conj_real_add_SNo__10__7)
u ≠ x
In Proofgold the corresponding term root is 0e8314... and proposition id is 1dbb14...
Proof:
Proof not loaded.
End of Section Conj_real_add_SNo__10__7
Beginning of Section Conj_real_add_SNo__13__2
L2590
Variable x : set
(*** Conj_real_add_SNo__13__2 TMbubgBGQb56FjAFGY6Sk9MsTyEFYBd92f4 bounty of about 25 bars ***)
L2591
Variable y : set
L2592
Variable z : set
L2593
Variable w : set
L2594
Variable u : set
L2595
Variable v : set
L2596
Variable x2 : set
L2597
Hypothesis H0 : x ∈ real
L2598
Hypothesis H1 : SNo x
L2599
Hypothesis H3 : SNo (x + y)
L2600
Hypothesis H4 : (∀y2 : set, y2 ∈ SNoS_ ω → (∀z2 : set, z2 ∈ ω → abs_SNo (y2 + - x) < eps_ z2) → y2 = x)
L2601
Hypothesis H5 : (∀y2 : set, y2 ∈ ω → SNo (ap (Sigma ω (λz2 : set ⇒ ap w (ordsucc z2) + ap v (ordsucc z2))) y2))
L2602
Hypothesis H6 : (∀y2 : set, y2 ∈ ω → (ap (Sigma ω (λz2 : set ⇒ ap w (ordsucc z2) + ap v (ordsucc z2))) y2 + - (eps_ y2)) < x + y)
L2603
Hypothesis H7 : SNo (SNoCut (Repl ω (ap (Sigma ω (λy2 : set ⇒ ap z (ordsucc y2) + ap u (ordsucc y2))))) (Repl ω (ap (Sigma ω (λy2 : set ⇒ ap w (ordsucc y2) + ap v (ordsucc y2))))))
L2604
Hypothesis H8 : (∀y2 : set, y2 ∈ Repl ω (ap (Sigma ω (λz2 : set ⇒ ap w (ordsucc z2) + ap v (ordsucc z2)))) → SNoCut (Repl ω (ap (Sigma ω (λz2 : set ⇒ ap z (ordsucc z2) + ap u (ordsucc z2))))) (Repl ω (ap (Sigma ω (λz2 : set ⇒ ap w (ordsucc z2) + ap v (ordsucc z2))))) < y2)
L2605
Hypothesis H9 : SNo x2
L2606
Hypothesis H10 : SNoLev x2 ∈ SNoLev x
L2607
Hypothesis H11 : x < x2
L2608
Theorem. (Conj_real_add_SNo__13__2)
x2 ∈ SNoS_ ω → SNoCut (Repl ω (ap (Sigma ω (λy2 : set ⇒ ap z (ordsucc y2) + ap u (ordsucc y2))))) (Repl ω (ap (Sigma ω (λy2 : set ⇒ ap w (ordsucc y2) + ap v (ordsucc y2))))) < x2 + y
In Proofgold the corresponding term root is b1d3ef... and proposition id is a6e591...
Proof:
Proof not loaded.
End of Section Conj_real_add_SNo__13__2
Beginning of Section Conj_real_add_SNo__13__3
L2614
Variable x : set
(*** Conj_real_add_SNo__13__3 TMJjAfBJR2BCEBpjDvfZJcHAPeoXeYFCGsH bounty of about 25 bars ***)
L2615
Variable y : set
L2616
Variable z : set
L2617
Variable w : set
L2618
Variable u : set
L2619
Variable v : set
L2620
Variable x2 : set
L2621
Hypothesis H0 : x ∈ real
L2622
Hypothesis H1 : SNo x
L2623
Hypothesis H2 : SNo y
L2624
Hypothesis H4 : (∀y2 : set, y2 ∈ SNoS_ ω → (∀z2 : set, z2 ∈ ω → abs_SNo (y2 + - x) < eps_ z2) → y2 = x)
L2625
Hypothesis H5 : (∀y2 : set, y2 ∈ ω → SNo (ap (Sigma ω (λz2 : set ⇒ ap w (ordsucc z2) + ap v (ordsucc z2))) y2))
L2626
Hypothesis H6 : (∀y2 : set, y2 ∈ ω → (ap (Sigma ω (λz2 : set ⇒ ap w (ordsucc z2) + ap v (ordsucc z2))) y2 + - (eps_ y2)) < x + y)
L2627
Hypothesis H7 : SNo (SNoCut (Repl ω (ap (Sigma ω (λy2 : set ⇒ ap z (ordsucc y2) + ap u (ordsucc y2))))) (Repl ω (ap (Sigma ω (λy2 : set ⇒ ap w (ordsucc y2) + ap v (ordsucc y2))))))
L2628
Hypothesis H8 : (∀y2 : set, y2 ∈ Repl ω (ap (Sigma ω (λz2 : set ⇒ ap w (ordsucc z2) + ap v (ordsucc z2)))) → SNoCut (Repl ω (ap (Sigma ω (λz2 : set ⇒ ap z (ordsucc z2) + ap u (ordsucc z2))))) (Repl ω (ap (Sigma ω (λz2 : set ⇒ ap w (ordsucc z2) + ap v (ordsucc z2))))) < y2)
L2629
Hypothesis H9 : SNo x2
L2630
Hypothesis H10 : SNoLev x2 ∈ SNoLev x
L2631
Hypothesis H11 : x < x2
L2632
Theorem. (Conj_real_add_SNo__13__3)
x2 ∈ SNoS_ ω → SNoCut (Repl ω (ap (Sigma ω (λy2 : set ⇒ ap z (ordsucc y2) + ap u (ordsucc y2))))) (Repl ω (ap (Sigma ω (λy2 : set ⇒ ap w (ordsucc y2) + ap v (ordsucc y2))))) < x2 + y
In Proofgold the corresponding term root is c788ff... and proposition id is b04447...
Proof:
Proof not loaded.
End of Section Conj_real_add_SNo__13__3
Beginning of Section Conj_real_add_SNo__18__2
L2638
Variable x : set
(*** Conj_real_add_SNo__18__2 TMNgBu2JEvCZcB9wgFUetHRjWVnSmsQgvWj bounty of about 25 bars ***)
L2639
Variable y : set
L2640
Variable z : set
L2641
Variable w : set
L2642
Variable u : set
L2643
Hypothesis H0 : SNo x
L2644
Hypothesis H1 : SNo y
L2645
Hypothesis H3 : (∀v : set, v ∈ SNoS_ ω → (∀x2 : set, x2 ∈ ω → abs_SNo (v + - x) < eps_ x2) → v = x)
L2646
Hypothesis H4 : (∀v : set, v ∈ ω → SNo (ap (Sigma ω (λx2 : set ⇒ ap z (ordsucc x2) + ap w (ordsucc x2))) v))
L2647
Hypothesis H5 : (∀v : set, v ∈ ω → (x + y) < ap (Sigma ω (λx2 : set ⇒ ap z (ordsucc x2) + ap w (ordsucc x2))) v + eps_ v)
L2648
Hypothesis H6 : SNo u
L2649
Hypothesis H7 : u < x
L2650
Hypothesis H8 : u ∈ SNoS_ ω
L2651
Hypothesis H9 : ¬ (∃v : set, v ∈ ω ∧ (u + y) ≤ ap (Sigma ω (λx2 : set ⇒ ap z (ordsucc x2) + ap w (ordsucc x2))) v)
L2652
Hypothesis H10 : Empty < x + - u
L2653
Theorem. (Conj_real_add_SNo__18__2)
u ≠ x
In Proofgold the corresponding term root is c5e72f... and proposition id is fe17b4...
Proof:
Proof not loaded.
End of Section Conj_real_add_SNo__18__2
Beginning of Section Conj_real_add_SNo__18__6
L2659
Variable x : set
(*** Conj_real_add_SNo__18__6 TMZUF197iJ9oJKmRN5swpsLonb89982xgBh bounty of about 25 bars ***)
L2660
Variable y : set
L2661
Variable z : set
L2662
Variable w : set
L2663
Variable u : set
L2664
Hypothesis H0 : SNo x
L2665
Hypothesis H1 : SNo y
L2666
Hypothesis H2 : SNo (x + y)
L2667
Hypothesis H3 : (∀v : set, v ∈ SNoS_ ω → (∀x2 : set, x2 ∈ ω → abs_SNo (v + - x) < eps_ x2) → v = x)
L2668
Hypothesis H4 : (∀v : set, v ∈ ω → SNo (ap (Sigma ω (λx2 : set ⇒ ap z (ordsucc x2) + ap w (ordsucc x2))) v))
L2669
Hypothesis H5 : (∀v : set, v ∈ ω → (x + y) < ap (Sigma ω (λx2 : set ⇒ ap z (ordsucc x2) + ap w (ordsucc x2))) v + eps_ v)
L2670
Hypothesis H7 : u < x
L2671
Hypothesis H8 : u ∈ SNoS_ ω
L2672
Hypothesis H9 : ¬ (∃v : set, v ∈ ω ∧ (u + y) ≤ ap (Sigma ω (λx2 : set ⇒ ap z (ordsucc x2) + ap w (ordsucc x2))) v)
L2673
Hypothesis H10 : Empty < x + - u
L2674
Theorem. (Conj_real_add_SNo__18__6)
u ≠ x
In Proofgold the corresponding term root is fd9878... and proposition id is 355567...
Proof:
Proof not loaded.
End of Section Conj_real_add_SNo__18__6
Beginning of Section Conj_real_add_SNo__22__6
L2680
Variable x : set
(*** Conj_real_add_SNo__22__6 TMMfHY9DHkkQLhfuL385WD4Pd4PNjpGi2py bounty of about 25 bars ***)
L2681
Variable y : set
L2682
Variable z : set
L2683
Variable w : set
L2684
Variable u : set
L2685
Variable v : set
L2686
Hypothesis H0 : x ∈ real
L2687
Hypothesis H1 : y ∈ real
L2688
Hypothesis H2 : SNo x
L2689
Hypothesis H3 : SNo y
L2690
Hypothesis H4 : SNo (x + y)
L2691
Hypothesis H5 : (∀x2 : set, x2 ∈ SNoS_ ω → (∀y2 : set, y2 ∈ ω → abs_SNo (x2 + - x) < eps_ y2) → x2 = x)
L2692
Hypothesis H7 : (∀x2 : set, x2 ∈ ω → SNo (ap (Sigma ω (λy2 : set ⇒ ap z (ordsucc y2) + ap u (ordsucc y2))) x2))
L2693
Hypothesis H8 : (∀x2 : set, x2 ∈ ω → SNo (ap (Sigma ω (λy2 : set ⇒ ap w (ordsucc y2) + ap v (ordsucc y2))) x2))
L2694
Hypothesis H9 : Sigma ω (λx2 : set ⇒ ap z (ordsucc x2) + ap u (ordsucc x2)) ∈ setexp (SNoS_ ω) ω
L2695
Hypothesis H10 : Sigma ω (λx2 : set ⇒ ap w (ordsucc x2) + ap v (ordsucc x2)) ∈ setexp (SNoS_ ω) ω
L2696
Hypothesis H11 : (∀x2 : set, x2 ∈ ω → ap (Sigma ω (λy2 : set ⇒ ap z (ordsucc y2) + ap u (ordsucc y2))) x2 < x + y)
L2697
Hypothesis H12 : (∀x2 : set, x2 ∈ ω → (x + y) < ap (Sigma ω (λy2 : set ⇒ ap z (ordsucc y2) + ap u (ordsucc y2))) x2 + eps_ x2)
L2698
Hypothesis H13 : (∀x2 : set, x2 ∈ ω → (∀y2 : set, y2 ∈ x2 → ap (Sigma ω (λz2 : set ⇒ ap z (ordsucc z2) + ap u (ordsucc z2))) y2 < ap (Sigma ω (λz2 : set ⇒ ap z (ordsucc z2) + ap u (ordsucc z2))) x2))
L2699
Hypothesis H14 : (∀x2 : set, x2 ∈ ω → (ap (Sigma ω (λy2 : set ⇒ ap w (ordsucc y2) + ap v (ordsucc y2))) x2 + - (eps_ x2)) < x + y)
L2700
Hypothesis H15 : (∀x2 : set, x2 ∈ ω → (x + y) < ap (Sigma ω (λy2 : set ⇒ ap w (ordsucc y2) + ap v (ordsucc y2))) x2)
L2701
Hypothesis H16 : (∀x2 : set, x2 ∈ ω → (∀y2 : set, y2 ∈ x2 → ap (Sigma ω (λz2 : set ⇒ ap w (ordsucc z2) + ap v (ordsucc z2))) x2 < ap (Sigma ω (λz2 : set ⇒ ap w (ordsucc z2) + ap v (ordsucc z2))) y2))
L2702
Hypothesis H17 : SNoCutP (Repl ω (ap (Sigma ω (λx2 : set ⇒ ap z (ordsucc x2) + ap u (ordsucc x2))))) (Repl ω (ap (Sigma ω (λx2 : set ⇒ ap w (ordsucc x2) + ap v (ordsucc x2)))))
L2703
Hypothesis H18 : SNo (SNoCut (Repl ω (ap (Sigma ω (λx2 : set ⇒ ap z (ordsucc x2) + ap u (ordsucc x2))))) (Repl ω (ap (Sigma ω (λx2 : set ⇒ ap w (ordsucc x2) + ap v (ordsucc x2))))))
L2704
Hypothesis H19 : (∀x2 : set, x2 ∈ Repl ω (ap (Sigma ω (λy2 : set ⇒ ap z (ordsucc y2) + ap u (ordsucc y2)))) → x2 < SNoCut (Repl ω (ap (Sigma ω (λy2 : set ⇒ ap z (ordsucc y2) + ap u (ordsucc y2))))) (Repl ω (ap (Sigma ω (λy2 : set ⇒ ap w (ordsucc y2) + ap v (ordsucc y2))))))
L2705
Hypothesis H20 : (∀x2 : set, x2 ∈ Repl ω (ap (Sigma ω (λy2 : set ⇒ ap w (ordsucc y2) + ap v (ordsucc y2)))) → SNoCut (Repl ω (ap (Sigma ω (λy2 : set ⇒ ap z (ordsucc y2) + ap u (ordsucc y2))))) (Repl ω (ap (Sigma ω (λy2 : set ⇒ ap w (ordsucc y2) + ap v (ordsucc y2))))) < x2)
L2706
Theorem. (Conj_real_add_SNo__22__6)
x + y = SNoCut (Repl ω (ap (Sigma ω (λx2 : set ⇒ ap z (ordsucc x2) + ap u (ordsucc x2))))) (Repl ω (ap (Sigma ω (λx2 : set ⇒ ap w (ordsucc x2) + ap v (ordsucc x2))))) → x + y ∈ real
In Proofgold the corresponding term root is f14600... and proposition id is 633adf...
Proof:
Proof not loaded.
End of Section Conj_real_add_SNo__22__6
Beginning of Section Conj_real_add_SNo__22__8
L2712
Variable x : set
(*** Conj_real_add_SNo__22__8 TMRE8qyWbSNbQhjhtxHrBmNEJWaW3aiz3QQ bounty of about 25 bars ***)
L2713
Variable y : set
L2714
Variable z : set
L2715
Variable w : set
L2716
Variable u : set
L2717
Variable v : set
L2718
Hypothesis H0 : x ∈ real
L2719
Hypothesis H1 : y ∈ real
L2720
Hypothesis H2 : SNo x
L2721
Hypothesis H3 : SNo y
L2722
Hypothesis H4 : SNo (x + y)
L2723
Hypothesis H5 : (∀x2 : set, x2 ∈ SNoS_ ω → (∀y2 : set, y2 ∈ ω → abs_SNo (x2 + - x) < eps_ y2) → x2 = x)
L2724
Hypothesis H6 : (∀x2 : set, x2 ∈ SNoS_ ω → (∀y2 : set, y2 ∈ ω → abs_SNo (x2 + - y) < eps_ y2) → x2 = y)
L2725
Hypothesis H7 : (∀x2 : set, x2 ∈ ω → SNo (ap (Sigma ω (λy2 : set ⇒ ap z (ordsucc y2) + ap u (ordsucc y2))) x2))
L2726
Hypothesis H9 : Sigma ω (λx2 : set ⇒ ap z (ordsucc x2) + ap u (ordsucc x2)) ∈ setexp (SNoS_ ω) ω
L2727
Hypothesis H10 : Sigma ω (λx2 : set ⇒ ap w (ordsucc x2) + ap v (ordsucc x2)) ∈ setexp (SNoS_ ω) ω
L2728
Hypothesis H11 : (∀x2 : set, x2 ∈ ω → ap (Sigma ω (λy2 : set ⇒ ap z (ordsucc y2) + ap u (ordsucc y2))) x2 < x + y)
L2729
Hypothesis H12 : (∀x2 : set, x2 ∈ ω → (x + y) < ap (Sigma ω (λy2 : set ⇒ ap z (ordsucc y2) + ap u (ordsucc y2))) x2 + eps_ x2)
L2730
Hypothesis H13 : (∀x2 : set, x2 ∈ ω → (∀y2 : set, y2 ∈ x2 → ap (Sigma ω (λz2 : set ⇒ ap z (ordsucc z2) + ap u (ordsucc z2))) y2 < ap (Sigma ω (λz2 : set ⇒ ap z (ordsucc z2) + ap u (ordsucc z2))) x2))
L2731
Hypothesis H14 : (∀x2 : set, x2 ∈ ω → (ap (Sigma ω (λy2 : set ⇒ ap w (ordsucc y2) + ap v (ordsucc y2))) x2 + - (eps_ x2)) < x + y)
L2732
Hypothesis H15 : (∀x2 : set, x2 ∈ ω → (x + y) < ap (Sigma ω (λy2 : set ⇒ ap w (ordsucc y2) + ap v (ordsucc y2))) x2)
L2733
Hypothesis H16 : (∀x2 : set, x2 ∈ ω → (∀y2 : set, y2 ∈ x2 → ap (Sigma ω (λz2 : set ⇒ ap w (ordsucc z2) + ap v (ordsucc z2))) x2 < ap (Sigma ω (λz2 : set ⇒ ap w (ordsucc z2) + ap v (ordsucc z2))) y2))
L2734
Hypothesis H17 : SNoCutP (Repl ω (ap (Sigma ω (λx2 : set ⇒ ap z (ordsucc x2) + ap u (ordsucc x2))))) (Repl ω (ap (Sigma ω (λx2 : set ⇒ ap w (ordsucc x2) + ap v (ordsucc x2)))))
L2735
Hypothesis H18 : SNo (SNoCut (Repl ω (ap (Sigma ω (λx2 : set ⇒ ap z (ordsucc x2) + ap u (ordsucc x2))))) (Repl ω (ap (Sigma ω (λx2 : set ⇒ ap w (ordsucc x2) + ap v (ordsucc x2))))))
L2736
Hypothesis H19 : (∀x2 : set, x2 ∈ Repl ω (ap (Sigma ω (λy2 : set ⇒ ap z (ordsucc y2) + ap u (ordsucc y2)))) → x2 < SNoCut (Repl ω (ap (Sigma ω (λy2 : set ⇒ ap z (ordsucc y2) + ap u (ordsucc y2))))) (Repl ω (ap (Sigma ω (λy2 : set ⇒ ap w (ordsucc y2) + ap v (ordsucc y2))))))
L2737
Hypothesis H20 : (∀x2 : set, x2 ∈ Repl ω (ap (Sigma ω (λy2 : set ⇒ ap w (ordsucc y2) + ap v (ordsucc y2)))) → SNoCut (Repl ω (ap (Sigma ω (λy2 : set ⇒ ap z (ordsucc y2) + ap u (ordsucc y2))))) (Repl ω (ap (Sigma ω (λy2 : set ⇒ ap w (ordsucc y2) + ap v (ordsucc y2))))) < x2)
L2738
Theorem. (Conj_real_add_SNo__22__8)
x + y = SNoCut (Repl ω (ap (Sigma ω (λx2 : set ⇒ ap z (ordsucc x2) + ap u (ordsucc x2))))) (Repl ω (ap (Sigma ω (λx2 : set ⇒ ap w (ordsucc x2) + ap v (ordsucc x2))))) → x + y ∈ real
In Proofgold the corresponding term root is e2c1b2... and proposition id is bc21b3...
Proof:
Proof not loaded.
End of Section Conj_real_add_SNo__22__8
Beginning of Section Conj_real_add_SNo__23__0
L2744
Variable x : set
(*** Conj_real_add_SNo__23__0 TMHfk5V5QAMcSuyEq9sd48Zx5om76E74M4b bounty of about 25 bars ***)
L2745
Variable y : set
L2746
Variable z : set
L2747
Variable w : set
L2748
Variable u : set
L2749
Variable v : set
L2750
Hypothesis H1 : y ∈ real
L2751
Hypothesis H2 : SNo x
L2752
Hypothesis H3 : SNo y
L2753
Hypothesis H4 : SNo (x + y)
L2754
Hypothesis H5 : (∀x2 : set, x2 ∈ SNoS_ ω → (∀y2 : set, y2 ∈ ω → abs_SNo (x2 + - x) < eps_ y2) → x2 = x)
L2755
Hypothesis H6 : (∀x2 : set, x2 ∈ SNoS_ ω → (∀y2 : set, y2 ∈ ω → abs_SNo (x2 + - y) < eps_ y2) → x2 = y)
L2756
Hypothesis H7 : (∀x2 : set, x2 ∈ ω → SNo (ap (Sigma ω (λy2 : set ⇒ ap z (ordsucc y2) + ap u (ordsucc y2))) x2))
L2757
Hypothesis H8 : (∀x2 : set, x2 ∈ ω → SNo (ap (Sigma ω (λy2 : set ⇒ ap w (ordsucc y2) + ap v (ordsucc y2))) x2))
L2758
Hypothesis H9 : Sigma ω (λx2 : set ⇒ ap z (ordsucc x2) + ap u (ordsucc x2)) ∈ setexp (SNoS_ ω) ω
L2759
Hypothesis H10 : Sigma ω (λx2 : set ⇒ ap w (ordsucc x2) + ap v (ordsucc x2)) ∈ setexp (SNoS_ ω) ω
L2760
Hypothesis H11 : (∀x2 : set, x2 ∈ ω → ap (Sigma ω (λy2 : set ⇒ ap z (ordsucc y2) + ap u (ordsucc y2))) x2 < x + y)
L2761
Hypothesis H12 : (∀x2 : set, x2 ∈ ω → (x + y) < ap (Sigma ω (λy2 : set ⇒ ap z (ordsucc y2) + ap u (ordsucc y2))) x2 + eps_ x2)
L2762
Hypothesis H13 : (∀x2 : set, x2 ∈ ω → (∀y2 : set, y2 ∈ x2 → ap (Sigma ω (λz2 : set ⇒ ap z (ordsucc z2) + ap u (ordsucc z2))) y2 < ap (Sigma ω (λz2 : set ⇒ ap z (ordsucc z2) + ap u (ordsucc z2))) x2))
L2763
Hypothesis H14 : (∀x2 : set, x2 ∈ ω → (ap (Sigma ω (λy2 : set ⇒ ap w (ordsucc y2) + ap v (ordsucc y2))) x2 + - (eps_ x2)) < x + y)
L2764
Hypothesis H15 : (∀x2 : set, x2 ∈ ω → (x + y) < ap (Sigma ω (λy2 : set ⇒ ap w (ordsucc y2) + ap v (ordsucc y2))) x2)
L2765
Hypothesis H16 : (∀x2 : set, x2 ∈ ω → (∀y2 : set, y2 ∈ x2 → ap (Sigma ω (λz2 : set ⇒ ap w (ordsucc z2) + ap v (ordsucc z2))) x2 < ap (Sigma ω (λz2 : set ⇒ ap w (ordsucc z2) + ap v (ordsucc z2))) y2))
L2766
Theorem. (Conj_real_add_SNo__23__0)
SNoCutP (Repl ω (ap (Sigma ω (λx2 : set ⇒ ap z (ordsucc x2) + ap u (ordsucc x2))))) (Repl ω (ap (Sigma ω (λx2 : set ⇒ ap w (ordsucc x2) + ap v (ordsucc x2))))) → x + y ∈ real
In Proofgold the corresponding term root is 9b193e... and proposition id is ec1d04...
Proof:
Proof not loaded.
End of Section Conj_real_add_SNo__23__0
Beginning of Section Conj_real_add_SNo__23__14
L2772
Variable x : set
(*** Conj_real_add_SNo__23__14 TMP8AGa3dn6TjgAXCz4z6sWPX44PJ7TRRdy bounty of about 25 bars ***)
L2773
Variable y : set
L2774
Variable z : set
L2775
Variable w : set
L2776
Variable u : set
L2777
Variable v : set
L2778
Hypothesis H0 : x ∈ real
L2779
Hypothesis H1 : y ∈ real
L2780
Hypothesis H2 : SNo x
L2781
Hypothesis H3 : SNo y
L2782
Hypothesis H4 : SNo (x + y)
L2783
Hypothesis H5 : (∀x2 : set, x2 ∈ SNoS_ ω → (∀y2 : set, y2 ∈ ω → abs_SNo (x2 + - x) < eps_ y2) → x2 = x)
L2784
Hypothesis H6 : (∀x2 : set, x2 ∈ SNoS_ ω → (∀y2 : set, y2 ∈ ω → abs_SNo (x2 + - y) < eps_ y2) → x2 = y)
L2785
Hypothesis H7 : (∀x2 : set, x2 ∈ ω → SNo (ap (Sigma ω (λy2 : set ⇒ ap z (ordsucc y2) + ap u (ordsucc y2))) x2))
L2786
Hypothesis H8 : (∀x2 : set, x2 ∈ ω → SNo (ap (Sigma ω (λy2 : set ⇒ ap w (ordsucc y2) + ap v (ordsucc y2))) x2))
L2787
Hypothesis H9 : Sigma ω (λx2 : set ⇒ ap z (ordsucc x2) + ap u (ordsucc x2)) ∈ setexp (SNoS_ ω) ω
L2788
Hypothesis H10 : Sigma ω (λx2 : set ⇒ ap w (ordsucc x2) + ap v (ordsucc x2)) ∈ setexp (SNoS_ ω) ω
L2789
Hypothesis H11 : (∀x2 : set, x2 ∈ ω → ap (Sigma ω (λy2 : set ⇒ ap z (ordsucc y2) + ap u (ordsucc y2))) x2 < x + y)
L2790
Hypothesis H12 : (∀x2 : set, x2 ∈ ω → (x + y) < ap (Sigma ω (λy2 : set ⇒ ap z (ordsucc y2) + ap u (ordsucc y2))) x2 + eps_ x2)
L2791
Hypothesis H13 : (∀x2 : set, x2 ∈ ω → (∀y2 : set, y2 ∈ x2 → ap (Sigma ω (λz2 : set ⇒ ap z (ordsucc z2) + ap u (ordsucc z2))) y2 < ap (Sigma ω (λz2 : set ⇒ ap z (ordsucc z2) + ap u (ordsucc z2))) x2))
L2792
Hypothesis H15 : (∀x2 : set, x2 ∈ ω → (x + y) < ap (Sigma ω (λy2 : set ⇒ ap w (ordsucc y2) + ap v (ordsucc y2))) x2)
L2793
Hypothesis H16 : (∀x2 : set, x2 ∈ ω → (∀y2 : set, y2 ∈ x2 → ap (Sigma ω (λz2 : set ⇒ ap w (ordsucc z2) + ap v (ordsucc z2))) x2 < ap (Sigma ω (λz2 : set ⇒ ap w (ordsucc z2) + ap v (ordsucc z2))) y2))
L2794
Theorem. (Conj_real_add_SNo__23__14)
SNoCutP (Repl ω (ap (Sigma ω (λx2 : set ⇒ ap z (ordsucc x2) + ap u (ordsucc x2))))) (Repl ω (ap (Sigma ω (λx2 : set ⇒ ap w (ordsucc x2) + ap v (ordsucc x2))))) → x + y ∈ real
In Proofgold the corresponding term root is 853eb5... and proposition id is 3aca55...
Proof:
Proof not loaded.
End of Section Conj_real_add_SNo__23__14
Beginning of Section Conj_real_add_SNo__25__18
L2800
Variable x : set
(*** Conj_real_add_SNo__25__18 TMFZ684aFGcC14gyHxsH4mnmS9fddMK7BfV bounty of about 25 bars ***)
L2801
Variable y : set
L2802
Variable z : set
L2803
Variable w : set
L2804
Variable u : set
L2805
Variable v : set
L2806
Hypothesis H0 : x ∈ real
L2807
Hypothesis H1 : y ∈ real
L2808
Hypothesis H2 : (∀x2 : set, x2 ∈ ω → x < ap w x2)
L2809
Hypothesis H3 : (∀x2 : set, x2 ∈ ω → (∀y2 : set, y2 ∈ x2 → ap w x2 < ap w y2))
L2810
Hypothesis H4 : (∀x2 : set, x2 ∈ ω → y < ap v x2)
L2811
Hypothesis H5 : (∀x2 : set, x2 ∈ ω → (∀y2 : set, y2 ∈ x2 → ap v x2 < ap v y2))
L2812
Hypothesis H6 : SNo x
L2813
Hypothesis H7 : SNo y
L2814
Hypothesis H8 : SNo (x + y)
L2815
Hypothesis H9 : (∀x2 : set, x2 ∈ SNoS_ ω → (∀y2 : set, y2 ∈ ω → abs_SNo (x2 + - x) < eps_ y2) → x2 = x)
L2816
Hypothesis H10 : (∀x2 : set, x2 ∈ SNoS_ ω → (∀y2 : set, y2 ∈ ω → abs_SNo (x2 + - y) < eps_ y2) → x2 = y)
L2817
Hypothesis H11 : (∀x2 : set, x2 ∈ ω → SNo (ap w (ordsucc x2)))
L2818
Hypothesis H12 : (∀x2 : set, x2 ∈ ω → SNo (ap v (ordsucc x2)))
L2819
Hypothesis H13 : (∀x2 : set, x2 ∈ ω → ap (Sigma ω (λy2 : set ⇒ ap w (ordsucc y2) + ap v (ordsucc y2))) x2 = ap w (ordsucc x2) + ap v (ordsucc x2))
L2820
Hypothesis H14 : (∀x2 : set, x2 ∈ ω → SNo (ap (Sigma ω (λy2 : set ⇒ ap z (ordsucc y2) + ap u (ordsucc y2))) x2))
L2821
Hypothesis H15 : (∀x2 : set, x2 ∈ ω → SNo (ap (Sigma ω (λy2 : set ⇒ ap w (ordsucc y2) + ap v (ordsucc y2))) x2))
L2822
Hypothesis H16 : Sigma ω (λx2 : set ⇒ ap z (ordsucc x2) + ap u (ordsucc x2)) ∈ setexp (SNoS_ ω) ω
L2823
Hypothesis H17 : Sigma ω (λx2 : set ⇒ ap w (ordsucc x2) + ap v (ordsucc x2)) ∈ setexp (SNoS_ ω) ω
L2824
Hypothesis H19 : (∀x2 : set, x2 ∈ ω → (x + y) < ap (Sigma ω (λy2 : set ⇒ ap z (ordsucc y2) + ap u (ordsucc y2))) x2 + eps_ x2)
L2825
Hypothesis H20 : (∀x2 : set, x2 ∈ ω → (∀y2 : set, y2 ∈ x2 → ap (Sigma ω (λz2 : set ⇒ ap z (ordsucc z2) + ap u (ordsucc z2))) y2 < ap (Sigma ω (λz2 : set ⇒ ap z (ordsucc z2) + ap u (ordsucc z2))) x2))
L2826
Hypothesis H21 : (∀x2 : set, x2 ∈ ω → (ap (Sigma ω (λy2 : set ⇒ ap w (ordsucc y2) + ap v (ordsucc y2))) x2 + - (eps_ x2)) < x + y)
L2827
Theorem. (Conj_real_add_SNo__25__18)
(∀x2 : set, x2 ∈ ω → (x + y) < ap (Sigma ω (λy2 : set ⇒ ap w (ordsucc y2) + ap v (ordsucc y2))) x2) → x + y ∈ real
In Proofgold the corresponding term root is 180fa0... and proposition id is fe42b4...
Proof:
Proof not loaded.
End of Section Conj_real_add_SNo__25__18
Beginning of Section Conj_real_add_SNo__29__8
L2833
Variable x : set
(*** Conj_real_add_SNo__29__8 TMat6shLMqqFVM477dyDXw2kK5voMr3tL9z bounty of about 25 bars ***)
L2834
Variable y : set
L2835
Variable z : set
L2836
Variable w : set
L2837
Variable u : set
L2838
Variable v : set
L2839
Hypothesis H0 : x ∈ real
L2840
Hypothesis H1 : y ∈ real
L2841
Hypothesis H2 : (∀x2 : set, x2 ∈ ω → ap z x2 < x)
L2842
Hypothesis H3 : (∀x2 : set, x2 ∈ ω → x < ap z x2 + eps_ x2)
L2843
Hypothesis H4 : (∀x2 : set, x2 ∈ ω → (∀y2 : set, y2 ∈ x2 → ap z y2 < ap z x2))
L2844
Hypothesis H5 : (∀x2 : set, x2 ∈ ω → (ap w x2 + - (eps_ x2)) < x)
L2845
Hypothesis H6 : (∀x2 : set, x2 ∈ ω → x < ap w x2)
L2846
Hypothesis H7 : (∀x2 : set, x2 ∈ ω → (∀y2 : set, y2 ∈ x2 → ap w x2 < ap w y2))
L2847
Hypothesis H9 : (∀x2 : set, x2 ∈ ω → y < ap u x2 + eps_ x2)
L2848
Hypothesis H10 : (∀x2 : set, x2 ∈ ω → (∀y2 : set, y2 ∈ x2 → ap u y2 < ap u x2))
L2849
Hypothesis H11 : (∀x2 : set, x2 ∈ ω → (ap v x2 + - (eps_ x2)) < y)
L2850
Hypothesis H12 : (∀x2 : set, x2 ∈ ω → y < ap v x2)
L2851
Hypothesis H13 : (∀x2 : set, x2 ∈ ω → (∀y2 : set, y2 ∈ x2 → ap v x2 < ap v y2))
L2852
Hypothesis H14 : SNo x
L2853
Hypothesis H15 : SNo y
L2854
Hypothesis H16 : SNo (x + y)
L2855
Hypothesis H17 : (∀x2 : set, x2 ∈ SNoS_ ω → (∀y2 : set, y2 ∈ ω → abs_SNo (x2 + - x) < eps_ y2) → x2 = x)
L2856
Hypothesis H18 : (∀x2 : set, x2 ∈ SNoS_ ω → (∀y2 : set, y2 ∈ ω → abs_SNo (x2 + - y) < eps_ y2) → x2 = y)
L2857
Hypothesis H19 : (∀x2 : set, x2 ∈ ω → SNo (ap z (ordsucc x2)))
L2858
Hypothesis H20 : (∀x2 : set, x2 ∈ ω → SNo (ap u (ordsucc x2)))
L2859
Hypothesis H21 : (∀x2 : set, x2 ∈ ω → SNo (ap w (ordsucc x2)))
L2860
Hypothesis H22 : (∀x2 : set, x2 ∈ ω → SNo (ap v (ordsucc x2)))
L2861
Hypothesis H23 : (∀x2 : set, x2 ∈ ω → ap (Sigma ω (λy2 : set ⇒ ap z (ordsucc y2) + ap u (ordsucc y2))) x2 = ap z (ordsucc x2) + ap u (ordsucc x2))
L2862
Hypothesis H24 : (∀x2 : set, x2 ∈ ω → ap (Sigma ω (λy2 : set ⇒ ap w (ordsucc y2) + ap v (ordsucc y2))) x2 = ap w (ordsucc x2) + ap v (ordsucc x2))
L2863
Hypothesis H25 : (∀x2 : set, x2 ∈ ω → SNo (ap (Sigma ω (λy2 : set ⇒ ap z (ordsucc y2) + ap u (ordsucc y2))) x2))
L2864
Hypothesis H26 : (∀x2 : set, x2 ∈ ω → SNo (ap (Sigma ω (λy2 : set ⇒ ap w (ordsucc y2) + ap v (ordsucc y2))) x2))
L2865
Hypothesis H27 : Sigma ω (λx2 : set ⇒ ap z (ordsucc x2) + ap u (ordsucc x2)) ∈ setexp (SNoS_ ω) ω
L2866
Hypothesis H28 : Sigma ω (λx2 : set ⇒ ap w (ordsucc x2) + ap v (ordsucc x2)) ∈ setexp (SNoS_ ω) ω
L2867
Theorem. (Conj_real_add_SNo__29__8)
(∀x2 : set, x2 ∈ ω → ap (Sigma ω (λy2 : set ⇒ ap z (ordsucc y2) + ap u (ordsucc y2))) x2 < x + y) → x + y ∈ real
In Proofgold the corresponding term root is aefff8... and proposition id is 940e8b...
Proof:
Proof not loaded.
End of Section Conj_real_add_SNo__29__8
Beginning of Section Conj_real_add_SNo__29__22
L2873
Variable x : set
(*** Conj_real_add_SNo__29__22 TMHG7ebV2M3ohAivbc5fb6xN5HdV12rWTmj bounty of about 25 bars ***)
L2874
Variable y : set
L2875
Variable z : set
L2876
Variable w : set
L2877
Variable u : set
L2878
Variable v : set
L2879
Hypothesis H0 : x ∈ real
L2880
Hypothesis H1 : y ∈ real
L2881
Hypothesis H2 : (∀x2 : set, x2 ∈ ω → ap z x2 < x)
L2882
Hypothesis H3 : (∀x2 : set, x2 ∈ ω → x < ap z x2 + eps_ x2)
L2883
Hypothesis H4 : (∀x2 : set, x2 ∈ ω → (∀y2 : set, y2 ∈ x2 → ap z y2 < ap z x2))
L2884
Hypothesis H5 : (∀x2 : set, x2 ∈ ω → (ap w x2 + - (eps_ x2)) < x)
L2885
Hypothesis H6 : (∀x2 : set, x2 ∈ ω → x < ap w x2)
L2886
Hypothesis H7 : (∀x2 : set, x2 ∈ ω → (∀y2 : set, y2 ∈ x2 → ap w x2 < ap w y2))
L2887
Hypothesis H8 : (∀x2 : set, x2 ∈ ω → ap u x2 < y)
L2888
Hypothesis H9 : (∀x2 : set, x2 ∈ ω → y < ap u x2 + eps_ x2)
L2889
Hypothesis H10 : (∀x2 : set, x2 ∈ ω → (∀y2 : set, y2 ∈ x2 → ap u y2 < ap u x2))
L2890
Hypothesis H11 : (∀x2 : set, x2 ∈ ω → (ap v x2 + - (eps_ x2)) < y)
L2891
Hypothesis H12 : (∀x2 : set, x2 ∈ ω → y < ap v x2)
L2892
Hypothesis H13 : (∀x2 : set, x2 ∈ ω → (∀y2 : set, y2 ∈ x2 → ap v x2 < ap v y2))
L2893
Hypothesis H14 : SNo x
L2894
Hypothesis H15 : SNo y
L2895
Hypothesis H16 : SNo (x + y)
L2896
Hypothesis H17 : (∀x2 : set, x2 ∈ SNoS_ ω → (∀y2 : set, y2 ∈ ω → abs_SNo (x2 + - x) < eps_ y2) → x2 = x)
L2897
Hypothesis H18 : (∀x2 : set, x2 ∈ SNoS_ ω → (∀y2 : set, y2 ∈ ω → abs_SNo (x2 + - y) < eps_ y2) → x2 = y)
L2898
Hypothesis H19 : (∀x2 : set, x2 ∈ ω → SNo (ap z (ordsucc x2)))
L2899
Hypothesis H20 : (∀x2 : set, x2 ∈ ω → SNo (ap u (ordsucc x2)))
L2900
Hypothesis H21 : (∀x2 : set, x2 ∈ ω → SNo (ap w (ordsucc x2)))
L2901
Hypothesis H23 : (∀x2 : set, x2 ∈ ω → ap (Sigma ω (λy2 : set ⇒ ap z (ordsucc y2) + ap u (ordsucc y2))) x2 = ap z (ordsucc x2) + ap u (ordsucc x2))
L2902
Hypothesis H24 : (∀x2 : set, x2 ∈ ω → ap (Sigma ω (λy2 : set ⇒ ap w (ordsucc y2) + ap v (ordsucc y2))) x2 = ap w (ordsucc x2) + ap v (ordsucc x2))
L2903
Hypothesis H25 : (∀x2 : set, x2 ∈ ω → SNo (ap (Sigma ω (λy2 : set ⇒ ap z (ordsucc y2) + ap u (ordsucc y2))) x2))
L2904
Hypothesis H26 : (∀x2 : set, x2 ∈ ω → SNo (ap (Sigma ω (λy2 : set ⇒ ap w (ordsucc y2) + ap v (ordsucc y2))) x2))
L2905
Hypothesis H27 : Sigma ω (λx2 : set ⇒ ap z (ordsucc x2) + ap u (ordsucc x2)) ∈ setexp (SNoS_ ω) ω
L2906
Hypothesis H28 : Sigma ω (λx2 : set ⇒ ap w (ordsucc x2) + ap v (ordsucc x2)) ∈ setexp (SNoS_ ω) ω
L2907
Theorem. (Conj_real_add_SNo__29__22)
(∀x2 : set, x2 ∈ ω → ap (Sigma ω (λy2 : set ⇒ ap z (ordsucc y2) + ap u (ordsucc y2))) x2 < x + y) → x + y ∈ real
In Proofgold the corresponding term root is 625dd4... and proposition id is 88ba1a...
Proof:
Proof not loaded.
End of Section Conj_real_add_SNo__29__22
Beginning of Section Conj_real_add_SNo__30__3
L2913
Variable x : set
(*** Conj_real_add_SNo__30__3 TMPv6ZcMeAauNtjJDDYXdy6WYmh7ae1r7G6 bounty of about 25 bars ***)
L2914
Variable y : set
L2915
Variable z : set
L2916
Variable w : set
L2917
Variable u : set
L2918
Variable v : set
L2919
Hypothesis H0 : x ∈ real
L2920
Hypothesis H1 : y ∈ real
L2921
Hypothesis H2 : (∀x2 : set, x2 ∈ ω → ap z x2 < x)
L2922
Hypothesis H4 : (∀x2 : set, x2 ∈ ω → (∀y2 : set, y2 ∈ x2 → ap z y2 < ap z x2))
L2923
Hypothesis H5 : (∀x2 : set, x2 ∈ ω → (ap w x2 + - (eps_ x2)) < x)
L2924
Hypothesis H6 : (∀x2 : set, x2 ∈ ω → x < ap w x2)
L2925
Hypothesis H7 : (∀x2 : set, x2 ∈ ω → (∀y2 : set, y2 ∈ x2 → ap w x2 < ap w y2))
L2926
Hypothesis H8 : (∀x2 : set, x2 ∈ ω → ap u x2 < y)
L2927
Hypothesis H9 : (∀x2 : set, x2 ∈ ω → y < ap u x2 + eps_ x2)
L2928
Hypothesis H10 : (∀x2 : set, x2 ∈ ω → (∀y2 : set, y2 ∈ x2 → ap u y2 < ap u x2))
L2929
Hypothesis H11 : (∀x2 : set, x2 ∈ ω → (ap v x2 + - (eps_ x2)) < y)
L2930
Hypothesis H12 : (∀x2 : set, x2 ∈ ω → y < ap v x2)
L2931
Hypothesis H13 : (∀x2 : set, x2 ∈ ω → (∀y2 : set, y2 ∈ x2 → ap v x2 < ap v y2))
L2932
Hypothesis H14 : SNo x
L2933
Hypothesis H15 : SNo y
L2934
Hypothesis H16 : SNo (x + y)
L2935
Hypothesis H17 : (∀x2 : set, x2 ∈ SNoS_ ω → (∀y2 : set, y2 ∈ ω → abs_SNo (x2 + - x) < eps_ y2) → x2 = x)
L2936
Hypothesis H18 : (∀x2 : set, x2 ∈ SNoS_ ω → (∀y2 : set, y2 ∈ ω → abs_SNo (x2 + - y) < eps_ y2) → x2 = y)
L2937
Hypothesis H19 : (∀x2 : set, x2 ∈ ω → SNo (ap z (ordsucc x2)))
L2938
Hypothesis H20 : (∀x2 : set, x2 ∈ ω → SNo (ap u (ordsucc x2)))
L2939
Hypothesis H21 : (∀x2 : set, x2 ∈ ω → ap w (ordsucc x2) ∈ SNoS_ ω)
L2940
Hypothesis H22 : (∀x2 : set, x2 ∈ ω → SNo (ap w (ordsucc x2)))
L2941
Hypothesis H23 : (∀x2 : set, x2 ∈ ω → ap v (ordsucc x2) ∈ SNoS_ ω)
L2942
Hypothesis H24 : (∀x2 : set, x2 ∈ ω → SNo (ap v (ordsucc x2)))
L2943
Hypothesis H25 : (∀x2 : set, x2 ∈ ω → ap (Sigma ω (λy2 : set ⇒ ap z (ordsucc y2) + ap u (ordsucc y2))) x2 = ap z (ordsucc x2) + ap u (ordsucc x2))
L2944
Hypothesis H26 : (∀x2 : set, x2 ∈ ω → ap (Sigma ω (λy2 : set ⇒ ap w (ordsucc y2) + ap v (ordsucc y2))) x2 = ap w (ordsucc x2) + ap v (ordsucc x2))
L2945
Hypothesis H27 : (∀x2 : set, x2 ∈ ω → SNo (ap (Sigma ω (λy2 : set ⇒ ap z (ordsucc y2) + ap u (ordsucc y2))) x2))
L2946
Hypothesis H28 : (∀x2 : set, x2 ∈ ω → SNo (ap (Sigma ω (λy2 : set ⇒ ap w (ordsucc y2) + ap v (ordsucc y2))) x2))
L2947
Hypothesis H29 : Sigma ω (λx2 : set ⇒ ap z (ordsucc x2) + ap u (ordsucc x2)) ∈ setexp (SNoS_ ω) ω
L2948
Theorem. (Conj_real_add_SNo__30__3)
Sigma ω (λx2 : set ⇒ ap w (ordsucc x2) + ap v (ordsucc x2)) ∈ setexp (SNoS_ ω) ω → x + y ∈ real
In Proofgold the corresponding term root is ca1130... and proposition id is d0238c...
Proof:
Proof not loaded.
End of Section Conj_real_add_SNo__30__3
Beginning of Section Conj_real_add_SNo__30__7
L2954
Variable x : set
(*** Conj_real_add_SNo__30__7 TMU8GgMdeAeWc4FaLPrRbCFG5jRWPRw5i52 bounty of about 25 bars ***)
L2955
Variable y : set
L2956
Variable z : set
L2957
Variable w : set
L2958
Variable u : set
L2959
Variable v : set
L2960
Hypothesis H0 : x ∈ real
L2961
Hypothesis H1 : y ∈ real
L2962
Hypothesis H2 : (∀x2 : set, x2 ∈ ω → ap z x2 < x)
L2963
Hypothesis H3 : (∀x2 : set, x2 ∈ ω → x < ap z x2 + eps_ x2)
L2964
Hypothesis H4 : (∀x2 : set, x2 ∈ ω → (∀y2 : set, y2 ∈ x2 → ap z y2 < ap z x2))
L2965
Hypothesis H5 : (∀x2 : set, x2 ∈ ω → (ap w x2 + - (eps_ x2)) < x)
L2966
Hypothesis H6 : (∀x2 : set, x2 ∈ ω → x < ap w x2)
L2967
Hypothesis H8 : (∀x2 : set, x2 ∈ ω → ap u x2 < y)
L2968
Hypothesis H9 : (∀x2 : set, x2 ∈ ω → y < ap u x2 + eps_ x2)
L2969
Hypothesis H10 : (∀x2 : set, x2 ∈ ω → (∀y2 : set, y2 ∈ x2 → ap u y2 < ap u x2))
L2970
Hypothesis H11 : (∀x2 : set, x2 ∈ ω → (ap v x2 + - (eps_ x2)) < y)
L2971
Hypothesis H12 : (∀x2 : set, x2 ∈ ω → y < ap v x2)
L2972
Hypothesis H13 : (∀x2 : set, x2 ∈ ω → (∀y2 : set, y2 ∈ x2 → ap v x2 < ap v y2))
L2973
Hypothesis H14 : SNo x
L2974
Hypothesis H15 : SNo y
L2975
Hypothesis H16 : SNo (x + y)
L2976
Hypothesis H17 : (∀x2 : set, x2 ∈ SNoS_ ω → (∀y2 : set, y2 ∈ ω → abs_SNo (x2 + - x) < eps_ y2) → x2 = x)
L2977
Hypothesis H18 : (∀x2 : set, x2 ∈ SNoS_ ω → (∀y2 : set, y2 ∈ ω → abs_SNo (x2 + - y) < eps_ y2) → x2 = y)
L2978
Hypothesis H19 : (∀x2 : set, x2 ∈ ω → SNo (ap z (ordsucc x2)))
L2979
Hypothesis H20 : (∀x2 : set, x2 ∈ ω → SNo (ap u (ordsucc x2)))
L2980
Hypothesis H21 : (∀x2 : set, x2 ∈ ω → ap w (ordsucc x2) ∈ SNoS_ ω)
L2981
Hypothesis H22 : (∀x2 : set, x2 ∈ ω → SNo (ap w (ordsucc x2)))
L2982
Hypothesis H23 : (∀x2 : set, x2 ∈ ω → ap v (ordsucc x2) ∈ SNoS_ ω)
L2983
Hypothesis H24 : (∀x2 : set, x2 ∈ ω → SNo (ap v (ordsucc x2)))
L2984
Hypothesis H25 : (∀x2 : set, x2 ∈ ω → ap (Sigma ω (λy2 : set ⇒ ap z (ordsucc y2) + ap u (ordsucc y2))) x2 = ap z (ordsucc x2) + ap u (ordsucc x2))
L2985
Hypothesis H26 : (∀x2 : set, x2 ∈ ω → ap (Sigma ω (λy2 : set ⇒ ap w (ordsucc y2) + ap v (ordsucc y2))) x2 = ap w (ordsucc x2) + ap v (ordsucc x2))
L2986
Hypothesis H27 : (∀x2 : set, x2 ∈ ω → SNo (ap (Sigma ω (λy2 : set ⇒ ap z (ordsucc y2) + ap u (ordsucc y2))) x2))
L2987
Hypothesis H28 : (∀x2 : set, x2 ∈ ω → SNo (ap (Sigma ω (λy2 : set ⇒ ap w (ordsucc y2) + ap v (ordsucc y2))) x2))
L2988
Hypothesis H29 : Sigma ω (λx2 : set ⇒ ap z (ordsucc x2) + ap u (ordsucc x2)) ∈ setexp (SNoS_ ω) ω
L2989
Theorem. (Conj_real_add_SNo__30__7)
Sigma ω (λx2 : set ⇒ ap w (ordsucc x2) + ap v (ordsucc x2)) ∈ setexp (SNoS_ ω) ω → x + y ∈ real
In Proofgold the corresponding term root is 870747... and proposition id is 56dcca...
Proof:
Proof not loaded.
End of Section Conj_real_add_SNo__30__7
Beginning of Section Conj_real_add_SNo__30__28
L2995
Variable x : set
(*** Conj_real_add_SNo__30__28 TMY8mPx97j8budqs7BPUDFcPiFvvm8nVBer bounty of about 25 bars ***)
L2996
Variable y : set
L2997
Variable z : set
L2998
Variable w : set
L2999
Variable u : set
L3000
Variable v : set
L3001
Hypothesis H0 : x ∈ real
L3002
Hypothesis H1 : y ∈ real
L3003
Hypothesis H2 : (∀x2 : set, x2 ∈ ω → ap z x2 < x)
L3004
Hypothesis H3 : (∀x2 : set, x2 ∈ ω → x < ap z x2 + eps_ x2)
L3005
Hypothesis H4 : (∀x2 : set, x2 ∈ ω → (∀y2 : set, y2 ∈ x2 → ap z y2 < ap z x2))
L3006
Hypothesis H5 : (∀x2 : set, x2 ∈ ω → (ap w x2 + - (eps_ x2)) < x)
L3007
Hypothesis H6 : (∀x2 : set, x2 ∈ ω → x < ap w x2)
L3008
Hypothesis H7 : (∀x2 : set, x2 ∈ ω → (∀y2 : set, y2 ∈ x2 → ap w x2 < ap w y2))
L3009
Hypothesis H8 : (∀x2 : set, x2 ∈ ω → ap u x2 < y)
L3010
Hypothesis H9 : (∀x2 : set, x2 ∈ ω → y < ap u x2 + eps_ x2)
L3011
Hypothesis H10 : (∀x2 : set, x2 ∈ ω → (∀y2 : set, y2 ∈ x2 → ap u y2 < ap u x2))
L3012
Hypothesis H11 : (∀x2 : set, x2 ∈ ω → (ap v x2 + - (eps_ x2)) < y)
L3013
Hypothesis H12 : (∀x2 : set, x2 ∈ ω → y < ap v x2)
L3014
Hypothesis H13 : (∀x2 : set, x2 ∈ ω → (∀y2 : set, y2 ∈ x2 → ap v x2 < ap v y2))
L3015
Hypothesis H14 : SNo x
L3016
Hypothesis H15 : SNo y
L3017
Hypothesis H16 : SNo (x + y)
L3018
Hypothesis H17 : (∀x2 : set, x2 ∈ SNoS_ ω → (∀y2 : set, y2 ∈ ω → abs_SNo (x2 + - x) < eps_ y2) → x2 = x)
L3019
Hypothesis H18 : (∀x2 : set, x2 ∈ SNoS_ ω → (∀y2 : set, y2 ∈ ω → abs_SNo (x2 + - y) < eps_ y2) → x2 = y)
L3020
Hypothesis H19 : (∀x2 : set, x2 ∈ ω → SNo (ap z (ordsucc x2)))
L3021
Hypothesis H20 : (∀x2 : set, x2 ∈ ω → SNo (ap u (ordsucc x2)))
L3022
Hypothesis H21 : (∀x2 : set, x2 ∈ ω → ap w (ordsucc x2) ∈ SNoS_ ω)
L3023
Hypothesis H22 : (∀x2 : set, x2 ∈ ω → SNo (ap w (ordsucc x2)))
L3024
Hypothesis H23 : (∀x2 : set, x2 ∈ ω → ap v (ordsucc x2) ∈ SNoS_ ω)
L3025
Hypothesis H24 : (∀x2 : set, x2 ∈ ω → SNo (ap v (ordsucc x2)))
L3026
Hypothesis H25 : (∀x2 : set, x2 ∈ ω → ap (Sigma ω (λy2 : set ⇒ ap z (ordsucc y2) + ap u (ordsucc y2))) x2 = ap z (ordsucc x2) + ap u (ordsucc x2))
L3027
Hypothesis H26 : (∀x2 : set, x2 ∈ ω → ap (Sigma ω (λy2 : set ⇒ ap w (ordsucc y2) + ap v (ordsucc y2))) x2 = ap w (ordsucc x2) + ap v (ordsucc x2))
L3028
Hypothesis H27 : (∀x2 : set, x2 ∈ ω → SNo (ap (Sigma ω (λy2 : set ⇒ ap z (ordsucc y2) + ap u (ordsucc y2))) x2))
L3029
Hypothesis H29 : Sigma ω (λx2 : set ⇒ ap z (ordsucc x2) + ap u (ordsucc x2)) ∈ setexp (SNoS_ ω) ω
L3030
Theorem. (Conj_real_add_SNo__30__28)
Sigma ω (λx2 : set ⇒ ap w (ordsucc x2) + ap v (ordsucc x2)) ∈ setexp (SNoS_ ω) ω → x + y ∈ real
In Proofgold the corresponding term root is f10aea... and proposition id is 79361a...
Proof:
Proof not loaded.
End of Section Conj_real_add_SNo__30__28
Beginning of Section Conj_real_add_SNo__31__24
L3036
Variable x : set
(*** Conj_real_add_SNo__31__24 TMcukGxZvH5Gt5hxymjC34kVidJdBSYWzYT bounty of about 25 bars ***)
L3037
Variable y : set
L3038
Variable z : set
L3039
Variable w : set
L3040
Variable u : set
L3041
Variable v : set
L3042
Hypothesis H0 : x ∈ real
L3043
Hypothesis H1 : y ∈ real
L3044
Hypothesis H2 : (∀x2 : set, x2 ∈ ω → ap z x2 < x)
L3045
Hypothesis H3 : (∀x2 : set, x2 ∈ ω → x < ap z x2 + eps_ x2)
L3046
Hypothesis H4 : (∀x2 : set, x2 ∈ ω → (∀y2 : set, y2 ∈ x2 → ap z y2 < ap z x2))
L3047
Hypothesis H5 : (∀x2 : set, x2 ∈ ω → (ap w x2 + - (eps_ x2)) < x)
L3048
Hypothesis H6 : (∀x2 : set, x2 ∈ ω → x < ap w x2)
L3049
Hypothesis H7 : (∀x2 : set, x2 ∈ ω → (∀y2 : set, y2 ∈ x2 → ap w x2 < ap w y2))
L3050
Hypothesis H8 : (∀x2 : set, x2 ∈ ω → ap u x2 < y)
L3051
Hypothesis H9 : (∀x2 : set, x2 ∈ ω → y < ap u x2 + eps_ x2)
L3052
Hypothesis H10 : (∀x2 : set, x2 ∈ ω → (∀y2 : set, y2 ∈ x2 → ap u y2 < ap u x2))
L3053
Hypothesis H11 : (∀x2 : set, x2 ∈ ω → (ap v x2 + - (eps_ x2)) < y)
L3054
Hypothesis H12 : (∀x2 : set, x2 ∈ ω → y < ap v x2)
L3055
Hypothesis H13 : (∀x2 : set, x2 ∈ ω → (∀y2 : set, y2 ∈ x2 → ap v x2 < ap v y2))
L3056
Hypothesis H14 : SNo x
L3057
Hypothesis H15 : SNo y
L3058
Hypothesis H16 : SNo (x + y)
L3059
Hypothesis H17 : (∀x2 : set, x2 ∈ SNoS_ ω → (∀y2 : set, y2 ∈ ω → abs_SNo (x2 + - x) < eps_ y2) → x2 = x)
L3060
Hypothesis H18 : (∀x2 : set, x2 ∈ SNoS_ ω → (∀y2 : set, y2 ∈ ω → abs_SNo (x2 + - y) < eps_ y2) → x2 = y)
L3061
Hypothesis H19 : (∀x2 : set, x2 ∈ ω → ap z (ordsucc x2) ∈ SNoS_ ω)
L3062
Hypothesis H20 : (∀x2 : set, x2 ∈ ω → SNo (ap z (ordsucc x2)))
L3063
Hypothesis H21 : (∀x2 : set, x2 ∈ ω → ap u (ordsucc x2) ∈ SNoS_ ω)
L3064
Hypothesis H22 : (∀x2 : set, x2 ∈ ω → SNo (ap u (ordsucc x2)))
L3065
Hypothesis H23 : (∀x2 : set, x2 ∈ ω → ap w (ordsucc x2) ∈ SNoS_ ω)
L3066
Hypothesis H25 : (∀x2 : set, x2 ∈ ω → ap v (ordsucc x2) ∈ SNoS_ ω)
L3067
Hypothesis H26 : (∀x2 : set, x2 ∈ ω → SNo (ap v (ordsucc x2)))
L3068
Hypothesis H27 : (∀x2 : set, x2 ∈ ω → ap (Sigma ω (λy2 : set ⇒ ap z (ordsucc y2) + ap u (ordsucc y2))) x2 = ap z (ordsucc x2) + ap u (ordsucc x2))
L3069
Hypothesis H28 : (∀x2 : set, x2 ∈ ω → ap (Sigma ω (λy2 : set ⇒ ap w (ordsucc y2) + ap v (ordsucc y2))) x2 = ap w (ordsucc x2) + ap v (ordsucc x2))
L3070
Hypothesis H29 : (∀x2 : set, x2 ∈ ω → SNo (ap (Sigma ω (λy2 : set ⇒ ap z (ordsucc y2) + ap u (ordsucc y2))) x2))
L3071
Hypothesis H30 : (∀x2 : set, x2 ∈ ω → SNo (ap (Sigma ω (λy2 : set ⇒ ap w (ordsucc y2) + ap v (ordsucc y2))) x2))
L3072
Theorem. (Conj_real_add_SNo__31__24)
Sigma ω (λx2 : set ⇒ ap z (ordsucc x2) + ap u (ordsucc x2)) ∈ setexp (SNoS_ ω) ω → x + y ∈ real
In Proofgold the corresponding term root is 5cb212... and proposition id is c487e7...
Proof:
Proof not loaded.
End of Section Conj_real_add_SNo__31__24
Beginning of Section Conj_real_add_SNo__32__17
L3078
Variable x : set
(*** Conj_real_add_SNo__32__17 TMYhmQBdaNCQ5YvMyKYfhaqoyGa4X46LPfK bounty of about 25 bars ***)
L3079
Variable y : set
L3080
Variable z : set
L3081
Variable w : set
L3082
Variable u : set
L3083
Variable v : set
L3084
Hypothesis H0 : x ∈ real
L3085
Hypothesis H1 : y ∈ real
L3086
Hypothesis H2 : (∀x2 : set, x2 ∈ ω → ap z x2 < x)
L3087
Hypothesis H3 : (∀x2 : set, x2 ∈ ω → x < ap z x2 + eps_ x2)
L3088
Hypothesis H4 : (∀x2 : set, x2 ∈ ω → (∀y2 : set, y2 ∈ x2 → ap z y2 < ap z x2))
L3089
Hypothesis H5 : (∀x2 : set, x2 ∈ ω → (ap w x2 + - (eps_ x2)) < x)
L3090
Hypothesis H6 : (∀x2 : set, x2 ∈ ω → x < ap w x2)
L3091
Hypothesis H7 : (∀x2 : set, x2 ∈ ω → (∀y2 : set, y2 ∈ x2 → ap w x2 < ap w y2))
L3092
Hypothesis H8 : (∀x2 : set, x2 ∈ ω → ap u x2 < y)
L3093
Hypothesis H9 : (∀x2 : set, x2 ∈ ω → y < ap u x2 + eps_ x2)
L3094
Hypothesis H10 : (∀x2 : set, x2 ∈ ω → (∀y2 : set, y2 ∈ x2 → ap u y2 < ap u x2))
L3095
Hypothesis H11 : (∀x2 : set, x2 ∈ ω → (ap v x2 + - (eps_ x2)) < y)
L3096
Hypothesis H12 : (∀x2 : set, x2 ∈ ω → y < ap v x2)
L3097
Hypothesis H13 : (∀x2 : set, x2 ∈ ω → (∀y2 : set, y2 ∈ x2 → ap v x2 < ap v y2))
L3098
Hypothesis H14 : SNo x
L3099
Hypothesis H15 : SNo y
L3100
Hypothesis H16 : SNo (x + y)
L3101
Hypothesis H18 : (∀x2 : set, x2 ∈ SNoS_ ω → (∀y2 : set, y2 ∈ ω → abs_SNo (x2 + - y) < eps_ y2) → x2 = y)
L3102
Hypothesis H19 : (∀x2 : set, x2 ∈ ω → ap z (ordsucc x2) ∈ SNoS_ ω)
L3103
Hypothesis H20 : (∀x2 : set, x2 ∈ ω → SNo (ap z (ordsucc x2)))
L3104
Hypothesis H21 : (∀x2 : set, x2 ∈ ω → ap u (ordsucc x2) ∈ SNoS_ ω)
L3105
Hypothesis H22 : (∀x2 : set, x2 ∈ ω → SNo (ap u (ordsucc x2)))
L3106
Hypothesis H23 : (∀x2 : set, x2 ∈ ω → ap w (ordsucc x2) ∈ SNoS_ ω)
L3107
Hypothesis H24 : (∀x2 : set, x2 ∈ ω → SNo (ap w (ordsucc x2)))
L3108
Hypothesis H25 : (∀x2 : set, x2 ∈ ω → ap v (ordsucc x2) ∈ SNoS_ ω)
L3109
Hypothesis H26 : (∀x2 : set, x2 ∈ ω → SNo (ap v (ordsucc x2)))
L3110
Hypothesis H27 : (∀x2 : set, x2 ∈ ω → ap (Sigma ω (λy2 : set ⇒ ap z (ordsucc y2) + ap u (ordsucc y2))) x2 = ap z (ordsucc x2) + ap u (ordsucc x2))
L3111
Hypothesis H28 : (∀x2 : set, x2 ∈ ω → ap (Sigma ω (λy2 : set ⇒ ap w (ordsucc y2) + ap v (ordsucc y2))) x2 = ap w (ordsucc x2) + ap v (ordsucc x2))
L3112
Hypothesis H29 : (∀x2 : set, x2 ∈ ω → SNo (ap (Sigma ω (λy2 : set ⇒ ap z (ordsucc y2) + ap u (ordsucc y2))) x2))
L3113
Theorem. (Conj_real_add_SNo__32__17)
(∀x2 : set, x2 ∈ ω → SNo (ap (Sigma ω (λy2 : set ⇒ ap w (ordsucc y2) + ap v (ordsucc y2))) x2)) → x + y ∈ real
In Proofgold the corresponding term root is 149622... and proposition id is f4caf0...
Proof:
Proof not loaded.
End of Section Conj_real_add_SNo__32__17
Beginning of Section Conj_real_add_SNo__33__2
L3119
Variable x : set
(*** Conj_real_add_SNo__33__2 TMcV9f1TofHDH1Ap51HwQfmgPRXqyYD9yi6 bounty of about 25 bars ***)
L3120
Variable y : set
L3121
Variable z : set
L3122
Variable w : set
L3123
Variable u : set
L3124
Variable v : set
L3125
Hypothesis H0 : x ∈ real
L3126
Hypothesis H1 : y ∈ real
L3127
Hypothesis H3 : (∀x2 : set, x2 ∈ ω → x < ap z x2 + eps_ x2)
L3128
Hypothesis H4 : (∀x2 : set, x2 ∈ ω → (∀y2 : set, y2 ∈ x2 → ap z y2 < ap z x2))
L3129
Hypothesis H5 : (∀x2 : set, x2 ∈ ω → (ap w x2 + - (eps_ x2)) < x)
L3130
Hypothesis H6 : (∀x2 : set, x2 ∈ ω → x < ap w x2)
L3131
Hypothesis H7 : (∀x2 : set, x2 ∈ ω → (∀y2 : set, y2 ∈ x2 → ap w x2 < ap w y2))
L3132
Hypothesis H8 : (∀x2 : set, x2 ∈ ω → ap u x2 < y)
L3133
Hypothesis H9 : (∀x2 : set, x2 ∈ ω → y < ap u x2 + eps_ x2)
L3134
Hypothesis H10 : (∀x2 : set, x2 ∈ ω → (∀y2 : set, y2 ∈ x2 → ap u y2 < ap u x2))
L3135
Hypothesis H11 : (∀x2 : set, x2 ∈ ω → (ap v x2 + - (eps_ x2)) < y)
L3136
Hypothesis H12 : (∀x2 : set, x2 ∈ ω → y < ap v x2)
L3137
Hypothesis H13 : (∀x2 : set, x2 ∈ ω → (∀y2 : set, y2 ∈ x2 → ap v x2 < ap v y2))
L3138
Hypothesis H14 : SNo x
L3139
Hypothesis H15 : SNo y
L3140
Hypothesis H16 : SNo (x + y)
L3141
Hypothesis H17 : (∀x2 : set, x2 ∈ SNoS_ ω → (∀y2 : set, y2 ∈ ω → abs_SNo (x2 + - x) < eps_ y2) → x2 = x)
L3142
Hypothesis H18 : (∀x2 : set, x2 ∈ SNoS_ ω → (∀y2 : set, y2 ∈ ω → abs_SNo (x2 + - y) < eps_ y2) → x2 = y)
L3143
Hypothesis H19 : (∀x2 : set, x2 ∈ ω → ap z (ordsucc x2) ∈ SNoS_ ω)
L3144
Hypothesis H20 : (∀x2 : set, x2 ∈ ω → SNo (ap z (ordsucc x2)))
L3145
Hypothesis H21 : (∀x2 : set, x2 ∈ ω → ap u (ordsucc x2) ∈ SNoS_ ω)
L3146
Hypothesis H22 : (∀x2 : set, x2 ∈ ω → SNo (ap u (ordsucc x2)))
L3147
Hypothesis H23 : (∀x2 : set, x2 ∈ ω → ap w (ordsucc x2) ∈ SNoS_ ω)
L3148
Hypothesis H24 : (∀x2 : set, x2 ∈ ω → SNo (ap w (ordsucc x2)))
L3149
Hypothesis H25 : (∀x2 : set, x2 ∈ ω → ap v (ordsucc x2) ∈ SNoS_ ω)
L3150
Hypothesis H26 : (∀x2 : set, x2 ∈ ω → SNo (ap v (ordsucc x2)))
L3151
Hypothesis H27 : (∀x2 : set, x2 ∈ ω → ap (Sigma ω (λy2 : set ⇒ ap z (ordsucc y2) + ap u (ordsucc y2))) x2 = ap z (ordsucc x2) + ap u (ordsucc x2))
L3152
Hypothesis H28 : (∀x2 : set, x2 ∈ ω → ap (Sigma ω (λy2 : set ⇒ ap w (ordsucc y2) + ap v (ordsucc y2))) x2 = ap w (ordsucc x2) + ap v (ordsucc x2))
L3153
Theorem. (Conj_real_add_SNo__33__2)
(∀x2 : set, x2 ∈ ω → SNo (ap (Sigma ω (λy2 : set ⇒ ap z (ordsucc y2) + ap u (ordsucc y2))) x2)) → x + y ∈ real
In Proofgold the corresponding term root is f6f911... and proposition id is 0e33e5...
Proof:
Proof not loaded.
End of Section Conj_real_add_SNo__33__2
Beginning of Section Conj_real_add_SNo__33__11
L3159
Variable x : set
(*** Conj_real_add_SNo__33__11 TMHaXqMyFUrjWzLSLqpjxXMkTkAyLeuZt5K bounty of about 25 bars ***)
L3160
Variable y : set
L3161
Variable z : set
L3162
Variable w : set
L3163
Variable u : set
L3164
Variable v : set
L3165
Hypothesis H0 : x ∈ real
L3166
Hypothesis H1 : y ∈ real
L3167
Hypothesis H2 : (∀x2 : set, x2 ∈ ω → ap z x2 < x)
L3168
Hypothesis H3 : (∀x2 : set, x2 ∈ ω → x < ap z x2 + eps_ x2)
L3169
Hypothesis H4 : (∀x2 : set, x2 ∈ ω → (∀y2 : set, y2 ∈ x2 → ap z y2 < ap z x2))
L3170
Hypothesis H5 : (∀x2 : set, x2 ∈ ω → (ap w x2 + - (eps_ x2)) < x)
L3171
Hypothesis H6 : (∀x2 : set, x2 ∈ ω → x < ap w x2)
L3172
Hypothesis H7 : (∀x2 : set, x2 ∈ ω → (∀y2 : set, y2 ∈ x2 → ap w x2 < ap w y2))
L3173
Hypothesis H8 : (∀x2 : set, x2 ∈ ω → ap u x2 < y)
L3174
Hypothesis H9 : (∀x2 : set, x2 ∈ ω → y < ap u x2 + eps_ x2)
L3175
Hypothesis H10 : (∀x2 : set, x2 ∈ ω → (∀y2 : set, y2 ∈ x2 → ap u y2 < ap u x2))
L3176
Hypothesis H12 : (∀x2 : set, x2 ∈ ω → y < ap v x2)
L3177
Hypothesis H13 : (∀x2 : set, x2 ∈ ω → (∀y2 : set, y2 ∈ x2 → ap v x2 < ap v y2))
L3178
Hypothesis H14 : SNo x
L3179
Hypothesis H15 : SNo y
L3180
Hypothesis H16 : SNo (x + y)
L3181
Hypothesis H17 : (∀x2 : set, x2 ∈ SNoS_ ω → (∀y2 : set, y2 ∈ ω → abs_SNo (x2 + - x) < eps_ y2) → x2 = x)
L3182
Hypothesis H18 : (∀x2 : set, x2 ∈ SNoS_ ω → (∀y2 : set, y2 ∈ ω → abs_SNo (x2 + - y) < eps_ y2) → x2 = y)
L3183
Hypothesis H19 : (∀x2 : set, x2 ∈ ω → ap z (ordsucc x2) ∈ SNoS_ ω)
L3184
Hypothesis H20 : (∀x2 : set, x2 ∈ ω → SNo (ap z (ordsucc x2)))
L3185
Hypothesis H21 : (∀x2 : set, x2 ∈ ω → ap u (ordsucc x2) ∈ SNoS_ ω)
L3186
Hypothesis H22 : (∀x2 : set, x2 ∈ ω → SNo (ap u (ordsucc x2)))
L3187
Hypothesis H23 : (∀x2 : set, x2 ∈ ω → ap w (ordsucc x2) ∈ SNoS_ ω)
L3188
Hypothesis H24 : (∀x2 : set, x2 ∈ ω → SNo (ap w (ordsucc x2)))
L3189
Hypothesis H25 : (∀x2 : set, x2 ∈ ω → ap v (ordsucc x2) ∈ SNoS_ ω)
L3190
Hypothesis H26 : (∀x2 : set, x2 ∈ ω → SNo (ap v (ordsucc x2)))
L3191
Hypothesis H27 : (∀x2 : set, x2 ∈ ω → ap (Sigma ω (λy2 : set ⇒ ap z (ordsucc y2) + ap u (ordsucc y2))) x2 = ap z (ordsucc x2) + ap u (ordsucc x2))
L3192
Hypothesis H28 : (∀x2 : set, x2 ∈ ω → ap (Sigma ω (λy2 : set ⇒ ap w (ordsucc y2) + ap v (ordsucc y2))) x2 = ap w (ordsucc x2) + ap v (ordsucc x2))
L3193
Theorem. (Conj_real_add_SNo__33__11)
(∀x2 : set, x2 ∈ ω → SNo (ap (Sigma ω (λy2 : set ⇒ ap z (ordsucc y2) + ap u (ordsucc y2))) x2)) → x + y ∈ real
In Proofgold the corresponding term root is 319ac2... and proposition id is 8c1e1c...
Proof:
Proof not loaded.
End of Section Conj_real_add_SNo__33__11
Beginning of Section Conj_real_add_SNo__34__4
L3199
Variable x : set
(*** Conj_real_add_SNo__34__4 TMdKdU8EM1qpbuN9iiQwg6cKoRV25mabNK6 bounty of about 25 bars ***)
L3200
Variable y : set
L3201
Variable z : set
L3202
Variable w : set
L3203
Variable u : set
L3204
Variable v : set
L3205
Hypothesis H0 : x ∈ real
L3206
Hypothesis H1 : y ∈ real
L3207
Hypothesis H2 : (∀x2 : set, x2 ∈ ω → ap z x2 < x)
L3208
Hypothesis H3 : (∀x2 : set, x2 ∈ ω → x < ap z x2 + eps_ x2)
L3209
Hypothesis H5 : (∀x2 : set, x2 ∈ ω → (ap w x2 + - (eps_ x2)) < x)
L3210
Hypothesis H6 : (∀x2 : set, x2 ∈ ω → x < ap w x2)
L3211
Hypothesis H7 : (∀x2 : set, x2 ∈ ω → (∀y2 : set, y2 ∈ x2 → ap w x2 < ap w y2))
L3212
Hypothesis H8 : (∀x2 : set, x2 ∈ ω → ap u x2 < y)
L3213
Hypothesis H9 : (∀x2 : set, x2 ∈ ω → y < ap u x2 + eps_ x2)
L3214
Hypothesis H10 : (∀x2 : set, x2 ∈ ω → (∀y2 : set, y2 ∈ x2 → ap u y2 < ap u x2))
L3215
Hypothesis H11 : (∀x2 : set, x2 ∈ ω → (ap v x2 + - (eps_ x2)) < y)
L3216
Hypothesis H12 : (∀x2 : set, x2 ∈ ω → y < ap v x2)
L3217
Hypothesis H13 : (∀x2 : set, x2 ∈ ω → (∀y2 : set, y2 ∈ x2 → ap v x2 < ap v y2))
L3218
Hypothesis H14 : SNo x
L3219
Hypothesis H15 : SNo y
L3220
Hypothesis H16 : SNo (x + y)
L3221
Hypothesis H17 : (∀x2 : set, x2 ∈ SNoS_ ω → (∀y2 : set, y2 ∈ ω → abs_SNo (x2 + - x) < eps_ y2) → x2 = x)
L3222
Hypothesis H18 : (∀x2 : set, x2 ∈ SNoS_ ω → (∀y2 : set, y2 ∈ ω → abs_SNo (x2 + - y) < eps_ y2) → x2 = y)
L3223
Hypothesis H19 : (∀x2 : set, x2 ∈ ω → ap z (ordsucc x2) ∈ SNoS_ ω)
L3224
Hypothesis H20 : (∀x2 : set, x2 ∈ ω → SNo (ap z (ordsucc x2)))
L3225
Hypothesis H21 : (∀x2 : set, x2 ∈ ω → ap u (ordsucc x2) ∈ SNoS_ ω)
L3226
Hypothesis H22 : (∀x2 : set, x2 ∈ ω → SNo (ap u (ordsucc x2)))
L3227
Hypothesis H23 : (∀x2 : set, x2 ∈ ω → ap w (ordsucc x2) ∈ SNoS_ ω)
L3228
Hypothesis H24 : (∀x2 : set, x2 ∈ ω → SNo (ap w (ordsucc x2)))
L3229
Hypothesis H25 : (∀x2 : set, x2 ∈ ω → ap v (ordsucc x2) ∈ SNoS_ ω)
L3230
Hypothesis H26 : (∀x2 : set, x2 ∈ ω → SNo (ap v (ordsucc x2)))
L3231
Hypothesis H27 : (∀x2 : set, x2 ∈ ω → ap (Sigma ω (λy2 : set ⇒ ap z (ordsucc y2) + ap u (ordsucc y2))) x2 = ap z (ordsucc x2) + ap u (ordsucc x2))
L3232
Theorem. (Conj_real_add_SNo__34__4)
(∀x2 : set, x2 ∈ ω → ap (Sigma ω (λy2 : set ⇒ ap w (ordsucc y2) + ap v (ordsucc y2))) x2 = ap w (ordsucc x2) + ap v (ordsucc x2)) → x + y ∈ real
In Proofgold the corresponding term root is cad3da... and proposition id is 1c4598...
Proof:
Proof not loaded.
End of Section Conj_real_add_SNo__34__4
Beginning of Section Conj_real_add_SNo__35__11
L3238
Variable x : set
(*** Conj_real_add_SNo__35__11 TMVVp78Q2iManYwP482mjH8hyRNRBzyLpqd bounty of about 25 bars ***)
L3239
Variable y : set
L3240
Variable z : set
L3241
Variable w : set
L3242
Variable u : set
L3243
Variable v : set
L3244
Hypothesis H0 : x ∈ real
L3245
Hypothesis H1 : y ∈ real
L3246
Hypothesis H2 : (∀x2 : set, x2 ∈ ω → ap z x2 < x)
L3247
Hypothesis H3 : (∀x2 : set, x2 ∈ ω → x < ap z x2 + eps_ x2)
L3248
Hypothesis H4 : (∀x2 : set, x2 ∈ ω → (∀y2 : set, y2 ∈ x2 → ap z y2 < ap z x2))
L3249
Hypothesis H5 : (∀x2 : set, x2 ∈ ω → (ap w x2 + - (eps_ x2)) < x)
L3250
Hypothesis H6 : (∀x2 : set, x2 ∈ ω → x < ap w x2)
L3251
Hypothesis H7 : (∀x2 : set, x2 ∈ ω → (∀y2 : set, y2 ∈ x2 → ap w x2 < ap w y2))
L3252
Hypothesis H8 : (∀x2 : set, x2 ∈ ω → ap u x2 < y)
L3253
Hypothesis H9 : (∀x2 : set, x2 ∈ ω → y < ap u x2 + eps_ x2)
L3254
Hypothesis H10 : (∀x2 : set, x2 ∈ ω → (∀y2 : set, y2 ∈ x2 → ap u y2 < ap u x2))
L3255
Hypothesis H12 : (∀x2 : set, x2 ∈ ω → y < ap v x2)
L3256
Hypothesis H13 : (∀x2 : set, x2 ∈ ω → (∀y2 : set, y2 ∈ x2 → ap v x2 < ap v y2))
L3257
Hypothesis H14 : SNo x
L3258
Hypothesis H15 : SNo y
L3259
Hypothesis H16 : SNo (x + y)
L3260
Hypothesis H17 : (∀x2 : set, x2 ∈ SNoS_ ω → (∀y2 : set, y2 ∈ ω → abs_SNo (x2 + - x) < eps_ y2) → x2 = x)
L3261
Hypothesis H18 : (∀x2 : set, x2 ∈ SNoS_ ω → (∀y2 : set, y2 ∈ ω → abs_SNo (x2 + - y) < eps_ y2) → x2 = y)
L3262
Hypothesis H19 : (∀x2 : set, x2 ∈ ω → ap z (ordsucc x2) ∈ SNoS_ ω)
L3263
Hypothesis H20 : (∀x2 : set, x2 ∈ ω → SNo (ap z (ordsucc x2)))
L3264
Hypothesis H21 : (∀x2 : set, x2 ∈ ω → ap u (ordsucc x2) ∈ SNoS_ ω)
L3265
Hypothesis H22 : (∀x2 : set, x2 ∈ ω → SNo (ap u (ordsucc x2)))
L3266
Hypothesis H23 : (∀x2 : set, x2 ∈ ω → ap w (ordsucc x2) ∈ SNoS_ ω)
L3267
Hypothesis H24 : (∀x2 : set, x2 ∈ ω → SNo (ap w (ordsucc x2)))
L3268
Hypothesis H25 : (∀x2 : set, x2 ∈ ω → ap v (ordsucc x2) ∈ SNoS_ ω)
L3269
Hypothesis H26 : (∀x2 : set, x2 ∈ ω → SNo (ap v (ordsucc x2)))
L3270
Theorem. (Conj_real_add_SNo__35__11)
(∀x2 : set, x2 ∈ ω → ap (Sigma ω (λy2 : set ⇒ ap z (ordsucc y2) + ap u (ordsucc y2))) x2 = ap z (ordsucc x2) + ap u (ordsucc x2)) → x + y ∈ real
In Proofgold the corresponding term root is d832c9... and proposition id is 6e0eb5...
Proof:
Proof not loaded.
End of Section Conj_real_add_SNo__35__11
Beginning of Section Conj_real_add_SNo__35__13
L3276
Variable x : set
(*** Conj_real_add_SNo__35__13 TMRZpkXjbKFxG2xpgu3XgZVmendfUCK8cKC bounty of about 25 bars ***)
L3277
Variable y : set
L3278
Variable z : set
L3279
Variable w : set
L3280
Variable u : set
L3281
Variable v : set
L3282
Hypothesis H0 : x ∈ real
L3283
Hypothesis H1 : y ∈ real
L3284
Hypothesis H2 : (∀x2 : set, x2 ∈ ω → ap z x2 < x)
L3285
Hypothesis H3 : (∀x2 : set, x2 ∈ ω → x < ap z x2 + eps_ x2)
L3286
Hypothesis H4 : (∀x2 : set, x2 ∈ ω → (∀y2 : set, y2 ∈ x2 → ap z y2 < ap z x2))
L3287
Hypothesis H5 : (∀x2 : set, x2 ∈ ω → (ap w x2 + - (eps_ x2)) < x)
L3288
Hypothesis H6 : (∀x2 : set, x2 ∈ ω → x < ap w x2)
L3289
Hypothesis H7 : (∀x2 : set, x2 ∈ ω → (∀y2 : set, y2 ∈ x2 → ap w x2 < ap w y2))
L3290
Hypothesis H8 : (∀x2 : set, x2 ∈ ω → ap u x2 < y)
L3291
Hypothesis H9 : (∀x2 : set, x2 ∈ ω → y < ap u x2 + eps_ x2)
L3292
Hypothesis H10 : (∀x2 : set, x2 ∈ ω → (∀y2 : set, y2 ∈ x2 → ap u y2 < ap u x2))
L3293
Hypothesis H11 : (∀x2 : set, x2 ∈ ω → (ap v x2 + - (eps_ x2)) < y)
L3294
Hypothesis H12 : (∀x2 : set, x2 ∈ ω → y < ap v x2)
L3295
Hypothesis H14 : SNo x
L3296
Hypothesis H15 : SNo y
L3297
Hypothesis H16 : SNo (x + y)
L3298
Hypothesis H17 : (∀x2 : set, x2 ∈ SNoS_ ω → (∀y2 : set, y2 ∈ ω → abs_SNo (x2 + - x) < eps_ y2) → x2 = x)
L3299
Hypothesis H18 : (∀x2 : set, x2 ∈ SNoS_ ω → (∀y2 : set, y2 ∈ ω → abs_SNo (x2 + - y) < eps_ y2) → x2 = y)
L3300
Hypothesis H19 : (∀x2 : set, x2 ∈ ω → ap z (ordsucc x2) ∈ SNoS_ ω)
L3301
Hypothesis H20 : (∀x2 : set, x2 ∈ ω → SNo (ap z (ordsucc x2)))
L3302
Hypothesis H21 : (∀x2 : set, x2 ∈ ω → ap u (ordsucc x2) ∈ SNoS_ ω)
L3303
Hypothesis H22 : (∀x2 : set, x2 ∈ ω → SNo (ap u (ordsucc x2)))
L3304
Hypothesis H23 : (∀x2 : set, x2 ∈ ω → ap w (ordsucc x2) ∈ SNoS_ ω)
L3305
Hypothesis H24 : (∀x2 : set, x2 ∈ ω → SNo (ap w (ordsucc x2)))
L3306
Hypothesis H25 : (∀x2 : set, x2 ∈ ω → ap v (ordsucc x2) ∈ SNoS_ ω)
L3307
Hypothesis H26 : (∀x2 : set, x2 ∈ ω → SNo (ap v (ordsucc x2)))
L3308
Theorem. (Conj_real_add_SNo__35__13)
(∀x2 : set, x2 ∈ ω → ap (Sigma ω (λy2 : set ⇒ ap z (ordsucc y2) + ap u (ordsucc y2))) x2 = ap z (ordsucc x2) + ap u (ordsucc x2)) → x + y ∈ real
In Proofgold the corresponding term root is f007a6... and proposition id is c87d9c...
Proof:
Proof not loaded.
End of Section Conj_real_add_SNo__35__13
Beginning of Section Conj_real_add_SNo__36__12
L3314
Variable x : set
(*** Conj_real_add_SNo__36__12 TMN9Lrb9YoSKjsJvRQAJFtnFgBE2YHWUDvx bounty of about 25 bars ***)
L3315
Variable y : set
L3316
Variable z : set
L3317
Variable w : set
L3318
Variable u : set
L3319
Variable v : set
L3320
Hypothesis H0 : x ∈ real
L3321
Hypothesis H1 : y ∈ real
L3322
Hypothesis H2 : (∀x2 : set, x2 ∈ ω → ap z x2 < x)
L3323
Hypothesis H3 : (∀x2 : set, x2 ∈ ω → x < ap z x2 + eps_ x2)
L3324
Hypothesis H4 : (∀x2 : set, x2 ∈ ω → (∀y2 : set, y2 ∈ x2 → ap z y2 < ap z x2))
L3325
Hypothesis H5 : (∀x2 : set, x2 ∈ ω → (ap w x2 + - (eps_ x2)) < x)
L3326
Hypothesis H6 : (∀x2 : set, x2 ∈ ω → x < ap w x2)
L3327
Hypothesis H7 : (∀x2 : set, x2 ∈ ω → (∀y2 : set, y2 ∈ x2 → ap w x2 < ap w y2))
L3328
Hypothesis H8 : (∀x2 : set, x2 ∈ ω → ap u x2 < y)
L3329
Hypothesis H9 : (∀x2 : set, x2 ∈ ω → y < ap u x2 + eps_ x2)
L3330
Hypothesis H10 : (∀x2 : set, x2 ∈ ω → (∀y2 : set, y2 ∈ x2 → ap u y2 < ap u x2))
L3331
Hypothesis H11 : (∀x2 : set, x2 ∈ ω → (ap v x2 + - (eps_ x2)) < y)
L3332
Hypothesis H13 : (∀x2 : set, x2 ∈ ω → (∀y2 : set, y2 ∈ x2 → ap v x2 < ap v y2))
L3333
Hypothesis H14 : SNo x
L3334
Hypothesis H15 : SNo y
L3335
Hypothesis H16 : SNo (x + y)
L3336
Hypothesis H17 : (∀x2 : set, x2 ∈ SNoS_ ω → (∀y2 : set, y2 ∈ ω → abs_SNo (x2 + - x) < eps_ y2) → x2 = x)
L3337
Hypothesis H18 : (∀x2 : set, x2 ∈ SNoS_ ω → (∀y2 : set, y2 ∈ ω → abs_SNo (x2 + - y) < eps_ y2) → x2 = y)
L3338
Hypothesis H19 : (∀x2 : set, x2 ∈ ω → ap z (ordsucc x2) ∈ SNoS_ ω)
L3339
Hypothesis H20 : (∀x2 : set, x2 ∈ ω → SNo (ap z (ordsucc x2)))
L3340
Hypothesis H21 : (∀x2 : set, x2 ∈ ω → ap u (ordsucc x2) ∈ SNoS_ ω)
L3341
Hypothesis H22 : (∀x2 : set, x2 ∈ ω → SNo (ap u (ordsucc x2)))
L3342
Hypothesis H23 : (∀x2 : set, x2 ∈ ω → ap w (ordsucc x2) ∈ SNoS_ ω)
L3343
Hypothesis H24 : (∀x2 : set, x2 ∈ ω → SNo (ap w (ordsucc x2)))
L3344
Hypothesis H25 : (∀x2 : set, x2 ∈ ω → ap v (ordsucc x2) ∈ SNoS_ ω)
L3345
Theorem. (Conj_real_add_SNo__36__12)
(∀x2 : set, x2 ∈ ω → SNo (ap v (ordsucc x2))) → x + y ∈ real
In Proofgold the corresponding term root is bda1ab... and proposition id is 5c476c...
Proof:
Proof not loaded.
End of Section Conj_real_add_SNo__36__12
Beginning of Section Conj_real_add_SNo__36__15
L3351
Variable x : set
(*** Conj_real_add_SNo__36__15 TMUJHzSboRvs5p3twNn5rWCcUy8HvtzPP8U bounty of about 25 bars ***)
L3352
Variable y : set
L3353
Variable z : set
L3354
Variable w : set
L3355
Variable u : set
L3356
Variable v : set
L3357
Hypothesis H0 : x ∈ real
L3358
Hypothesis H1 : y ∈ real
L3359
Hypothesis H2 : (∀x2 : set, x2 ∈ ω → ap z x2 < x)
L3360
Hypothesis H3 : (∀x2 : set, x2 ∈ ω → x < ap z x2 + eps_ x2)
L3361
Hypothesis H4 : (∀x2 : set, x2 ∈ ω → (∀y2 : set, y2 ∈ x2 → ap z y2 < ap z x2))
L3362
Hypothesis H5 : (∀x2 : set, x2 ∈ ω → (ap w x2 + - (eps_ x2)) < x)
L3363
Hypothesis H6 : (∀x2 : set, x2 ∈ ω → x < ap w x2)
L3364
Hypothesis H7 : (∀x2 : set, x2 ∈ ω → (∀y2 : set, y2 ∈ x2 → ap w x2 < ap w y2))
L3365
Hypothesis H8 : (∀x2 : set, x2 ∈ ω → ap u x2 < y)
L3366
Hypothesis H9 : (∀x2 : set, x2 ∈ ω → y < ap u x2 + eps_ x2)
L3367
Hypothesis H10 : (∀x2 : set, x2 ∈ ω → (∀y2 : set, y2 ∈ x2 → ap u y2 < ap u x2))
L3368
Hypothesis H11 : (∀x2 : set, x2 ∈ ω → (ap v x2 + - (eps_ x2)) < y)
L3369
Hypothesis H12 : (∀x2 : set, x2 ∈ ω → y < ap v x2)
L3370
Hypothesis H13 : (∀x2 : set, x2 ∈ ω → (∀y2 : set, y2 ∈ x2 → ap v x2 < ap v y2))
L3371
Hypothesis H14 : SNo x
L3372
Hypothesis H16 : SNo (x + y)
L3373
Hypothesis H17 : (∀x2 : set, x2 ∈ SNoS_ ω → (∀y2 : set, y2 ∈ ω → abs_SNo (x2 + - x) < eps_ y2) → x2 = x)
L3374
Hypothesis H18 : (∀x2 : set, x2 ∈ SNoS_ ω → (∀y2 : set, y2 ∈ ω → abs_SNo (x2 + - y) < eps_ y2) → x2 = y)
L3375
Hypothesis H19 : (∀x2 : set, x2 ∈ ω → ap z (ordsucc x2) ∈ SNoS_ ω)
L3376
Hypothesis H20 : (∀x2 : set, x2 ∈ ω → SNo (ap z (ordsucc x2)))
L3377
Hypothesis H21 : (∀x2 : set, x2 ∈ ω → ap u (ordsucc x2) ∈ SNoS_ ω)
L3378
Hypothesis H22 : (∀x2 : set, x2 ∈ ω → SNo (ap u (ordsucc x2)))
L3379
Hypothesis H23 : (∀x2 : set, x2 ∈ ω → ap w (ordsucc x2) ∈ SNoS_ ω)
L3380
Hypothesis H24 : (∀x2 : set, x2 ∈ ω → SNo (ap w (ordsucc x2)))
L3381
Hypothesis H25 : (∀x2 : set, x2 ∈ ω → ap v (ordsucc x2) ∈ SNoS_ ω)
L3382
Theorem. (Conj_real_add_SNo__36__15)
(∀x2 : set, x2 ∈ ω → SNo (ap v (ordsucc x2))) → x + y ∈ real
In Proofgold the corresponding term root is c77bf7... and proposition id is ec53b5...
Proof:
Proof not loaded.
End of Section Conj_real_add_SNo__36__15
Beginning of Section Conj_real_add_SNo__37__9
L3388
Variable x : set
(*** Conj_real_add_SNo__37__9 TMXa8dVQdR2FU851xv7aUWFv4U6fwEq9KDW bounty of about 25 bars ***)
L3389
Variable y : set
L3390
Variable z : set
L3391
Variable w : set
L3392
Variable u : set
L3393
Variable v : set
L3394
Hypothesis H0 : x ∈ real
L3395
Hypothesis H1 : y ∈ real
L3396
Hypothesis H2 : (∀x2 : set, x2 ∈ ω → ap z x2 < x)
L3397
Hypothesis H3 : (∀x2 : set, x2 ∈ ω → x < ap z x2 + eps_ x2)
L3398
Hypothesis H4 : (∀x2 : set, x2 ∈ ω → (∀y2 : set, y2 ∈ x2 → ap z y2 < ap z x2))
L3399
Hypothesis H5 : (∀x2 : set, x2 ∈ ω → (ap w x2 + - (eps_ x2)) < x)
L3400
Hypothesis H6 : (∀x2 : set, x2 ∈ ω → x < ap w x2)
L3401
Hypothesis H7 : (∀x2 : set, x2 ∈ ω → (∀y2 : set, y2 ∈ x2 → ap w x2 < ap w y2))
L3402
Hypothesis H8 : v ∈ setexp (SNoS_ ω) ω
L3403
Hypothesis H10 : (∀x2 : set, x2 ∈ ω → y < ap u x2 + eps_ x2)
L3404
Hypothesis H11 : (∀x2 : set, x2 ∈ ω → (∀y2 : set, y2 ∈ x2 → ap u y2 < ap u x2))
L3405
Hypothesis H12 : (∀x2 : set, x2 ∈ ω → (ap v x2 + - (eps_ x2)) < y)
L3406
Hypothesis H13 : (∀x2 : set, x2 ∈ ω → y < ap v x2)
L3407
Hypothesis H14 : (∀x2 : set, x2 ∈ ω → (∀y2 : set, y2 ∈ x2 → ap v x2 < ap v y2))
L3408
Hypothesis H15 : SNo x
L3409
Hypothesis H16 : SNo y
L3410
Hypothesis H17 : SNo (x + y)
L3411
Hypothesis H18 : (∀x2 : set, x2 ∈ SNoS_ ω → (∀y2 : set, y2 ∈ ω → abs_SNo (x2 + - x) < eps_ y2) → x2 = x)
L3412
Hypothesis H19 : (∀x2 : set, x2 ∈ SNoS_ ω → (∀y2 : set, y2 ∈ ω → abs_SNo (x2 + - y) < eps_ y2) → x2 = y)
L3413
Hypothesis H20 : (∀x2 : set, x2 ∈ ω → ap z (ordsucc x2) ∈ SNoS_ ω)
L3414
Hypothesis H21 : (∀x2 : set, x2 ∈ ω → SNo (ap z (ordsucc x2)))
L3415
Hypothesis H22 : (∀x2 : set, x2 ∈ ω → ap u (ordsucc x2) ∈ SNoS_ ω)
L3416
Hypothesis H23 : (∀x2 : set, x2 ∈ ω → SNo (ap u (ordsucc x2)))
L3417
Hypothesis H24 : (∀x2 : set, x2 ∈ ω → ap w (ordsucc x2) ∈ SNoS_ ω)
L3418
Hypothesis H25 : (∀x2 : set, x2 ∈ ω → SNo (ap w (ordsucc x2)))
L3419
Theorem. (Conj_real_add_SNo__37__9)
(∀x2 : set, x2 ∈ ω → ap v (ordsucc x2) ∈ SNoS_ ω) → x + y ∈ real
In Proofgold the corresponding term root is 6d4d9d... and proposition id is 786e0d...
Proof:
Proof not loaded.
End of Section Conj_real_add_SNo__37__9
Beginning of Section Conj_real_add_SNo__40__2
L3425
Variable x : set
(*** Conj_real_add_SNo__40__2 TMYC36yyWQW49XmeDviv565hfZycrUiCJCk bounty of about 25 bars ***)
L3426
Variable y : set
L3427
Variable z : set
L3428
Variable w : set
L3429
Variable u : set
L3430
Variable v : set
L3431
Hypothesis H0 : x ∈ real
L3432
Hypothesis H1 : y ∈ real
L3433
Hypothesis H3 : (∀x2 : set, x2 ∈ ω → ap z x2 < x)
L3434
Hypothesis H4 : (∀x2 : set, x2 ∈ ω → x < ap z x2 + eps_ x2)
L3435
Hypothesis H5 : (∀x2 : set, x2 ∈ ω → (∀y2 : set, y2 ∈ x2 → ap z y2 < ap z x2))
L3436
Hypothesis H6 : (∀x2 : set, x2 ∈ ω → (ap w x2 + - (eps_ x2)) < x)
L3437
Hypothesis H7 : (∀x2 : set, x2 ∈ ω → x < ap w x2)
L3438
Hypothesis H8 : (∀x2 : set, x2 ∈ ω → (∀y2 : set, y2 ∈ x2 → ap w x2 < ap w y2))
L3439
Hypothesis H9 : v ∈ setexp (SNoS_ ω) ω
L3440
Hypothesis H10 : (∀x2 : set, x2 ∈ ω → ap u x2 < y)
L3441
Hypothesis H11 : (∀x2 : set, x2 ∈ ω → y < ap u x2 + eps_ x2)
L3442
Hypothesis H12 : (∀x2 : set, x2 ∈ ω → (∀y2 : set, y2 ∈ x2 → ap u y2 < ap u x2))
L3443
Hypothesis H13 : (∀x2 : set, x2 ∈ ω → (ap v x2 + - (eps_ x2)) < y)
L3444
Hypothesis H14 : (∀x2 : set, x2 ∈ ω → y < ap v x2)
L3445
Hypothesis H15 : (∀x2 : set, x2 ∈ ω → (∀y2 : set, y2 ∈ x2 → ap v x2 < ap v y2))
L3446
Hypothesis H16 : SNo x
L3447
Hypothesis H17 : SNo y
L3448
Hypothesis H18 : SNo (x + y)
L3449
Hypothesis H19 : (∀x2 : set, x2 ∈ SNoS_ ω → (∀y2 : set, y2 ∈ ω → abs_SNo (x2 + - x) < eps_ y2) → x2 = x)
L3450
Hypothesis H20 : (∀x2 : set, x2 ∈ SNoS_ ω → (∀y2 : set, y2 ∈ ω → abs_SNo (x2 + - y) < eps_ y2) → x2 = y)
L3451
Hypothesis H21 : (∀x2 : set, x2 ∈ ω → ap z (ordsucc x2) ∈ SNoS_ ω)
L3452
Hypothesis H22 : (∀x2 : set, x2 ∈ ω → SNo (ap z (ordsucc x2)))
L3453
Hypothesis H23 : (∀x2 : set, x2 ∈ ω → ap u (ordsucc x2) ∈ SNoS_ ω)
L3454
Theorem. (Conj_real_add_SNo__40__2)
(∀x2 : set, x2 ∈ ω → SNo (ap u (ordsucc x2))) → x + y ∈ real
In Proofgold the corresponding term root is 51f8bb... and proposition id is d6042b...
Proof:
Proof not loaded.
End of Section Conj_real_add_SNo__40__2
Beginning of Section Conj_real_add_SNo__40__19
L3460
Variable x : set
(*** Conj_real_add_SNo__40__19 TMZfkmk6NcG8xR5Ud7AWKGjhDQP8iSkvRK1 bounty of about 25 bars ***)
L3461
Variable y : set
L3462
Variable z : set
L3463
Variable w : set
L3464
Variable u : set
L3465
Variable v : set
L3466
Hypothesis H0 : x ∈ real
L3467
Hypothesis H1 : y ∈ real
L3468
Hypothesis H2 : w ∈ setexp (SNoS_ ω) ω
L3469
Hypothesis H3 : (∀x2 : set, x2 ∈ ω → ap z x2 < x)
L3470
Hypothesis H4 : (∀x2 : set, x2 ∈ ω → x < ap z x2 + eps_ x2)
L3471
Hypothesis H5 : (∀x2 : set, x2 ∈ ω → (∀y2 : set, y2 ∈ x2 → ap z y2 < ap z x2))
L3472
Hypothesis H6 : (∀x2 : set, x2 ∈ ω → (ap w x2 + - (eps_ x2)) < x)
L3473
Hypothesis H7 : (∀x2 : set, x2 ∈ ω → x < ap w x2)
L3474
Hypothesis H8 : (∀x2 : set, x2 ∈ ω → (∀y2 : set, y2 ∈ x2 → ap w x2 < ap w y2))
L3475
Hypothesis H9 : v ∈ setexp (SNoS_ ω) ω
L3476
Hypothesis H10 : (∀x2 : set, x2 ∈ ω → ap u x2 < y)
L3477
Hypothesis H11 : (∀x2 : set, x2 ∈ ω → y < ap u x2 + eps_ x2)
L3478
Hypothesis H12 : (∀x2 : set, x2 ∈ ω → (∀y2 : set, y2 ∈ x2 → ap u y2 < ap u x2))
L3479
Hypothesis H13 : (∀x2 : set, x2 ∈ ω → (ap v x2 + - (eps_ x2)) < y)
L3480
Hypothesis H14 : (∀x2 : set, x2 ∈ ω → y < ap v x2)
L3481
Hypothesis H15 : (∀x2 : set, x2 ∈ ω → (∀y2 : set, y2 ∈ x2 → ap v x2 < ap v y2))
L3482
Hypothesis H16 : SNo x
L3483
Hypothesis H17 : SNo y
L3484
Hypothesis H18 : SNo (x + y)
L3485
Hypothesis H20 : (∀x2 : set, x2 ∈ SNoS_ ω → (∀y2 : set, y2 ∈ ω → abs_SNo (x2 + - y) < eps_ y2) → x2 = y)
L3486
Hypothesis H21 : (∀x2 : set, x2 ∈ ω → ap z (ordsucc x2) ∈ SNoS_ ω)
L3487
Hypothesis H22 : (∀x2 : set, x2 ∈ ω → SNo (ap z (ordsucc x2)))
L3488
Hypothesis H23 : (∀x2 : set, x2 ∈ ω → ap u (ordsucc x2) ∈ SNoS_ ω)
L3489
Theorem. (Conj_real_add_SNo__40__19)
(∀x2 : set, x2 ∈ ω → SNo (ap u (ordsucc x2))) → x + y ∈ real
In Proofgold the corresponding term root is 3e007c... and proposition id is 59aa99...
Proof:
Proof not loaded.
End of Section Conj_real_add_SNo__40__19
Beginning of Section Conj_real_add_SNo__41__9
L3495
Variable x : set
(*** Conj_real_add_SNo__41__9 TMNyvLxZjov9Rr7KWfZ6VyyLZXXfDxtbfaF bounty of about 25 bars ***)
L3496
Variable y : set
L3497
Variable z : set
L3498
Variable w : set
L3499
Variable u : set
L3500
Variable v : set
L3501
Hypothesis H0 : x ∈ real
L3502
Hypothesis H1 : y ∈ real
L3503
Hypothesis H2 : w ∈ setexp (SNoS_ ω) ω
L3504
Hypothesis H3 : (∀x2 : set, x2 ∈ ω → ap z x2 < x)
L3505
Hypothesis H4 : (∀x2 : set, x2 ∈ ω → x < ap z x2 + eps_ x2)
L3506
Hypothesis H5 : (∀x2 : set, x2 ∈ ω → (∀y2 : set, y2 ∈ x2 → ap z y2 < ap z x2))
L3507
Hypothesis H6 : (∀x2 : set, x2 ∈ ω → (ap w x2 + - (eps_ x2)) < x)
L3508
Hypothesis H7 : (∀x2 : set, x2 ∈ ω → x < ap w x2)
L3509
Hypothesis H8 : (∀x2 : set, x2 ∈ ω → (∀y2 : set, y2 ∈ x2 → ap w x2 < ap w y2))
L3510
Hypothesis H10 : v ∈ setexp (SNoS_ ω) ω
L3511
Hypothesis H11 : (∀x2 : set, x2 ∈ ω → ap u x2 < y)
L3512
Hypothesis H12 : (∀x2 : set, x2 ∈ ω → y < ap u x2 + eps_ x2)
L3513
Hypothesis H13 : (∀x2 : set, x2 ∈ ω → (∀y2 : set, y2 ∈ x2 → ap u y2 < ap u x2))
L3514
Hypothesis H14 : (∀x2 : set, x2 ∈ ω → (ap v x2 + - (eps_ x2)) < y)
L3515
Hypothesis H15 : (∀x2 : set, x2 ∈ ω → y < ap v x2)
L3516
Hypothesis H16 : (∀x2 : set, x2 ∈ ω → (∀y2 : set, y2 ∈ x2 → ap v x2 < ap v y2))
L3517
Hypothesis H17 : SNo x
L3518
Hypothesis H18 : SNo y
L3519
Hypothesis H19 : SNo (x + y)
L3520
Hypothesis H20 : (∀x2 : set, x2 ∈ SNoS_ ω → (∀y2 : set, y2 ∈ ω → abs_SNo (x2 + - x) < eps_ y2) → x2 = x)
L3521
Hypothesis H21 : (∀x2 : set, x2 ∈ SNoS_ ω → (∀y2 : set, y2 ∈ ω → abs_SNo (x2 + - y) < eps_ y2) → x2 = y)
L3522
Hypothesis H22 : (∀x2 : set, x2 ∈ ω → ap z (ordsucc x2) ∈ SNoS_ ω)
L3523
Hypothesis H23 : (∀x2 : set, x2 ∈ ω → SNo (ap z (ordsucc x2)))
L3524
Theorem. (Conj_real_add_SNo__41__9)
(∀x2 : set, x2 ∈ ω → ap u (ordsucc x2) ∈ SNoS_ ω) → x + y ∈ real
In Proofgold the corresponding term root is d9772a... and proposition id is bce5db...
Proof:
Proof not loaded.
End of Section Conj_real_add_SNo__41__9
Beginning of Section Conj_real_add_SNo__43__10
L3530
Variable x : set
(*** Conj_real_add_SNo__43__10 TMdcsc7WySVWmkDzvUMW2koxhrFHDWv1peg bounty of about 25 bars ***)
L3531
Variable y : set
L3532
Variable z : set
L3533
Variable w : set
L3534
Variable u : set
L3535
Variable v : set
L3536
Hypothesis H0 : x ∈ real
L3537
Hypothesis H1 : y ∈ real
L3538
Hypothesis H2 : z ∈ setexp (SNoS_ ω) ω
L3539
Hypothesis H3 : w ∈ setexp (SNoS_ ω) ω
L3540
Hypothesis H4 : (∀x2 : set, x2 ∈ ω → ap z x2 < x)
L3541
Hypothesis H5 : (∀x2 : set, x2 ∈ ω → x < ap z x2 + eps_ x2)
L3542
Hypothesis H6 : (∀x2 : set, x2 ∈ ω → (∀y2 : set, y2 ∈ x2 → ap z y2 < ap z x2))
L3543
Hypothesis H7 : (∀x2 : set, x2 ∈ ω → (ap w x2 + - (eps_ x2)) < x)
L3544
Hypothesis H8 : (∀x2 : set, x2 ∈ ω → x < ap w x2)
L3545
Hypothesis H9 : (∀x2 : set, x2 ∈ ω → (∀y2 : set, y2 ∈ x2 → ap w x2 < ap w y2))
L3546
Hypothesis H11 : v ∈ setexp (SNoS_ ω) ω
L3547
Hypothesis H12 : (∀x2 : set, x2 ∈ ω → ap u x2 < y)
L3548
Hypothesis H13 : (∀x2 : set, x2 ∈ ω → y < ap u x2 + eps_ x2)
L3549
Hypothesis H14 : (∀x2 : set, x2 ∈ ω → (∀y2 : set, y2 ∈ x2 → ap u y2 < ap u x2))
L3550
Hypothesis H15 : (∀x2 : set, x2 ∈ ω → (ap v x2 + - (eps_ x2)) < y)
L3551
Hypothesis H16 : (∀x2 : set, x2 ∈ ω → y < ap v x2)
L3552
Hypothesis H17 : (∀x2 : set, x2 ∈ ω → (∀y2 : set, y2 ∈ x2 → ap v x2 < ap v y2))
L3553
Hypothesis H18 : SNo x
L3554
Hypothesis H19 : SNo y
L3555
Hypothesis H20 : SNo (x + y)
L3556
Hypothesis H21 : (∀x2 : set, x2 ∈ SNoS_ ω → (∀y2 : set, y2 ∈ ω → abs_SNo (x2 + - x) < eps_ y2) → x2 = x)
L3557
Hypothesis H22 : (∀x2 : set, x2 ∈ SNoS_ ω → (∀y2 : set, y2 ∈ ω → abs_SNo (x2 + - y) < eps_ y2) → x2 = y)
L3558
Theorem. (Conj_real_add_SNo__43__10)
(∀x2 : set, x2 ∈ ω → ap z (ordsucc x2) ∈ SNoS_ ω) → x + y ∈ real
In Proofgold the corresponding term root is 969c0b... and proposition id is c4ebc4...
Proof:
Proof not loaded.
End of Section Conj_real_add_SNo__43__10
Beginning of Section Conj_real_add_SNo__44__7
L3564
Variable x : set
(*** Conj_real_add_SNo__44__7 TMWLEHzGuRdxrrLLxYp758pqDH36E1hiHuR bounty of about 25 bars ***)
L3565
Variable y : set
L3566
Variable z : set
L3567
Variable w : set
L3568
Variable u : set
L3569
Variable v : set
L3570
Hypothesis H0 : x ∈ real
L3571
Hypothesis H1 : y ∈ real
L3572
Hypothesis H2 : z ∈ setexp (SNoS_ ω) ω
L3573
Hypothesis H3 : w ∈ setexp (SNoS_ ω) ω
L3574
Hypothesis H4 : (∀x2 : set, x2 ∈ ω → ap z x2 < x)
L3575
Hypothesis H5 : (∀x2 : set, x2 ∈ ω → x < ap z x2 + eps_ x2)
L3576
Hypothesis H6 : (∀x2 : set, x2 ∈ ω → (∀y2 : set, y2 ∈ x2 → ap z y2 < ap z x2))
L3577
Hypothesis H8 : (∀x2 : set, x2 ∈ ω → x < ap w x2)
L3578
Hypothesis H9 : (∀x2 : set, x2 ∈ ω → (∀y2 : set, y2 ∈ x2 → ap w x2 < ap w y2))
L3579
Hypothesis H10 : u ∈ setexp (SNoS_ ω) ω
L3580
Hypothesis H11 : v ∈ setexp (SNoS_ ω) ω
L3581
Hypothesis H12 : (∀x2 : set, x2 ∈ ω → ap u x2 < y)
L3582
Hypothesis H13 : (∀x2 : set, x2 ∈ ω → y < ap u x2 + eps_ x2)
L3583
Hypothesis H14 : (∀x2 : set, x2 ∈ ω → (∀y2 : set, y2 ∈ x2 → ap u y2 < ap u x2))
L3584
Hypothesis H15 : (∀x2 : set, x2 ∈ ω → (ap v x2 + - (eps_ x2)) < y)
L3585
Hypothesis H16 : (∀x2 : set, x2 ∈ ω → y < ap v x2)
L3586
Hypothesis H17 : (∀x2 : set, x2 ∈ ω → (∀y2 : set, y2 ∈ x2 → ap v x2 < ap v y2))
L3587
Hypothesis H18 : SNo x
L3588
Hypothesis H19 : SNo y
L3589
Hypothesis H20 : SNo (x + y)
L3590
Hypothesis H21 : (∀x2 : set, x2 ∈ SNoS_ ω → (∀y2 : set, y2 ∈ ω → abs_SNo (x2 + - x) < eps_ y2) → x2 = x)
L3591
Theorem. (Conj_real_add_SNo__44__7)
(∀x2 : set, x2 ∈ SNoS_ ω → (∀y2 : set, y2 ∈ ω → abs_SNo (x2 + - y) < eps_ y2) → x2 = y) → x + y ∈ real
In Proofgold the corresponding term root is cb2b2e... and proposition id is 34bd0f...
Proof:
Proof not loaded.
End of Section Conj_real_add_SNo__44__7
Beginning of Section Conj_real_add_SNo__44__17
L3597
Variable x : set
(*** Conj_real_add_SNo__44__17 TMP1pdCmpoHmjLBEhtcJ1qU2476ayGsf6Rg bounty of about 25 bars ***)
L3598
Variable y : set
L3599
Variable z : set
L3600
Variable w : set
L3601
Variable u : set
L3602
Variable v : set
L3603
Hypothesis H0 : x ∈ real
L3604
Hypothesis H1 : y ∈ real
L3605
Hypothesis H2 : z ∈ setexp (SNoS_ ω) ω
L3606
Hypothesis H3 : w ∈ setexp (SNoS_ ω) ω
L3607
Hypothesis H4 : (∀x2 : set, x2 ∈ ω → ap z x2 < x)
L3608
Hypothesis H5 : (∀x2 : set, x2 ∈ ω → x < ap z x2 + eps_ x2)
L3609
Hypothesis H6 : (∀x2 : set, x2 ∈ ω → (∀y2 : set, y2 ∈ x2 → ap z y2 < ap z x2))
L3610
Hypothesis H7 : (∀x2 : set, x2 ∈ ω → (ap w x2 + - (eps_ x2)) < x)
L3611
Hypothesis H8 : (∀x2 : set, x2 ∈ ω → x < ap w x2)
L3612
Hypothesis H9 : (∀x2 : set, x2 ∈ ω → (∀y2 : set, y2 ∈ x2 → ap w x2 < ap w y2))
L3613
Hypothesis H10 : u ∈ setexp (SNoS_ ω) ω
L3614
Hypothesis H11 : v ∈ setexp (SNoS_ ω) ω
L3615
Hypothesis H12 : (∀x2 : set, x2 ∈ ω → ap u x2 < y)
L3616
Hypothesis H13 : (∀x2 : set, x2 ∈ ω → y < ap u x2 + eps_ x2)
L3617
Hypothesis H14 : (∀x2 : set, x2 ∈ ω → (∀y2 : set, y2 ∈ x2 → ap u y2 < ap u x2))
L3618
Hypothesis H15 : (∀x2 : set, x2 ∈ ω → (ap v x2 + - (eps_ x2)) < y)
L3619
Hypothesis H16 : (∀x2 : set, x2 ∈ ω → y < ap v x2)
L3620
Hypothesis H18 : SNo x
L3621
Hypothesis H19 : SNo y
L3622
Hypothesis H20 : SNo (x + y)
L3623
Hypothesis H21 : (∀x2 : set, x2 ∈ SNoS_ ω → (∀y2 : set, y2 ∈ ω → abs_SNo (x2 + - x) < eps_ y2) → x2 = x)
L3624
Theorem. (Conj_real_add_SNo__44__17)
(∀x2 : set, x2 ∈ SNoS_ ω → (∀y2 : set, y2 ∈ ω → abs_SNo (x2 + - y) < eps_ y2) → x2 = y) → x + y ∈ real
In Proofgold the corresponding term root is 75958e... and proposition id is 7312b0...
Proof:
Proof not loaded.
End of Section Conj_real_add_SNo__44__17
Beginning of Section Conj_real_add_SNo__45__16
L3630
Variable x : set
(*** Conj_real_add_SNo__45__16 TMF6wjz8SL3jYRfAj2p7WabkRVMuVk7UDL2 bounty of about 25 bars ***)
L3631
Variable y : set
L3632
Variable z : set
L3633
Variable w : set
L3634
Variable u : set
L3635
Variable v : set
L3636
Hypothesis H0 : x ∈ real
L3637
Hypothesis H1 : y ∈ real
L3638
Hypothesis H2 : z ∈ setexp (SNoS_ ω) ω
L3639
Hypothesis H3 : w ∈ setexp (SNoS_ ω) ω
L3640
Hypothesis H4 : (∀x2 : set, x2 ∈ ω → ap z x2 < x)
L3641
Hypothesis H5 : (∀x2 : set, x2 ∈ ω → x < ap z x2 + eps_ x2)
L3642
Hypothesis H6 : (∀x2 : set, x2 ∈ ω → (∀y2 : set, y2 ∈ x2 → ap z y2 < ap z x2))
L3643
Hypothesis H7 : (∀x2 : set, x2 ∈ ω → (ap w x2 + - (eps_ x2)) < x)
L3644
Hypothesis H8 : (∀x2 : set, x2 ∈ ω → x < ap w x2)
L3645
Hypothesis H9 : (∀x2 : set, x2 ∈ ω → (∀y2 : set, y2 ∈ x2 → ap w x2 < ap w y2))
L3646
Hypothesis H10 : u ∈ setexp (SNoS_ ω) ω
L3647
Hypothesis H11 : v ∈ setexp (SNoS_ ω) ω
L3648
Hypothesis H12 : (∀x2 : set, x2 ∈ ω → ap u x2 < y)
L3649
Hypothesis H13 : (∀x2 : set, x2 ∈ ω → y < ap u x2 + eps_ x2)
L3650
Hypothesis H14 : (∀x2 : set, x2 ∈ ω → (∀y2 : set, y2 ∈ x2 → ap u y2 < ap u x2))
L3651
Hypothesis H15 : (∀x2 : set, x2 ∈ ω → (ap v x2 + - (eps_ x2)) < y)
L3652
Hypothesis H17 : (∀x2 : set, x2 ∈ ω → (∀y2 : set, y2 ∈ x2 → ap v x2 < ap v y2))
L3653
Hypothesis H18 : SNo x
L3654
Hypothesis H19 : SNo y
L3655
Hypothesis H20 : SNo (x + y)
L3656
Theorem. (Conj_real_add_SNo__45__16)
(∀x2 : set, x2 ∈ SNoS_ ω → (∀y2 : set, y2 ∈ ω → abs_SNo (x2 + - x) < eps_ y2) → x2 = x) → x + y ∈ real
In Proofgold the corresponding term root is d38374... and proposition id is aa8d23...
Proof:
Proof not loaded.
End of Section Conj_real_add_SNo__45__16
Beginning of Section Conj_real_add_SNo__45__20
L3662
Variable x : set
(*** Conj_real_add_SNo__45__20 TMQEqcrSDRbzRZhrJ9mp34cWbshw9QNtM25 bounty of about 25 bars ***)
L3663
Variable y : set
L3664
Variable z : set
L3665
Variable w : set
L3666
Variable u : set
L3667
Variable v : set
L3668
Hypothesis H0 : x ∈ real
L3669
Hypothesis H1 : y ∈ real
L3670
Hypothesis H2 : z ∈ setexp (SNoS_ ω) ω
L3671
Hypothesis H3 : w ∈ setexp (SNoS_ ω) ω
L3672
Hypothesis H4 : (∀x2 : set, x2 ∈ ω → ap z x2 < x)
L3673
Hypothesis H5 : (∀x2 : set, x2 ∈ ω → x < ap z x2 + eps_ x2)
L3674
Hypothesis H6 : (∀x2 : set, x2 ∈ ω → (∀y2 : set, y2 ∈ x2 → ap z y2 < ap z x2))
L3675
Hypothesis H7 : (∀x2 : set, x2 ∈ ω → (ap w x2 + - (eps_ x2)) < x)
L3676
Hypothesis H8 : (∀x2 : set, x2 ∈ ω → x < ap w x2)
L3677
Hypothesis H9 : (∀x2 : set, x2 ∈ ω → (∀y2 : set, y2 ∈ x2 → ap w x2 < ap w y2))
L3678
Hypothesis H10 : u ∈ setexp (SNoS_ ω) ω
L3679
Hypothesis H11 : v ∈ setexp (SNoS_ ω) ω
L3680
Hypothesis H12 : (∀x2 : set, x2 ∈ ω → ap u x2 < y)
L3681
Hypothesis H13 : (∀x2 : set, x2 ∈ ω → y < ap u x2 + eps_ x2)
L3682
Hypothesis H14 : (∀x2 : set, x2 ∈ ω → (∀y2 : set, y2 ∈ x2 → ap u y2 < ap u x2))
L3683
Hypothesis H15 : (∀x2 : set, x2 ∈ ω → (ap v x2 + - (eps_ x2)) < y)
L3684
Hypothesis H16 : (∀x2 : set, x2 ∈ ω → y < ap v x2)
L3685
Hypothesis H17 : (∀x2 : set, x2 ∈ ω → (∀y2 : set, y2 ∈ x2 → ap v x2 < ap v y2))
L3686
Hypothesis H18 : SNo x
L3687
Hypothesis H19 : SNo y
L3688
Theorem. (Conj_real_add_SNo__45__20)
(∀x2 : set, x2 ∈ SNoS_ ω → (∀y2 : set, y2 ∈ ω → abs_SNo (x2 + - x) < eps_ y2) → x2 = x) → x + y ∈ real
In Proofgold the corresponding term root is 3a5511... and proposition id is 5103b9...
Proof:
Proof not loaded.
End of Section Conj_real_add_SNo__45__20
Beginning of Section Conj_real_add_SNo__47__0
(*** Conj_real_add_SNo__47__0 TMMkXNLbgNQec4JiiWQcHFeBpec3r7HMBXE bounty of about 25 bars ***)
L3695
Variable x : set
L3696
Variable y : set
L3697
Variable z : set
L3698
Variable w : set
L3699
Variable u : set
L3700
Variable v : set
L3701
Hypothesis H1 : y ∈ real
L3702
Hypothesis H2 : z ∈ setexp (SNoS_ ω) ω
L3703
Hypothesis H3 : w ∈ setexp (SNoS_ ω) ω
L3704
Hypothesis H4 : (∀x2 : set, x2 ∈ ω → ap z x2 < x)
L3705
Hypothesis H5 : (∀x2 : set, x2 ∈ ω → x < ap z x2 + eps_ x2)
L3706
Hypothesis H6 : (∀x2 : set, x2 ∈ ω → (∀y2 : set, y2 ∈ x2 → ap z y2 < ap z x2))
L3707
Hypothesis H7 : (∀x2 : set, x2 ∈ ω → (ap w x2 + - (eps_ x2)) < x)
L3708
Hypothesis H8 : (∀x2 : set, x2 ∈ ω → x < ap w x2)
L3709
Hypothesis H9 : (∀x2 : set, x2 ∈ ω → (∀y2 : set, y2 ∈ x2 → ap w x2 < ap w y2))
L3710
Hypothesis H10 : u ∈ setexp (SNoS_ ω) ω
L3711
Hypothesis H11 : v ∈ setexp (SNoS_ ω) ω
L3712
Hypothesis H12 : (∀x2 : set, x2 ∈ ω → ap u x2 < y)
L3713
Hypothesis H13 : (∀x2 : set, x2 ∈ ω → y < ap u x2 + eps_ x2)
L3714
Hypothesis H14 : (∀x2 : set, x2 ∈ ω → (∀y2 : set, y2 ∈ x2 → ap u y2 < ap u x2))
L3715
Hypothesis H15 : (∀x2 : set, x2 ∈ ω → (ap v x2 + - (eps_ x2)) < y)
L3716
Hypothesis H16 : (∀x2 : set, x2 ∈ ω → y < ap v x2)
L3717
Hypothesis H17 : (∀x2 : set, x2 ∈ ω → (∀y2 : set, y2 ∈ x2 → ap v x2 < ap v y2))
L3718
Hypothesis H18 : SNo x
L3719
Theorem. (Conj_real_add_SNo__47__0)
SNo y → x + y ∈ real
In Proofgold the corresponding term root is d8e7d5... and proposition id is 9478d5...
Proof:
Proof not loaded.
End of Section Conj_real_add_SNo__47__0
Beginning of Section Conj_real_mul_SNo_pos__2__2
L3725
Variable x : set
(*** Conj_real_mul_SNo_pos__2__2 TMb4ztu1T6aUoRsXHk5bjqZpAzV7JyhzaqD bounty of about 25 bars ***)
L3726
Variable y : set
L3727
Variable z : set
L3728
Variable w : set
L3729
Variable u : set
L3730
Variable v : set
L3731
Variable x2 : set
L3732
Hypothesis H0 : SNo x
L3733
Hypothesis H1 : SNo y
L3734
Hypothesis H3 : SNo (- (x * y))
L3735
Hypothesis H4 : (∀y2 : set, y2 ∈ ω → abs_SNo (z + - (x * y)) < eps_ y2)
L3736
Hypothesis H5 : SNo z
L3737
Hypothesis H6 : x * y < z
L3738
Hypothesis H7 : SNo w
L3739
Hypothesis H8 : SNo u
L3740
Hypothesis H9 : (w * y + x * u) ≤ z + w * u
L3741
Hypothesis H10 : SNo (w * y)
L3742
Hypothesis H11 : SNo (x * u)
L3743
Hypothesis H12 : SNo (w * u)
L3744
Hypothesis H13 : SNo (- (w * u))
L3745
Hypothesis H14 : SNo (w + - x)
L3746
Hypothesis H15 : SNo (y + - u)
L3747
Hypothesis H16 : v ∈ ω
L3748
Hypothesis H17 : eps_ v ≤ w + - x
L3749
Hypothesis H18 : x2 ∈ ω
L3750
Hypothesis H19 : eps_ x2 ≤ y + - u
L3751
Hypothesis H20 : SNo (eps_ v)
L3752
Hypothesis H21 : SNo (eps_ x2)
L3753
Hypothesis H22 : SNo (eps_ (v + x2))
L3754
Hypothesis H23 : SNo (eps_ v * eps_ x2)
L3755
Theorem. (Conj_real_mul_SNo_pos__2__2)
¬ abs_SNo (z + - (x * y)) < eps_ (v + x2)
In Proofgold the corresponding term root is c76aac... and proposition id is 8a8fc2...
Proof:
Proof not loaded.
End of Section Conj_real_mul_SNo_pos__2__2
Beginning of Section Conj_real_mul_SNo_pos__2__11
L3761
Variable x : set
(*** Conj_real_mul_SNo_pos__2__11 TMFM8ZbbSYeDy53rtbd5SSEbiDNEnsfz1BS bounty of about 25 bars ***)
L3762
Variable y : set
L3763
Variable z : set
L3764
Variable w : set
L3765
Variable u : set
L3766
Variable v : set
L3767
Variable x2 : set
L3768
Hypothesis H0 : SNo x
L3769
Hypothesis H1 : SNo y
L3770
Hypothesis H2 : SNo (x * y)
L3771
Hypothesis H3 : SNo (- (x * y))
L3772
Hypothesis H4 : (∀y2 : set, y2 ∈ ω → abs_SNo (z + - (x * y)) < eps_ y2)
L3773
Hypothesis H5 : SNo z
L3774
Hypothesis H6 : x * y < z
L3775
Hypothesis H7 : SNo w
L3776
Hypothesis H8 : SNo u
L3777
Hypothesis H9 : (w * y + x * u) ≤ z + w * u
L3778
Hypothesis H10 : SNo (w * y)
L3779
Hypothesis H12 : SNo (w * u)
L3780
Hypothesis H13 : SNo (- (w * u))
L3781
Hypothesis H14 : SNo (w + - x)
L3782
Hypothesis H15 : SNo (y + - u)
L3783
Hypothesis H16 : v ∈ ω
L3784
Hypothesis H17 : eps_ v ≤ w + - x
L3785
Hypothesis H18 : x2 ∈ ω
L3786
Hypothesis H19 : eps_ x2 ≤ y + - u
L3787
Hypothesis H20 : SNo (eps_ v)
L3788
Hypothesis H21 : SNo (eps_ x2)
L3789
Hypothesis H22 : SNo (eps_ (v + x2))
L3790
Hypothesis H23 : SNo (eps_ v * eps_ x2)
L3791
Theorem. (Conj_real_mul_SNo_pos__2__11)
¬ abs_SNo (z + - (x * y)) < eps_ (v + x2)
In Proofgold the corresponding term root is ad42d1... and proposition id is 32b097...
Proof:
Proof not loaded.
End of Section Conj_real_mul_SNo_pos__2__11
Beginning of Section Conj_real_mul_SNo_pos__2__21
L3797
Variable x : set
(*** Conj_real_mul_SNo_pos__2__21 TMWPj3bQPLDXep6XXWqJnJs6kKq2HvTmJYp bounty of about 25 bars ***)
L3798
Variable y : set
L3799
Variable z : set
L3800
Variable w : set
L3801
Variable u : set
L3802
Variable v : set
L3803
Variable x2 : set
L3804
Hypothesis H0 : SNo x
L3805
Hypothesis H1 : SNo y
L3806
Hypothesis H2 : SNo (x * y)
L3807
Hypothesis H3 : SNo (- (x * y))
L3808
Hypothesis H4 : (∀y2 : set, y2 ∈ ω → abs_SNo (z + - (x * y)) < eps_ y2)
L3809
Hypothesis H5 : SNo z
L3810
Hypothesis H6 : x * y < z
L3811
Hypothesis H7 : SNo w
L3812
Hypothesis H8 : SNo u
L3813
Hypothesis H9 : (w * y + x * u) ≤ z + w * u
L3814
Hypothesis H10 : SNo (w * y)
L3815
Hypothesis H11 : SNo (x * u)
L3816
Hypothesis H12 : SNo (w * u)
L3817
Hypothesis H13 : SNo (- (w * u))
L3818
Hypothesis H14 : SNo (w + - x)
L3819
Hypothesis H15 : SNo (y + - u)
L3820
Hypothesis H16 : v ∈ ω
L3821
Hypothesis H17 : eps_ v ≤ w + - x
L3822
Hypothesis H18 : x2 ∈ ω
L3823
Hypothesis H19 : eps_ x2 ≤ y + - u
L3824
Hypothesis H20 : SNo (eps_ v)
L3825
Hypothesis H22 : SNo (eps_ (v + x2))
L3826
Hypothesis H23 : SNo (eps_ v * eps_ x2)
L3827
Theorem. (Conj_real_mul_SNo_pos__2__21)
¬ abs_SNo (z + - (x * y)) < eps_ (v + x2)
In Proofgold the corresponding term root is 37b716... and proposition id is e93959...
Proof:
Proof not loaded.
End of Section Conj_real_mul_SNo_pos__2__21
Beginning of Section Conj_real_mul_SNo_pos__2__22
L3833
Variable x : set
(*** Conj_real_mul_SNo_pos__2__22 TMaBTxesneNG5zLQDtKz7BxTRpPkLow3kRB bounty of about 25 bars ***)
L3834
Variable y : set
L3835
Variable z : set
L3836
Variable w : set
L3837
Variable u : set
L3838
Variable v : set
L3839
Variable x2 : set
L3840
Hypothesis H0 : SNo x
L3841
Hypothesis H1 : SNo y
L3842
Hypothesis H2 : SNo (x * y)
L3843
Hypothesis H3 : SNo (- (x * y))
L3844
Hypothesis H4 : (∀y2 : set, y2 ∈ ω → abs_SNo (z + - (x * y)) < eps_ y2)
L3845
Hypothesis H5 : SNo z
L3846
Hypothesis H6 : x * y < z
L3847
Hypothesis H7 : SNo w
L3848
Hypothesis H8 : SNo u
L3849
Hypothesis H9 : (w * y + x * u) ≤ z + w * u
L3850
Hypothesis H10 : SNo (w * y)
L3851
Hypothesis H11 : SNo (x * u)
L3852
Hypothesis H12 : SNo (w * u)
L3853
Hypothesis H13 : SNo (- (w * u))
L3854
Hypothesis H14 : SNo (w + - x)
L3855
Hypothesis H15 : SNo (y + - u)
L3856
Hypothesis H16 : v ∈ ω
L3857
Hypothesis H17 : eps_ v ≤ w + - x
L3858
Hypothesis H18 : x2 ∈ ω
L3859
Hypothesis H19 : eps_ x2 ≤ y + - u
L3860
Hypothesis H20 : SNo (eps_ v)
L3861
Hypothesis H21 : SNo (eps_ x2)
L3862
Hypothesis H23 : SNo (eps_ v * eps_ x2)
L3863
Theorem. (Conj_real_mul_SNo_pos__2__22)
¬ abs_SNo (z + - (x * y)) < eps_ (v + x2)
In Proofgold the corresponding term root is 4e6443... and proposition id is 7d6896...
Proof:
Proof not loaded.
End of Section Conj_real_mul_SNo_pos__2__22
Beginning of Section Conj_real_mul_SNo_pos__4__10
L3869
Variable x : set
(*** Conj_real_mul_SNo_pos__4__10 TMLyZM9FMgRqhmRP4eGmH9Xp3X9gL6oQp73 bounty of about 25 bars ***)
L3870
Variable y : set
L3871
Variable z : set
L3872
Variable w : set
L3873
Variable u : set
L3874
Variable v : set
L3875
Variable x2 : set
L3876
Hypothesis H0 : SNo x
L3877
Hypothesis H1 : SNo y
L3878
Hypothesis H2 : SNo (x * y)
L3879
Hypothesis H3 : SNo (- (x * y))
L3880
Hypothesis H4 : (∀y2 : set, y2 ∈ ω → abs_SNo (z + - (x * y)) < eps_ y2)
L3881
Hypothesis H5 : SNo z
L3882
Hypothesis H6 : x * y < z
L3883
Hypothesis H7 : SNo w
L3884
Hypothesis H8 : SNo u
L3885
Hypothesis H9 : (w * y + x * u) ≤ z + w * u
L3886
Hypothesis H11 : SNo (x * u)
L3887
Hypothesis H12 : SNo (w * u)
L3888
Hypothesis H13 : SNo (- (w * u))
L3889
Hypothesis H14 : SNo (w + - x)
L3890
Hypothesis H15 : SNo (y + - u)
L3891
Hypothesis H16 : v ∈ ω
L3892
Hypothesis H17 : eps_ v ≤ w + - x
L3893
Hypothesis H18 : x2 ∈ ω
L3894
Hypothesis H19 : eps_ x2 ≤ y + - u
L3895
Hypothesis H20 : SNo (eps_ v)
L3896
Hypothesis H21 : SNo (eps_ x2)
L3897
Hypothesis H22 : v + x2 ∈ ω
L3898
Theorem. (Conj_real_mul_SNo_pos__4__10)
¬ SNo (eps_ (v + x2))
In Proofgold the corresponding term root is fc9c7d... and proposition id is 081117...
Proof:
Proof not loaded.
End of Section Conj_real_mul_SNo_pos__4__10
Beginning of Section Conj_real_mul_SNo_pos__5__1
L3904
Variable x : set
(*** Conj_real_mul_SNo_pos__5__1 TMGh3q8JL8T1Pe71mKC1zsxP6Q6ErhB1hyW bounty of about 25 bars ***)
L3905
Variable y : set
L3906
Variable z : set
L3907
Variable w : set
L3908
Variable u : set
L3909
Variable v : set
L3910
Variable x2 : set
L3911
Hypothesis H0 : SNo x
L3912
Hypothesis H2 : SNo (x * y)
L3913
Hypothesis H3 : SNo (- (x * y))
L3914
Hypothesis H4 : (∀y2 : set, y2 ∈ ω → abs_SNo (z + - (x * y)) < eps_ y2)
L3915
Hypothesis H5 : SNo z
L3916
Hypothesis H6 : x * y < z
L3917
Hypothesis H7 : SNo w
L3918
Hypothesis H8 : SNo u
L3919
Hypothesis H9 : (w * y + x * u) ≤ z + w * u
L3920
Hypothesis H10 : SNo (w * y)
L3921
Hypothesis H11 : SNo (x * u)
L3922
Hypothesis H12 : SNo (w * u)
L3923
Hypothesis H13 : SNo (- (w * u))
L3924
Hypothesis H14 : SNo (w + - x)
L3925
Hypothesis H15 : SNo (y + - u)
L3926
Hypothesis H16 : v ∈ ω
L3927
Hypothesis H17 : eps_ v ≤ w + - x
L3928
Hypothesis H18 : x2 ∈ ω
L3929
Hypothesis H19 : eps_ x2 ≤ y + - u
L3930
Hypothesis H20 : SNo (eps_ v)
L3931
Hypothesis H21 : SNo (eps_ x2)
L3932
Theorem. (Conj_real_mul_SNo_pos__5__1)
¬ v + x2 ∈ ω
In Proofgold the corresponding term root is a1420a... and proposition id is d8af21...
Proof:
Proof not loaded.
End of Section Conj_real_mul_SNo_pos__5__1
Beginning of Section Conj_real_mul_SNo_pos__5__5
L3938
Variable x : set
(*** Conj_real_mul_SNo_pos__5__5 TMbBxx9inTYvuet9seEqFKtvL67jbravsAQ bounty of about 25 bars ***)
L3939
Variable y : set
L3940
Variable z : set
L3941
Variable w : set
L3942
Variable u : set
L3943
Variable v : set
L3944
Variable x2 : set
L3945
Hypothesis H0 : SNo x
L3946
Hypothesis H1 : SNo y
L3947
Hypothesis H2 : SNo (x * y)
L3948
Hypothesis H3 : SNo (- (x * y))
L3949
Hypothesis H4 : (∀y2 : set, y2 ∈ ω → abs_SNo (z + - (x * y)) < eps_ y2)
L3950
Hypothesis H6 : x * y < z
L3951
Hypothesis H7 : SNo w
L3952
Hypothesis H8 : SNo u
L3953
Hypothesis H9 : (w * y + x * u) ≤ z + w * u
L3954
Hypothesis H10 : SNo (w * y)
L3955
Hypothesis H11 : SNo (x * u)
L3956
Hypothesis H12 : SNo (w * u)
L3957
Hypothesis H13 : SNo (- (w * u))
L3958
Hypothesis H14 : SNo (w + - x)
L3959
Hypothesis H15 : SNo (y + - u)
L3960
Hypothesis H16 : v ∈ ω
L3961
Hypothesis H17 : eps_ v ≤ w + - x
L3962
Hypothesis H18 : x2 ∈ ω
L3963
Hypothesis H19 : eps_ x2 ≤ y + - u
L3964
Hypothesis H20 : SNo (eps_ v)
L3965
Hypothesis H21 : SNo (eps_ x2)
L3966
Theorem. (Conj_real_mul_SNo_pos__5__5)
¬ v + x2 ∈ ω
In Proofgold the corresponding term root is 80a00e... and proposition id is 24fa55...
Proof:
Proof not loaded.
End of Section Conj_real_mul_SNo_pos__5__5
Beginning of Section Conj_real_mul_SNo_pos__5__21
L3972
Variable x : set
(*** Conj_real_mul_SNo_pos__5__21 TMUWSJ3sGfREd4hHR7JHvr2FLLQ8x7HhdNP bounty of about 25 bars ***)
L3973
Variable y : set
L3974
Variable z : set
L3975
Variable w : set
L3976
Variable u : set
L3977
Variable v : set
L3978
Variable x2 : set
L3979
Hypothesis H0 : SNo x
L3980
Hypothesis H1 : SNo y
L3981
Hypothesis H2 : SNo (x * y)
L3982
Hypothesis H3 : SNo (- (x * y))
L3983
Hypothesis H4 : (∀y2 : set, y2 ∈ ω → abs_SNo (z + - (x * y)) < eps_ y2)
L3984
Hypothesis H5 : SNo z
L3985
Hypothesis H6 : x * y < z
L3986
Hypothesis H7 : SNo w
L3987
Hypothesis H8 : SNo u
L3988
Hypothesis H9 : (w * y + x * u) ≤ z + w * u
L3989
Hypothesis H10 : SNo (w * y)
L3990
Hypothesis H11 : SNo (x * u)
L3991
Hypothesis H12 : SNo (w * u)
L3992
Hypothesis H13 : SNo (- (w * u))
L3993
Hypothesis H14 : SNo (w + - x)
L3994
Hypothesis H15 : SNo (y + - u)
L3995
Hypothesis H16 : v ∈ ω
L3996
Hypothesis H17 : eps_ v ≤ w + - x
L3997
Hypothesis H18 : x2 ∈ ω
L3998
Hypothesis H19 : eps_ x2 ≤ y + - u
L3999
Hypothesis H20 : SNo (eps_ v)
L4000
Theorem. (Conj_real_mul_SNo_pos__5__21)
¬ v + x2 ∈ ω
In Proofgold the corresponding term root is af54b2... and proposition id is c716d2...
Proof:
Proof not loaded.
End of Section Conj_real_mul_SNo_pos__5__21
Beginning of Section Conj_real_mul_SNo_pos__6__13
L4006
Variable x : set
(*** Conj_real_mul_SNo_pos__6__13 TMVFDSCjDVngrWZ8ybhbRjcLwaZ6RWAfYd4 bounty of about 25 bars ***)
L4007
Variable y : set
L4008
Variable z : set
L4009
Variable w : set
L4010
Variable u : set
L4011
Variable v : set
L4012
Variable x2 : set
L4013
Hypothesis H0 : SNo x
L4014
Hypothesis H1 : SNo y
L4015
Hypothesis H2 : SNo (x * y)
L4016
Hypothesis H3 : SNo (- (x * y))
L4017
Hypothesis H4 : (∀y2 : set, y2 ∈ ω → abs_SNo (z + - (x * y)) < eps_ y2)
L4018
Hypothesis H5 : SNo z
L4019
Hypothesis H6 : x * y < z
L4020
Hypothesis H7 : SNo w
L4021
Hypothesis H8 : SNo u
L4022
Hypothesis H9 : (w * y + x * u) ≤ z + w * u
L4023
Hypothesis H10 : SNo (w * y)
L4024
Hypothesis H11 : SNo (x * u)
L4025
Hypothesis H12 : SNo (w * u)
L4026
Hypothesis H14 : SNo (w + - x)
L4027
Hypothesis H15 : SNo (y + - u)
L4028
Hypothesis H16 : v ∈ ω
L4029
Hypothesis H17 : eps_ v ≤ w + - x
L4030
Hypothesis H18 : x2 ∈ ω
L4031
Hypothesis H19 : eps_ x2 ≤ y + - u
L4032
Hypothesis H20 : SNo (eps_ v)
L4033
Theorem. (Conj_real_mul_SNo_pos__6__13)
¬ SNo (eps_ x2)
In Proofgold the corresponding term root is b34438... and proposition id is 306f08...
Proof:
Proof not loaded.
End of Section Conj_real_mul_SNo_pos__6__13
Beginning of Section Conj_real_mul_SNo_pos__7__0
L4039
Variable x : set
(*** Conj_real_mul_SNo_pos__7__0 TMGPW4R8amVWgLbbWSAXsr5RuxxYEMR1M4o bounty of about 25 bars ***)
L4040
Variable y : set
L4041
Variable z : set
L4042
Variable w : set
L4043
Variable u : set
L4044
Variable v : set
L4045
Variable x2 : set
L4046
Hypothesis H1 : SNo y
L4047
Hypothesis H2 : SNo (x * y)
L4048
Hypothesis H3 : SNo (- (x * y))
L4049
Hypothesis H4 : (∀y2 : set, y2 ∈ ω → abs_SNo (z + - (x * y)) < eps_ y2)
L4050
Hypothesis H5 : SNo z
L4051
Hypothesis H6 : x * y < z
L4052
Hypothesis H7 : SNo w
L4053
Hypothesis H8 : SNo u
L4054
Hypothesis H9 : (w * y + x * u) ≤ z + w * u
L4055
Hypothesis H10 : SNo (w * y)
L4056
Hypothesis H11 : SNo (x * u)
L4057
Hypothesis H12 : SNo (w * u)
L4058
Hypothesis H13 : SNo (- (w * u))
L4059
Hypothesis H14 : SNo (w + - x)
L4060
Hypothesis H15 : SNo (y + - u)
L4061
Hypothesis H16 : v ∈ ω
L4062
Hypothesis H17 : eps_ v ≤ w + - x
L4063
Hypothesis H18 : x2 ∈ ω
L4064
Hypothesis H19 : eps_ x2 ≤ y + - u
L4065
Theorem. (Conj_real_mul_SNo_pos__7__0)
¬ SNo (eps_ v)
In Proofgold the corresponding term root is 65cd67... and proposition id is a39d9d...
Proof:
Proof not loaded.
End of Section Conj_real_mul_SNo_pos__7__0
Beginning of Section Conj_real_mul_SNo_pos__7__5
L4071
Variable x : set
(*** Conj_real_mul_SNo_pos__7__5 TMTgXSvTiZRooyMMizKBB4tvfP2vTL1yTc1 bounty of about 25 bars ***)
L4072
Variable y : set
L4073
Variable z : set
L4074
Variable w : set
L4075
Variable u : set
L4076
Variable v : set
L4077
Variable x2 : set
L4078
Hypothesis H0 : SNo x
L4079
Hypothesis H1 : SNo y
L4080
Hypothesis H2 : SNo (x * y)
L4081
Hypothesis H3 : SNo (- (x * y))
L4082
Hypothesis H4 : (∀y2 : set, y2 ∈ ω → abs_SNo (z + - (x * y)) < eps_ y2)
L4083
Hypothesis H6 : x * y < z
L4084
Hypothesis H7 : SNo w
L4085
Hypothesis H8 : SNo u
L4086
Hypothesis H9 : (w * y + x * u) ≤ z + w * u
L4087
Hypothesis H10 : SNo (w * y)
L4088
Hypothesis H11 : SNo (x * u)
L4089
Hypothesis H12 : SNo (w * u)
L4090
Hypothesis H13 : SNo (- (w * u))
L4091
Hypothesis H14 : SNo (w + - x)
L4092
Hypothesis H15 : SNo (y + - u)
L4093
Hypothesis H16 : v ∈ ω
L4094
Hypothesis H17 : eps_ v ≤ w + - x
L4095
Hypothesis H18 : x2 ∈ ω
L4096
Hypothesis H19 : eps_ x2 ≤ y + - u
L4097
Theorem. (Conj_real_mul_SNo_pos__7__5)
¬ SNo (eps_ v)
In Proofgold the corresponding term root is 22c660... and proposition id is dfd5b2...
Proof:
Proof not loaded.
End of Section Conj_real_mul_SNo_pos__7__5
Beginning of Section Conj_real_mul_SNo_pos__8__14
L4103
Variable x : set
(*** Conj_real_mul_SNo_pos__8__14 TMZ6hiRV6xT2hvHLa2FfnWjQp3eq7oiai5Z bounty of about 25 bars ***)
L4104
Variable y : set
L4105
Variable z : set
L4106
Variable w : set
L4107
Variable u : set
L4108
Hypothesis H0 : SNo x
L4109
Hypothesis H1 : SNo y
L4110
Hypothesis H2 : SNo (x * y)
L4111
Hypothesis H3 : SNo (- (x * y))
L4112
Hypothesis H4 : (∀v : set, v ∈ SNoR x → (∀P : prop, (∀x2 : set, x2 ∈ ω → eps_ x2 ≤ v + - x → P) → P))
L4113
Hypothesis H5 : (∀v : set, v ∈ SNoL y → (∀P : prop, (∀x2 : set, x2 ∈ ω → eps_ x2 ≤ y + - v → P) → P))
L4114
Hypothesis H6 : (∀v : set, v ∈ ω → abs_SNo (z + - (x * y)) < eps_ v)
L4115
Hypothesis H7 : SNo z
L4116
Hypothesis H8 : x * y < z
L4117
Hypothesis H9 : w ∈ SNoR x
L4118
Hypothesis H10 : u ∈ SNoL y
L4119
Hypothesis H11 : SNo w
L4120
Hypothesis H12 : SNo u
L4121
Hypothesis H13 : (w * y + x * u) ≤ z + w * u
L4122
Hypothesis H15 : SNo (x * u)
L4123
Hypothesis H16 : SNo (w * u)
L4124
Hypothesis H17 : SNo (- (w * u))
L4125
Hypothesis H18 : SNo (w + - x)
L4126
Theorem. (Conj_real_mul_SNo_pos__8__14)
¬ SNo (y + - u)
In Proofgold the corresponding term root is 3eb5a8... and proposition id is a36dc7...
Proof:
Proof not loaded.
End of Section Conj_real_mul_SNo_pos__8__14
Beginning of Section Conj_real_mul_SNo_pos__10__3
L4132
Variable x : set
(*** Conj_real_mul_SNo_pos__10__3 TMJk2xTp8Z1gdjk25uUkbgYSN6hzymi3HZF bounty of about 25 bars ***)
L4133
Variable y : set
L4134
Variable z : set
L4135
Variable w : set
L4136
Variable u : set
L4137
Hypothesis H0 : SNo x
L4138
Hypothesis H1 : SNo y
L4139
Hypothesis H2 : SNo (x * y)
L4140
Hypothesis H4 : (∀v : set, v ∈ SNoR x → (∀P : prop, (∀x2 : set, x2 ∈ ω → eps_ x2 ≤ v + - x → P) → P))
L4141
Hypothesis H5 : (∀v : set, v ∈ SNoL y → (∀P : prop, (∀x2 : set, x2 ∈ ω → eps_ x2 ≤ y + - v → P) → P))
L4142
Hypothesis H6 : (∀v : set, v ∈ ω → abs_SNo (z + - (x * y)) < eps_ v)
L4143
Hypothesis H7 : SNo z
L4144
Hypothesis H8 : x * y < z
L4145
Hypothesis H9 : w ∈ SNoR x
L4146
Hypothesis H10 : u ∈ SNoL y
L4147
Hypothesis H11 : SNo w
L4148
Hypothesis H12 : SNo u
L4149
Hypothesis H13 : (w * y + x * u) ≤ z + w * u
L4150
Hypothesis H14 : SNo (w * y)
L4151
Hypothesis H15 : SNo (x * u)
L4152
Hypothesis H16 : SNo (w * u)
L4153
Theorem. (Conj_real_mul_SNo_pos__10__3)
¬ SNo (- (w * u))
In Proofgold the corresponding term root is 8dd68a... and proposition id is 0781ed...
Proof:
Proof not loaded.
End of Section Conj_real_mul_SNo_pos__10__3
Beginning of Section Conj_real_mul_SNo_pos__10__12
L4159
Variable x : set
(*** Conj_real_mul_SNo_pos__10__12 TMZZUwten3tHmhroqv8p7yhinoi8xZn6rkY bounty of about 25 bars ***)
L4160
Variable y : set
L4161
Variable z : set
L4162
Variable w : set
L4163
Variable u : set
L4164
Hypothesis H0 : SNo x
L4165
Hypothesis H1 : SNo y
L4166
Hypothesis H2 : SNo (x * y)
L4167
Hypothesis H3 : SNo (- (x * y))
L4168
Hypothesis H4 : (∀v : set, v ∈ SNoR x → (∀P : prop, (∀x2 : set, x2 ∈ ω → eps_ x2 ≤ v + - x → P) → P))
L4169
Hypothesis H5 : (∀v : set, v ∈ SNoL y → (∀P : prop, (∀x2 : set, x2 ∈ ω → eps_ x2 ≤ y + - v → P) → P))
L4170
Hypothesis H6 : (∀v : set, v ∈ ω → abs_SNo (z + - (x * y)) < eps_ v)
L4171
Hypothesis H7 : SNo z
L4172
Hypothesis H8 : x * y < z
L4173
Hypothesis H9 : w ∈ SNoR x
L4174
Hypothesis H10 : u ∈ SNoL y
L4175
Hypothesis H11 : SNo w
L4176
Hypothesis H13 : (w * y + x * u) ≤ z + w * u
L4177
Hypothesis H14 : SNo (w * y)
L4178
Hypothesis H15 : SNo (x * u)
L4179
Hypothesis H16 : SNo (w * u)
L4180
Theorem. (Conj_real_mul_SNo_pos__10__12)
¬ SNo (- (w * u))
In Proofgold the corresponding term root is c16d64... and proposition id is 6e489c...
Proof:
Proof not loaded.
End of Section Conj_real_mul_SNo_pos__10__12
Beginning of Section Conj_real_mul_SNo_pos__14__9
L4186
Variable x : set
(*** Conj_real_mul_SNo_pos__14__9 TMY1tpSZgY6XRiPYNoCJDRCtn3mCZgeL7Nj bounty of about 25 bars ***)
L4187
Variable y : set
L4188
Variable z : set
L4189
Variable w : set
L4190
Variable u : set
L4191
Variable v : set
L4192
Variable x2 : set
L4193
Hypothesis H0 : SNo x
L4194
Hypothesis H1 : SNo y
L4195
Hypothesis H2 : SNo (x * y)
L4196
Hypothesis H3 : SNo (- (x * y))
L4197
Hypothesis H4 : SNo z
L4198
Hypothesis H5 : x * y < z
L4199
Hypothesis H6 : SNo w
L4200
Hypothesis H7 : SNo u
L4201
Hypothesis H8 : (w * y + x * u) ≤ z + w * u
L4202
Hypothesis H10 : SNo (x * u)
L4203
Hypothesis H11 : SNo (w * u)
L4204
Hypothesis H12 : SNo (- (w * u))
L4205
Hypothesis H13 : SNo (x + - w)
L4206
Hypothesis H14 : SNo (u + - y)
L4207
Hypothesis H15 : v ∈ ω
L4208
Hypothesis H16 : eps_ v ≤ x + - w
L4209
Hypothesis H17 : x2 ∈ ω
L4210
Hypothesis H18 : eps_ x2 ≤ u + - y
L4211
Hypothesis H19 : SNo (eps_ v)
L4212
Hypothesis H20 : SNo (eps_ x2)
L4213
Hypothesis H21 : SNo (eps_ (v + x2))
L4214
Hypothesis H22 : SNo (eps_ v * eps_ x2)
L4215
Hypothesis H23 : abs_SNo (z + - (x * y)) < eps_ (v + x2)
L4216
Theorem. (Conj_real_mul_SNo_pos__14__9)
¬ eps_ (v + x2) ≤ abs_SNo (z + - (x * y))
In Proofgold the corresponding term root is 2dc4b3... and proposition id is 090579...
Proof:
Proof not loaded.
End of Section Conj_real_mul_SNo_pos__14__9
Beginning of Section Conj_real_mul_SNo_pos__15__5
L4222
Variable x : set
(*** Conj_real_mul_SNo_pos__15__5 TMVTHJvz3E8kP8Q8KC7KS6ESFKnWLyXf8pQ bounty of about 25 bars ***)
L4223
Variable y : set
L4224
Variable z : set
L4225
Variable w : set
L4226
Variable u : set
L4227
Variable v : set
L4228
Variable x2 : set
L4229
Hypothesis H0 : SNo x
L4230
Hypothesis H1 : SNo y
L4231
Hypothesis H2 : SNo (x * y)
L4232
Hypothesis H3 : SNo (- (x * y))
L4233
Hypothesis H4 : (∀y2 : set, y2 ∈ ω → abs_SNo (z + - (x * y)) < eps_ y2)
L4234
Hypothesis H6 : x * y < z
L4235
Hypothesis H7 : SNo w
L4236
Hypothesis H8 : SNo u
L4237
Hypothesis H9 : (w * y + x * u) ≤ z + w * u
L4238
Hypothesis H10 : SNo (w * y)
L4239
Hypothesis H11 : SNo (x * u)
L4240
Hypothesis H12 : SNo (w * u)
L4241
Hypothesis H13 : SNo (- (w * u))
L4242
Hypothesis H14 : SNo (x + - w)
L4243
Hypothesis H15 : SNo (u + - y)
L4244
Hypothesis H16 : v ∈ ω
L4245
Hypothesis H17 : eps_ v ≤ x + - w
L4246
Hypothesis H18 : x2 ∈ ω
L4247
Hypothesis H19 : eps_ x2 ≤ u + - y
L4248
Hypothesis H20 : SNo (eps_ v)
L4249
Hypothesis H21 : SNo (eps_ x2)
L4250
Hypothesis H22 : SNo (eps_ (v + x2))
L4251
Hypothesis H23 : SNo (eps_ v * eps_ x2)
L4252
Theorem. (Conj_real_mul_SNo_pos__15__5)
¬ abs_SNo (z + - (x * y)) < eps_ (v + x2)
In Proofgold the corresponding term root is df0bc2... and proposition id is 86ea7d...
Proof:
Proof not loaded.
End of Section Conj_real_mul_SNo_pos__15__5
Beginning of Section Conj_real_mul_SNo_pos__15__11
L4258
Variable x : set
(*** Conj_real_mul_SNo_pos__15__11 TMY8rTwN7TPhJTjrC5UDuE1yZ1JxjaoScer bounty of about 25 bars ***)
L4259
Variable y : set
L4260
Variable z : set
L4261
Variable w : set
L4262
Variable u : set
L4263
Variable v : set
L4264
Variable x2 : set
L4265
Hypothesis H0 : SNo x
L4266
Hypothesis H1 : SNo y
L4267
Hypothesis H2 : SNo (x * y)
L4268
Hypothesis H3 : SNo (- (x * y))
L4269
Hypothesis H4 : (∀y2 : set, y2 ∈ ω → abs_SNo (z + - (x * y)) < eps_ y2)
L4270
Hypothesis H5 : SNo z
L4271
Hypothesis H6 : x * y < z
L4272
Hypothesis H7 : SNo w
L4273
Hypothesis H8 : SNo u
L4274
Hypothesis H9 : (w * y + x * u) ≤ z + w * u
L4275
Hypothesis H10 : SNo (w * y)
L4276
Hypothesis H12 : SNo (w * u)
L4277
Hypothesis H13 : SNo (- (w * u))
L4278
Hypothesis H14 : SNo (x + - w)
L4279
Hypothesis H15 : SNo (u + - y)
L4280
Hypothesis H16 : v ∈ ω
L4281
Hypothesis H17 : eps_ v ≤ x + - w
L4282
Hypothesis H18 : x2 ∈ ω
L4283
Hypothesis H19 : eps_ x2 ≤ u + - y
L4284
Hypothesis H20 : SNo (eps_ v)
L4285
Hypothesis H21 : SNo (eps_ x2)
L4286
Hypothesis H22 : SNo (eps_ (v + x2))
L4287
Hypothesis H23 : SNo (eps_ v * eps_ x2)
L4288
Theorem. (Conj_real_mul_SNo_pos__15__11)
¬ abs_SNo (z + - (x * y)) < eps_ (v + x2)
In Proofgold the corresponding term root is a94d56... and proposition id is 3993cc...
Proof:
Proof not loaded.
End of Section Conj_real_mul_SNo_pos__15__11
Beginning of Section Conj_real_mul_SNo_pos__15__12
L4294
Variable x : set
(*** Conj_real_mul_SNo_pos__15__12 TMRJ5fZJJiUNMLJxr7fEuPgWwF7yJ8sGoay bounty of about 25 bars ***)
L4295
Variable y : set
L4296
Variable z : set
L4297
Variable w : set
L4298
Variable u : set
L4299
Variable v : set
L4300
Variable x2 : set
L4301
Hypothesis H0 : SNo x
L4302
Hypothesis H1 : SNo y
L4303
Hypothesis H2 : SNo (x * y)
L4304
Hypothesis H3 : SNo (- (x * y))
L4305
Hypothesis H4 : (∀y2 : set, y2 ∈ ω → abs_SNo (z + - (x * y)) < eps_ y2)
L4306
Hypothesis H5 : SNo z
L4307
Hypothesis H6 : x * y < z
L4308
Hypothesis H7 : SNo w
L4309
Hypothesis H8 : SNo u
L4310
Hypothesis H9 : (w * y + x * u) ≤ z + w * u
L4311
Hypothesis H10 : SNo (w * y)
L4312
Hypothesis H11 : SNo (x * u)
L4313
Hypothesis H13 : SNo (- (w * u))
L4314
Hypothesis H14 : SNo (x + - w)
L4315
Hypothesis H15 : SNo (u + - y)
L4316
Hypothesis H16 : v ∈ ω
L4317
Hypothesis H17 : eps_ v ≤ x + - w
L4318
Hypothesis H18 : x2 ∈ ω
L4319
Hypothesis H19 : eps_ x2 ≤ u + - y
L4320
Hypothesis H20 : SNo (eps_ v)
L4321
Hypothesis H21 : SNo (eps_ x2)
L4322
Hypothesis H22 : SNo (eps_ (v + x2))
L4323
Hypothesis H23 : SNo (eps_ v * eps_ x2)
L4324
Theorem. (Conj_real_mul_SNo_pos__15__12)
¬ abs_SNo (z + - (x * y)) < eps_ (v + x2)
In Proofgold the corresponding term root is 2c88b6... and proposition id is cef247...
Proof:
Proof not loaded.
End of Section Conj_real_mul_SNo_pos__15__12
Beginning of Section Conj_real_mul_SNo_pos__17__10
L4330
Variable x : set
(*** Conj_real_mul_SNo_pos__17__10 TMTvNDCJynJbCyvB5gSCXQozGi1wJQNoZJK bounty of about 25 bars ***)
L4331
Variable y : set
L4332
Variable z : set
L4333
Variable w : set
L4334
Variable u : set
L4335
Variable v : set
L4336
Variable x2 : set
L4337
Hypothesis H0 : SNo x
L4338
Hypothesis H1 : SNo y
L4339
Hypothesis H2 : SNo (x * y)
L4340
Hypothesis H3 : SNo (- (x * y))
L4341
Hypothesis H4 : (∀y2 : set, y2 ∈ ω → abs_SNo (z + - (x * y)) < eps_ y2)
L4342
Hypothesis H5 : SNo z
L4343
Hypothesis H6 : x * y < z
L4344
Hypothesis H7 : SNo w
L4345
Hypothesis H8 : SNo u
L4346
Hypothesis H9 : (w * y + x * u) ≤ z + w * u
L4347
Hypothesis H11 : SNo (x * u)
L4348
Hypothesis H12 : SNo (w * u)
L4349
Hypothesis H13 : SNo (- (w * u))
L4350
Hypothesis H14 : SNo (x + - w)
L4351
Hypothesis H15 : SNo (u + - y)
L4352
Hypothesis H16 : v ∈ ω
L4353
Hypothesis H17 : eps_ v ≤ x + - w
L4354
Hypothesis H18 : x2 ∈ ω
L4355
Hypothesis H19 : eps_ x2 ≤ u + - y
L4356
Hypothesis H20 : SNo (eps_ v)
L4357
Hypothesis H21 : SNo (eps_ x2)
L4358
Hypothesis H22 : v + x2 ∈ ω
L4359
Theorem. (Conj_real_mul_SNo_pos__17__10)
¬ SNo (eps_ (v + x2))
In Proofgold the corresponding term root is e44429... and proposition id is 1a982a...
Proof:
Proof not loaded.
End of Section Conj_real_mul_SNo_pos__17__10
Beginning of Section Conj_real_mul_SNo_pos__18__5
L4365
Variable x : set
(*** Conj_real_mul_SNo_pos__18__5 TMJg9eWN7w96gpt7G9geW9bvV926d8YF8bt bounty of about 25 bars ***)
L4366
Variable y : set
L4367
Variable z : set
L4368
Variable w : set
L4369
Variable u : set
L4370
Variable v : set
L4371
Variable x2 : set
L4372
Hypothesis H0 : SNo x
L4373
Hypothesis H1 : SNo y
L4374
Hypothesis H2 : SNo (x * y)
L4375
Hypothesis H3 : SNo (- (x * y))
L4376
Hypothesis H4 : (∀y2 : set, y2 ∈ ω → abs_SNo (z + - (x * y)) < eps_ y2)
L4377
Hypothesis H6 : x * y < z
L4378
Hypothesis H7 : SNo w
L4379
Hypothesis H8 : SNo u
L4380
Hypothesis H9 : (w * y + x * u) ≤ z + w * u
L4381
Hypothesis H10 : SNo (w * y)
L4382
Hypothesis H11 : SNo (x * u)
L4383
Hypothesis H12 : SNo (w * u)
L4384
Hypothesis H13 : SNo (- (w * u))
L4385
Hypothesis H14 : SNo (x + - w)
L4386
Hypothesis H15 : SNo (u + - y)
L4387
Hypothesis H16 : v ∈ ω
L4388
Hypothesis H17 : eps_ v ≤ x + - w
L4389
Hypothesis H18 : x2 ∈ ω
L4390
Hypothesis H19 : eps_ x2 ≤ u + - y
L4391
Hypothesis H20 : SNo (eps_ v)
L4392
Hypothesis H21 : SNo (eps_ x2)
L4393
Theorem. (Conj_real_mul_SNo_pos__18__5)
¬ v + x2 ∈ ω
In Proofgold the corresponding term root is fca560... and proposition id is 67d8d4...
Proof:
Proof not loaded.
End of Section Conj_real_mul_SNo_pos__18__5
Beginning of Section Conj_real_mul_SNo_pos__18__21
L4399
Variable x : set
(*** Conj_real_mul_SNo_pos__18__21 TMdwoJGHjwa3Tpbb9Spo6kG2FoSpZEb1D8b bounty of about 25 bars ***)
L4400
Variable y : set
L4401
Variable z : set
L4402
Variable w : set
L4403
Variable u : set
L4404
Variable v : set
L4405
Variable x2 : set
L4406
Hypothesis H0 : SNo x
L4407
Hypothesis H1 : SNo y
L4408
Hypothesis H2 : SNo (x * y)
L4409
Hypothesis H3 : SNo (- (x * y))
L4410
Hypothesis H4 : (∀y2 : set, y2 ∈ ω → abs_SNo (z + - (x * y)) < eps_ y2)
L4411
Hypothesis H5 : SNo z
L4412
Hypothesis H6 : x * y < z
L4413
Hypothesis H7 : SNo w
L4414
Hypothesis H8 : SNo u
L4415
Hypothesis H9 : (w * y + x * u) ≤ z + w * u
L4416
Hypothesis H10 : SNo (w * y)
L4417
Hypothesis H11 : SNo (x * u)
L4418
Hypothesis H12 : SNo (w * u)
L4419
Hypothesis H13 : SNo (- (w * u))
L4420
Hypothesis H14 : SNo (x + - w)
L4421
Hypothesis H15 : SNo (u + - y)
L4422
Hypothesis H16 : v ∈ ω
L4423
Hypothesis H17 : eps_ v ≤ x + - w
L4424
Hypothesis H18 : x2 ∈ ω
L4425
Hypothesis H19 : eps_ x2 ≤ u + - y
L4426
Hypothesis H20 : SNo (eps_ v)
L4427
Theorem. (Conj_real_mul_SNo_pos__18__21)
¬ v + x2 ∈ ω
In Proofgold the corresponding term root is 6e4885... and proposition id is 2505dc...
Proof:
Proof not loaded.
End of Section Conj_real_mul_SNo_pos__18__21
Beginning of Section Conj_real_mul_SNo_pos__21__1
L4433
Variable x : set
(*** Conj_real_mul_SNo_pos__21__1 TMU8Uj3WpVgm1oRxdn2aZCqXW87rKBhF7VJ bounty of about 25 bars ***)
L4434
Variable y : set
L4435
Variable z : set
L4436
Variable w : set
L4437
Variable u : set
L4438
Hypothesis H0 : SNo x
L4439
Hypothesis H2 : SNo (x * y)
L4440
Hypothesis H3 : SNo (- (x * y))
L4441
Hypothesis H4 : (∀v : set, v ∈ SNoL x → (∀P : prop, (∀x2 : set, x2 ∈ ω → eps_ x2 ≤ x + - v → P) → P))
L4442
Hypothesis H5 : (∀v : set, v ∈ SNoR y → (∀P : prop, (∀x2 : set, x2 ∈ ω → eps_ x2 ≤ v + - y → P) → P))
L4443
Hypothesis H6 : (∀v : set, v ∈ ω → abs_SNo (z + - (x * y)) < eps_ v)
L4444
Hypothesis H7 : SNo z
L4445
Hypothesis H8 : x * y < z
L4446
Hypothesis H9 : w ∈ SNoL x
L4447
Hypothesis H10 : u ∈ SNoR y
L4448
Hypothesis H11 : SNo w
L4449
Hypothesis H12 : SNo u
L4450
Hypothesis H13 : (w * y + x * u) ≤ z + w * u
L4451
Hypothesis H14 : SNo (w * y)
L4452
Hypothesis H15 : SNo (x * u)
L4453
Hypothesis H16 : SNo (w * u)
L4454
Hypothesis H17 : SNo (- (w * u))
L4455
Hypothesis H18 : SNo (x + - w)
L4456
Theorem. (Conj_real_mul_SNo_pos__21__1)
¬ SNo (u + - y)
In Proofgold the corresponding term root is f20aca... and proposition id is 6f0115...
Proof:
Proof not loaded.
End of Section Conj_real_mul_SNo_pos__21__1
Beginning of Section Conj_real_mul_SNo_pos__21__2
L4462
Variable x : set
(*** Conj_real_mul_SNo_pos__21__2 TMJc1UE2We1XxAP1ChnE1hzVLoKxmC1ZqLm bounty of about 25 bars ***)
L4463
Variable y : set
L4464
Variable z : set
L4465
Variable w : set
L4466
Variable u : set
L4467
Hypothesis H0 : SNo x
L4468
Hypothesis H1 : SNo y
L4469
Hypothesis H3 : SNo (- (x * y))
L4470
Hypothesis H4 : (∀v : set, v ∈ SNoL x → (∀P : prop, (∀x2 : set, x2 ∈ ω → eps_ x2 ≤ x + - v → P) → P))
L4471
Hypothesis H5 : (∀v : set, v ∈ SNoR y → (∀P : prop, (∀x2 : set, x2 ∈ ω → eps_ x2 ≤ v + - y → P) → P))
L4472
Hypothesis H6 : (∀v : set, v ∈ ω → abs_SNo (z + - (x * y)) < eps_ v)
L4473
Hypothesis H7 : SNo z
L4474
Hypothesis H8 : x * y < z
L4475
Hypothesis H9 : w ∈ SNoL x
L4476
Hypothesis H10 : u ∈ SNoR y
L4477
Hypothesis H11 : SNo w
L4478
Hypothesis H12 : SNo u
L4479
Hypothesis H13 : (w * y + x * u) ≤ z + w * u
L4480
Hypothesis H14 : SNo (w * y)
L4481
Hypothesis H15 : SNo (x * u)
L4482
Hypothesis H16 : SNo (w * u)
L4483
Hypothesis H17 : SNo (- (w * u))
L4484
Hypothesis H18 : SNo (x + - w)
L4485
Theorem. (Conj_real_mul_SNo_pos__21__2)
¬ SNo (u + - y)
In Proofgold the corresponding term root is 348a8f... and proposition id is a50720...
Proof:
Proof not loaded.
End of Section Conj_real_mul_SNo_pos__21__2
Beginning of Section Conj_real_mul_SNo_pos__25__1
L4491
Variable x : set
(*** Conj_real_mul_SNo_pos__25__1 TMHqD2hPSYEnFQ1Uoe1fbZwJmPxZNMJX869 bounty of about 25 bars ***)
L4492
Variable y : set
L4493
Variable z : set
L4494
Variable w : set
L4495
Variable u : set
L4496
Hypothesis H0 : SNo x
L4497
Hypothesis H2 : SNo (x * y)
L4498
Hypothesis H3 : SNo (- (x * y))
L4499
Hypothesis H4 : (∀v : set, v ∈ SNoL x → (∀P : prop, (∀x2 : set, x2 ∈ ω → eps_ x2 ≤ x + - v → P) → P))
L4500
Hypothesis H5 : (∀v : set, v ∈ SNoR y → (∀P : prop, (∀x2 : set, x2 ∈ ω → eps_ x2 ≤ v + - y → P) → P))
L4501
Hypothesis H6 : (∀v : set, v ∈ ω → abs_SNo (z + - (x * y)) < eps_ v)
L4502
Hypothesis H7 : SNo z
L4503
Hypothesis H8 : x * y < z
L4504
Hypothesis H9 : w ∈ SNoL x
L4505
Hypothesis H10 : u ∈ SNoR y
L4506
Hypothesis H11 : SNo w
L4507
Hypothesis H12 : SNo u
L4508
Hypothesis H13 : (w * y + x * u) ≤ z + w * u
L4509
Hypothesis H14 : SNo (w * y)
L4510
Hypothesis H15 : SNo (x * u)
L4511
Theorem. (Conj_real_mul_SNo_pos__25__1)
¬ SNo (- (x * u))
In Proofgold the corresponding term root is 701f0f... and proposition id is 283777...
Proof:
Proof not loaded.
End of Section Conj_real_mul_SNo_pos__25__1
Beginning of Section Conj_real_mul_SNo_pos__29__9
L4517
Variable x : set
(*** Conj_real_mul_SNo_pos__29__9 TMdAwAJr93jDFZPDQTVqKbs76b8CwaYUmbk bounty of about 25 bars ***)
L4518
Variable y : set
L4519
Variable z : set
L4520
Variable w : set
L4521
Variable u : set
L4522
Variable v : set
L4523
Variable x2 : set
L4524
Hypothesis H0 : SNo x
L4525
Hypothesis H1 : SNo y
L4526
Hypothesis H2 : SNo (x * y)
L4527
Hypothesis H3 : SNo z
L4528
Hypothesis H4 : z < x * y
L4529
Hypothesis H5 : SNo w
L4530
Hypothesis H6 : SNo u
L4531
Hypothesis H7 : (z + w * u) ≤ w * y + x * u
L4532
Hypothesis H8 : SNo (w * u)
L4533
Hypothesis H10 : SNo (- (x * u))
L4534
Hypothesis H11 : SNo (w * y)
L4535
Hypothesis H12 : SNo (- (w * y))
L4536
Hypothesis H13 : SNo (w + - x)
L4537
Hypothesis H14 : SNo (u + - y)
L4538
Hypothesis H15 : v ∈ ω
L4539
Hypothesis H16 : eps_ v ≤ w + - x
L4540
Hypothesis H17 : x2 ∈ ω
L4541
Hypothesis H18 : eps_ x2 ≤ u + - y
L4542
Hypothesis H19 : SNo (eps_ v)
L4543
Hypothesis H20 : SNo (eps_ x2)
L4544
Hypothesis H21 : SNo (eps_ (v + x2))
L4545
Hypothesis H22 : SNo (eps_ v * eps_ x2)
L4546
Hypothesis H23 : abs_SNo (z + - (x * y)) < eps_ (v + x2)
L4547
Theorem. (Conj_real_mul_SNo_pos__29__9)
¬ eps_ (v + x2) ≤ abs_SNo (z + - (x * y))
In Proofgold the corresponding term root is f32bd5... and proposition id is 75ca59...
Proof:
Proof not loaded.
End of Section Conj_real_mul_SNo_pos__29__9
Beginning of Section Conj_real_mul_SNo_pos__30__12
L4553
Variable x : set
(*** Conj_real_mul_SNo_pos__30__12 TMbVKHnebTA6auFRf2Lj3UvPJVBMgVkSFus bounty of about 25 bars ***)
L4554
Variable y : set
L4555
Variable z : set
L4556
Variable w : set
L4557
Variable u : set
L4558
Variable v : set
L4559
Variable x2 : set
L4560
Hypothesis H0 : SNo x
L4561
Hypothesis H1 : SNo y
L4562
Hypothesis H2 : SNo (x * y)
L4563
Hypothesis H3 : (∀y2 : set, y2 ∈ ω → abs_SNo (z + - (x * y)) < eps_ y2)
L4564
Hypothesis H4 : SNo z
L4565
Hypothesis H5 : z < x * y
L4566
Hypothesis H6 : SNo w
L4567
Hypothesis H7 : SNo u
L4568
Hypothesis H8 : (z + w * u) ≤ w * y + x * u
L4569
Hypothesis H9 : SNo (w * u)
L4570
Hypothesis H10 : SNo (x * u)
L4571
Hypothesis H11 : SNo (- (x * u))
L4572
Hypothesis H13 : SNo (- (w * y))
L4573
Hypothesis H14 : SNo (w + - x)
L4574
Hypothesis H15 : SNo (u + - y)
L4575
Hypothesis H16 : v ∈ ω
L4576
Hypothesis H17 : eps_ v ≤ w + - x
L4577
Hypothesis H18 : x2 ∈ ω
L4578
Hypothesis H19 : eps_ x2 ≤ u + - y
L4579
Hypothesis H20 : SNo (eps_ v)
L4580
Hypothesis H21 : SNo (eps_ x2)
L4581
Hypothesis H22 : SNo (eps_ (v + x2))
L4582
Hypothesis H23 : SNo (eps_ v * eps_ x2)
L4583
Theorem. (Conj_real_mul_SNo_pos__30__12)
¬ abs_SNo (z + - (x * y)) < eps_ (v + x2)
In Proofgold the corresponding term root is 041b14... and proposition id is 43989f...
Proof:
Proof not loaded.
End of Section Conj_real_mul_SNo_pos__30__12
Beginning of Section Conj_real_mul_SNo_pos__30__15
L4589
Variable x : set
(*** Conj_real_mul_SNo_pos__30__15 TMGNvxnvGL2gNd6LPmhtDjstjLZmWk8RDRM bounty of about 25 bars ***)
L4590
Variable y : set
L4591
Variable z : set
L4592
Variable w : set
L4593
Variable u : set
L4594
Variable v : set
L4595
Variable x2 : set
L4596
Hypothesis H0 : SNo x
L4597
Hypothesis H1 : SNo y
L4598
Hypothesis H2 : SNo (x * y)
L4599
Hypothesis H3 : (∀y2 : set, y2 ∈ ω → abs_SNo (z + - (x * y)) < eps_ y2)
L4600
Hypothesis H4 : SNo z
L4601
Hypothesis H5 : z < x * y
L4602
Hypothesis H6 : SNo w
L4603
Hypothesis H7 : SNo u
L4604
Hypothesis H8 : (z + w * u) ≤ w * y + x * u
L4605
Hypothesis H9 : SNo (w * u)
L4606
Hypothesis H10 : SNo (x * u)
L4607
Hypothesis H11 : SNo (- (x * u))
L4608
Hypothesis H12 : SNo (w * y)
L4609
Hypothesis H13 : SNo (- (w * y))
L4610
Hypothesis H14 : SNo (w + - x)
L4611
Hypothesis H16 : v ∈ ω
L4612
Hypothesis H17 : eps_ v ≤ w + - x
L4613
Hypothesis H18 : x2 ∈ ω
L4614
Hypothesis H19 : eps_ x2 ≤ u + - y
L4615
Hypothesis H20 : SNo (eps_ v)
L4616
Hypothesis H21 : SNo (eps_ x2)
L4617
Hypothesis H22 : SNo (eps_ (v + x2))
L4618
Hypothesis H23 : SNo (eps_ v * eps_ x2)
L4619
Theorem. (Conj_real_mul_SNo_pos__30__15)
¬ abs_SNo (z + - (x * y)) < eps_ (v + x2)
In Proofgold the corresponding term root is c8149f... and proposition id is 475a14...
Proof:
Proof not loaded.
End of Section Conj_real_mul_SNo_pos__30__15
Beginning of Section Conj_real_mul_SNo_pos__33__1
L4625
Variable x : set
(*** Conj_real_mul_SNo_pos__33__1 TMGxdnezSHvuVpEkkijxJP1AqL4SD42CZZA bounty of about 25 bars ***)
L4626
Variable y : set
L4627
Variable z : set
L4628
Variable w : set
L4629
Variable u : set
L4630
Variable v : set
L4631
Variable x2 : set
L4632
Hypothesis H0 : SNo x
L4633
Hypothesis H2 : SNo (x * y)
L4634
Hypothesis H3 : (∀y2 : set, y2 ∈ ω → abs_SNo (z + - (x * y)) < eps_ y2)
L4635
Hypothesis H4 : SNo z
L4636
Hypothesis H5 : z < x * y
L4637
Hypothesis H6 : SNo w
L4638
Hypothesis H7 : SNo u
L4639
Hypothesis H8 : (z + w * u) ≤ w * y + x * u
L4640
Hypothesis H9 : SNo (w * u)
L4641
Hypothesis H10 : SNo (x * u)
L4642
Hypothesis H11 : SNo (- (x * u))
L4643
Hypothesis H12 : SNo (w * y)
L4644
Hypothesis H13 : SNo (- (w * y))
L4645
Hypothesis H14 : SNo (w + - x)
L4646
Hypothesis H15 : SNo (u + - y)
L4647
Hypothesis H16 : v ∈ ω
L4648
Hypothesis H17 : eps_ v ≤ w + - x
L4649
Hypothesis H18 : x2 ∈ ω
L4650
Hypothesis H19 : eps_ x2 ≤ u + - y
L4651
Hypothesis H20 : SNo (eps_ v)
L4652
Hypothesis H21 : SNo (eps_ x2)
L4653
Theorem. (Conj_real_mul_SNo_pos__33__1)
¬ v + x2 ∈ ω
In Proofgold the corresponding term root is a2d3e1... and proposition id is b06a21...
Proof:
Proof not loaded.
End of Section Conj_real_mul_SNo_pos__33__1
Beginning of Section Conj_real_mul_SNo_pos__33__18
L4659
Variable x : set
(*** Conj_real_mul_SNo_pos__33__18 TMbsMfp8uUjr35AURGbebxrLstwB8Qy8vbb bounty of about 25 bars ***)
L4660
Variable y : set
L4661
Variable z : set
L4662
Variable w : set
L4663
Variable u : set
L4664
Variable v : set
L4665
Variable x2 : set
L4666
Hypothesis H0 : SNo x
L4667
Hypothesis H1 : SNo y
L4668
Hypothesis H2 : SNo (x * y)
L4669
Hypothesis H3 : (∀y2 : set, y2 ∈ ω → abs_SNo (z + - (x * y)) < eps_ y2)
L4670
Hypothesis H4 : SNo z
L4671
Hypothesis H5 : z < x * y
L4672
Hypothesis H6 : SNo w
L4673
Hypothesis H7 : SNo u
L4674
Hypothesis H8 : (z + w * u) ≤ w * y + x * u
L4675
Hypothesis H9 : SNo (w * u)
L4676
Hypothesis H10 : SNo (x * u)
L4677
Hypothesis H11 : SNo (- (x * u))
L4678
Hypothesis H12 : SNo (w * y)
L4679
Hypothesis H13 : SNo (- (w * y))
L4680
Hypothesis H14 : SNo (w + - x)
L4681
Hypothesis H15 : SNo (u + - y)
L4682
Hypothesis H16 : v ∈ ω
L4683
Hypothesis H17 : eps_ v ≤ w + - x
L4684
Hypothesis H19 : eps_ x2 ≤ u + - y
L4685
Hypothesis H20 : SNo (eps_ v)
L4686
Hypothesis H21 : SNo (eps_ x2)
L4687
Theorem. (Conj_real_mul_SNo_pos__33__18)
¬ v + x2 ∈ ω
In Proofgold the corresponding term root is 73f931... and proposition id is ea4880...
Proof:
Proof not loaded.
End of Section Conj_real_mul_SNo_pos__33__18
Beginning of Section Conj_real_mul_SNo_pos__35__9
L4693
Variable x : set
(*** Conj_real_mul_SNo_pos__35__9 TMHX3c35GpCLQaC7FnwyH1dYAryEk5DHK7E bounty of about 25 bars ***)
L4694
Variable y : set
L4695
Variable z : set
L4696
Variable w : set
L4697
Variable u : set
L4698
Variable v : set
L4699
Variable x2 : set
L4700
Hypothesis H0 : SNo x
L4701
Hypothesis H1 : SNo y
L4702
Hypothesis H2 : SNo (x * y)
L4703
Hypothesis H3 : (∀y2 : set, y2 ∈ ω → abs_SNo (z + - (x * y)) < eps_ y2)
L4704
Hypothesis H4 : SNo z
L4705
Hypothesis H5 : z < x * y
L4706
Hypothesis H6 : SNo w
L4707
Hypothesis H7 : SNo u
L4708
Hypothesis H8 : (z + w * u) ≤ w * y + x * u
L4709
Hypothesis H10 : SNo (x * u)
L4710
Hypothesis H11 : SNo (- (x * u))
L4711
Hypothesis H12 : SNo (w * y)
L4712
Hypothesis H13 : SNo (- (w * y))
L4713
Hypothesis H14 : SNo (w + - x)
L4714
Hypothesis H15 : SNo (u + - y)
L4715
Hypothesis H16 : v ∈ ω
L4716
Hypothesis H17 : eps_ v ≤ w + - x
L4717
Hypothesis H18 : x2 ∈ ω
L4718
Hypothesis H19 : eps_ x2 ≤ u + - y
L4719
Theorem. (Conj_real_mul_SNo_pos__35__9)
¬ SNo (eps_ v)
In Proofgold the corresponding term root is 4579a4... and proposition id is 35fc75...
Proof:
Proof not loaded.
End of Section Conj_real_mul_SNo_pos__35__9
Beginning of Section Conj_real_mul_SNo_pos__36__0
L4725
Variable x : set
(*** Conj_real_mul_SNo_pos__36__0 TMUbRVYkPzwZB7HBhn899BWwEEBYw4oMAsD bounty of about 25 bars ***)
L4726
Variable y : set
L4727
Variable z : set
L4728
Variable w : set
L4729
Variable u : set
L4730
Hypothesis H1 : SNo y
L4731
Hypothesis H2 : SNo (x * y)
L4732
Hypothesis H3 : (∀v : set, v ∈ SNoR x → (∀P : prop, (∀x2 : set, x2 ∈ ω → eps_ x2 ≤ v + - x → P) → P))
L4733
Hypothesis H4 : (∀v : set, v ∈ SNoR y → (∀P : prop, (∀x2 : set, x2 ∈ ω → eps_ x2 ≤ v + - y → P) → P))
L4734
Hypothesis H5 : (∀v : set, v ∈ ω → abs_SNo (z + - (x * y)) < eps_ v)
L4735
Hypothesis H6 : SNo z
L4736
Hypothesis H7 : z < x * y
L4737
Hypothesis H8 : w ∈ SNoR x
L4738
Hypothesis H9 : u ∈ SNoR y
L4739
Hypothesis H10 : SNo w
L4740
Hypothesis H11 : SNo u
L4741
Hypothesis H12 : (z + w * u) ≤ w * y + x * u
L4742
Hypothesis H13 : SNo (w * u)
L4743
Hypothesis H14 : SNo (x * u)
L4744
Hypothesis H15 : SNo (- (x * u))
L4745
Hypothesis H16 : SNo (w * y)
L4746
Hypothesis H17 : SNo (- (w * y))
L4747
Hypothesis H18 : SNo (w + - x)
L4748
Theorem. (Conj_real_mul_SNo_pos__36__0)
¬ SNo (u + - y)
In Proofgold the corresponding term root is 60d510... and proposition id is 139626...
Proof:
Proof not loaded.
End of Section Conj_real_mul_SNo_pos__36__0
Beginning of Section Conj_real_mul_SNo_pos__39__9
L4754
Variable x : set
(*** Conj_real_mul_SNo_pos__39__9 TMHLqNZ98TqtpoUAp3NgQd55oe9zJzHAcWJ bounty of about 25 bars ***)
L4755
Variable y : set
L4756
Variable z : set
L4757
Variable w : set
L4758
Variable u : set
L4759
Hypothesis H0 : SNo x
L4760
Hypothesis H1 : SNo y
L4761
Hypothesis H2 : SNo (x * y)
L4762
Hypothesis H3 : (∀v : set, v ∈ SNoR x → (∀P : prop, (∀x2 : set, x2 ∈ ω → eps_ x2 ≤ v + - x → P) → P))
L4763
Hypothesis H4 : (∀v : set, v ∈ SNoR y → (∀P : prop, (∀x2 : set, x2 ∈ ω → eps_ x2 ≤ v + - y → P) → P))
L4764
Hypothesis H5 : (∀v : set, v ∈ ω → abs_SNo (z + - (x * y)) < eps_ v)
L4765
Hypothesis H6 : SNo z
L4766
Hypothesis H7 : z < x * y
L4767
Hypothesis H8 : w ∈ SNoR x
L4768
Hypothesis H10 : SNo w
L4769
Hypothesis H11 : SNo u
L4770
Hypothesis H12 : (z + w * u) ≤ w * y + x * u
L4771
Hypothesis H13 : SNo (w * u)
L4772
Hypothesis H14 : SNo (x * u)
L4773
Hypothesis H15 : SNo (- (x * u))
L4774
Theorem. (Conj_real_mul_SNo_pos__39__9)
¬ SNo (w * y)
In Proofgold the corresponding term root is a387ea... and proposition id is 8df934...
Proof:
Proof not loaded.
End of Section Conj_real_mul_SNo_pos__39__9
Beginning of Section Conj_real_mul_SNo_pos__39__12
L4780
Variable x : set
(*** Conj_real_mul_SNo_pos__39__12 TMTreUTv1VUcD9KsjMsDSa6TeFNnNraYgSZ bounty of about 25 bars ***)
L4781
Variable y : set
L4782
Variable z : set
L4783
Variable w : set
L4784
Variable u : set
L4785
Hypothesis H0 : SNo x
L4786
Hypothesis H1 : SNo y
L4787
Hypothesis H2 : SNo (x * y)
L4788
Hypothesis H3 : (∀v : set, v ∈ SNoR x → (∀P : prop, (∀x2 : set, x2 ∈ ω → eps_ x2 ≤ v + - x → P) → P))
L4789
Hypothesis H4 : (∀v : set, v ∈ SNoR y → (∀P : prop, (∀x2 : set, x2 ∈ ω → eps_ x2 ≤ v + - y → P) → P))
L4790
Hypothesis H5 : (∀v : set, v ∈ ω → abs_SNo (z + - (x * y)) < eps_ v)
L4791
Hypothesis H6 : SNo z
L4792
Hypothesis H7 : z < x * y
L4793
Hypothesis H8 : w ∈ SNoR x
L4794
Hypothesis H9 : u ∈ SNoR y
L4795
Hypothesis H10 : SNo w
L4796
Hypothesis H11 : SNo u
L4797
Hypothesis H13 : SNo (w * u)
L4798
Hypothesis H14 : SNo (x * u)
L4799
Hypothesis H15 : SNo (- (x * u))
L4800
Theorem. (Conj_real_mul_SNo_pos__39__12)
¬ SNo (w * y)
In Proofgold the corresponding term root is 448049... and proposition id is a80fa0...
Proof:
Proof not loaded.
End of Section Conj_real_mul_SNo_pos__39__12
Beginning of Section Conj_real_mul_SNo_pos__41__1
L4806
Variable x : set
(*** Conj_real_mul_SNo_pos__41__1 TMZhhNJ4CPyemyKZQA28mCyT6Jq4LN5FBzn bounty of about 25 bars ***)
L4807
Variable y : set
L4808
Variable z : set
L4809
Variable w : set
L4810
Variable u : set
L4811
Hypothesis H0 : SNo x
L4812
Hypothesis H2 : SNo (x * y)
L4813
Hypothesis H3 : (∀v : set, v ∈ SNoR x → (∀P : prop, (∀x2 : set, x2 ∈ ω → eps_ x2 ≤ v + - x → P) → P))
L4814
Hypothesis H4 : (∀v : set, v ∈ SNoR y → (∀P : prop, (∀x2 : set, x2 ∈ ω → eps_ x2 ≤ v + - y → P) → P))
L4815
Hypothesis H5 : (∀v : set, v ∈ ω → abs_SNo (z + - (x * y)) < eps_ v)
L4816
Hypothesis H6 : SNo z
L4817
Hypothesis H7 : z < x * y
L4818
Hypothesis H8 : w ∈ SNoR x
L4819
Hypothesis H9 : u ∈ SNoR y
L4820
Hypothesis H10 : SNo w
L4821
Hypothesis H11 : SNo u
L4822
Hypothesis H12 : (z + w * u) ≤ w * y + x * u
L4823
Hypothesis H13 : SNo (w * u)
L4824
Theorem. (Conj_real_mul_SNo_pos__41__1)
¬ SNo (x * u)
In Proofgold the corresponding term root is 9f9cfe... and proposition id is 2ee45d...
Proof:
Proof not loaded.
End of Section Conj_real_mul_SNo_pos__41__1
Beginning of Section Conj_real_mul_SNo_pos__42__3
L4830
Variable x : set
(*** Conj_real_mul_SNo_pos__42__3 TMdDayLHAHaYYYkgCRwvjvF5qVtCVZUuVEh bounty of about 25 bars ***)
L4831
Variable y : set
L4832
Variable z : set
L4833
Variable w : set
L4834
Variable u : set
L4835
Hypothesis H0 : SNo x
L4836
Hypothesis H1 : SNo y
L4837
Hypothesis H2 : SNo (x * y)
L4838
Hypothesis H4 : (∀v : set, v ∈ SNoR y → (∀P : prop, (∀x2 : set, x2 ∈ ω → eps_ x2 ≤ v + - y → P) → P))
L4839
Hypothesis H5 : (∀v : set, v ∈ ω → abs_SNo (z + - (x * y)) < eps_ v)
L4840
Hypothesis H6 : SNo z
L4841
Hypothesis H7 : z < x * y
L4842
Hypothesis H8 : w ∈ SNoR x
L4843
Hypothesis H9 : u ∈ SNoR y
L4844
Hypothesis H10 : SNo w
L4845
Hypothesis H11 : SNo u
L4846
Hypothesis H12 : (z + w * u) ≤ w * y + x * u
L4847
Theorem. (Conj_real_mul_SNo_pos__42__3)
¬ SNo (w * u)
In Proofgold the corresponding term root is 76355c... and proposition id is 698435...
Proof:
Proof not loaded.
End of Section Conj_real_mul_SNo_pos__42__3
Beginning of Section Conj_real_mul_SNo_pos__45__5
L4853
Variable x : set
(*** Conj_real_mul_SNo_pos__45__5 TMMaH8G3p6NukY7R89XeVXfbrVfp3FVEJt5 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
Variable x2 : set
L4860
Hypothesis H0 : SNo x
L4861
Hypothesis H1 : SNo y
L4862
Hypothesis H2 : SNo (x * y)
L4863
Hypothesis H3 : SNo (- (x * y))
L4864
Hypothesis H4 : (∀y2 : set, y2 ∈ ω → abs_SNo (z + - (x * y)) < eps_ y2)
L4865
Hypothesis H6 : z < x * y
L4866
Hypothesis H7 : SNo w
L4867
Hypothesis H8 : SNo u
L4868
Hypothesis H9 : (z + w * u) ≤ w * y + x * u
L4869
Hypothesis H10 : SNo (w * u)
L4870
Hypothesis H11 : SNo (x * u)
L4871
Hypothesis H12 : SNo (- (x * u))
L4872
Hypothesis H13 : SNo (w * y)
L4873
Hypothesis H14 : SNo (- (w * y))
L4874
Hypothesis H15 : SNo (x + - w)
L4875
Hypothesis H16 : SNo (y + - u)
L4876
Hypothesis H17 : v ∈ ω
L4877
Hypothesis H18 : eps_ v ≤ x + - w
L4878
Hypothesis H19 : x2 ∈ ω
L4879
Hypothesis H20 : eps_ x2 ≤ y + - u
L4880
Hypothesis H21 : SNo (eps_ v)
L4881
Hypothesis H22 : SNo (eps_ x2)
L4882
Hypothesis H23 : SNo (eps_ (v + x2))
L4883
Theorem. (Conj_real_mul_SNo_pos__45__5)
¬ SNo (eps_ v * eps_ x2)
In Proofgold the corresponding term root is d009a5... and proposition id is cf389d...
Proof:
Proof not loaded.
End of Section Conj_real_mul_SNo_pos__45__5
Beginning of Section Conj_real_mul_SNo_pos__45__14
L4889
Variable x : set
(*** Conj_real_mul_SNo_pos__45__14 TMWep4wb7N1D7DUhDbEUqQPLpt46uVxFQw3 bounty of about 25 bars ***)
L4890
Variable y : set
L4891
Variable z : set
L4892
Variable w : set
L4893
Variable u : set
L4894
Variable v : set
L4895
Variable x2 : set
L4896
Hypothesis H0 : SNo x
L4897
Hypothesis H1 : SNo y
L4898
Hypothesis H2 : SNo (x * y)
L4899
Hypothesis H3 : SNo (- (x * y))
L4900
Hypothesis H4 : (∀y2 : set, y2 ∈ ω → abs_SNo (z + - (x * y)) < eps_ y2)
L4901
Hypothesis H5 : SNo z
L4902
Hypothesis H6 : z < x * y
L4903
Hypothesis H7 : SNo w
L4904
Hypothesis H8 : SNo u
L4905
Hypothesis H9 : (z + w * u) ≤ w * y + x * u
L4906
Hypothesis H10 : SNo (w * u)
L4907
Hypothesis H11 : SNo (x * u)
L4908
Hypothesis H12 : SNo (- (x * u))
L4909
Hypothesis H13 : SNo (w * y)
L4910
Hypothesis H15 : SNo (x + - w)
L4911
Hypothesis H16 : SNo (y + - u)
L4912
Hypothesis H17 : v ∈ ω
L4913
Hypothesis H18 : eps_ v ≤ x + - w
L4914
Hypothesis H19 : x2 ∈ ω
L4915
Hypothesis H20 : eps_ x2 ≤ y + - u
L4916
Hypothesis H21 : SNo (eps_ v)
L4917
Hypothesis H22 : SNo (eps_ x2)
L4918
Hypothesis H23 : SNo (eps_ (v + x2))
L4919
Theorem. (Conj_real_mul_SNo_pos__45__14)
¬ SNo (eps_ v * eps_ x2)
In Proofgold the corresponding term root is 52db87... and proposition id is d47be8...
Proof:
Proof not loaded.
End of Section Conj_real_mul_SNo_pos__45__14
Beginning of Section Conj_real_mul_SNo_pos__46__12
L4925
Variable x : set
(*** Conj_real_mul_SNo_pos__46__12 TMZuHFCgKcUJSppYycU4VP78mJhFFbTpLcx bounty of about 25 bars ***)
L4926
Variable y : set
L4927
Variable z : set
L4928
Variable w : set
L4929
Variable u : set
L4930
Variable v : set
L4931
Variable x2 : set
L4932
Hypothesis H0 : SNo x
L4933
Hypothesis H1 : SNo y
L4934
Hypothesis H2 : SNo (x * y)
L4935
Hypothesis H3 : SNo (- (x * y))
L4936
Hypothesis H4 : (∀y2 : set, y2 ∈ ω → abs_SNo (z + - (x * y)) < eps_ y2)
L4937
Hypothesis H5 : SNo z
L4938
Hypothesis H6 : z < x * y
L4939
Hypothesis H7 : SNo w
L4940
Hypothesis H8 : SNo u
L4941
Hypothesis H9 : (z + w * u) ≤ w * y + x * u
L4942
Hypothesis H10 : SNo (w * u)
L4943
Hypothesis H11 : SNo (x * u)
L4944
Hypothesis H13 : SNo (w * y)
L4945
Hypothesis H14 : SNo (- (w * y))
L4946
Hypothesis H15 : SNo (x + - w)
L4947
Hypothesis H16 : SNo (y + - u)
L4948
Hypothesis H17 : v ∈ ω
L4949
Hypothesis H18 : eps_ v ≤ x + - w
L4950
Hypothesis H19 : x2 ∈ ω
L4951
Hypothesis H20 : eps_ x2 ≤ y + - u
L4952
Hypothesis H21 : SNo (eps_ v)
L4953
Hypothesis H22 : SNo (eps_ x2)
L4954
Hypothesis H23 : v + x2 ∈ ω
L4955
Theorem. (Conj_real_mul_SNo_pos__46__12)
¬ SNo (eps_ (v + x2))
In Proofgold the corresponding term root is fad55c... and proposition id is b61ebd...
Proof:
Proof not loaded.
End of Section Conj_real_mul_SNo_pos__46__12
Beginning of Section Conj_real_mul_SNo_pos__46__18
L4961
Variable x : set
(*** Conj_real_mul_SNo_pos__46__18 TMJPiMxCDJjw17ncKHMwSA17aWfa48YmMRj bounty of about 25 bars ***)
L4962
Variable y : set
L4963
Variable z : set
L4964
Variable w : set
L4965
Variable u : set
L4966
Variable v : set
L4967
Variable x2 : set
L4968
Hypothesis H0 : SNo x
L4969
Hypothesis H1 : SNo y
L4970
Hypothesis H2 : SNo (x * y)
L4971
Hypothesis H3 : SNo (- (x * y))
L4972
Hypothesis H4 : (∀y2 : set, y2 ∈ ω → abs_SNo (z + - (x * y)) < eps_ y2)
L4973
Hypothesis H5 : SNo z
L4974
Hypothesis H6 : z < x * y
L4975
Hypothesis H7 : SNo w
L4976
Hypothesis H8 : SNo u
L4977
Hypothesis H9 : (z + w * u) ≤ w * y + x * u
L4978
Hypothesis H10 : SNo (w * u)
L4979
Hypothesis H11 : SNo (x * u)
L4980
Hypothesis H12 : SNo (- (x * u))
L4981
Hypothesis H13 : SNo (w * y)
L4982
Hypothesis H14 : SNo (- (w * y))
L4983
Hypothesis H15 : SNo (x + - w)
L4984
Hypothesis H16 : SNo (y + - u)
L4985
Hypothesis H17 : v ∈ ω
L4986
Hypothesis H19 : x2 ∈ ω
L4987
Hypothesis H20 : eps_ x2 ≤ y + - u
L4988
Hypothesis H21 : SNo (eps_ v)
L4989
Hypothesis H22 : SNo (eps_ x2)
L4990
Hypothesis H23 : v + x2 ∈ ω
L4991
Theorem. (Conj_real_mul_SNo_pos__46__18)
¬ SNo (eps_ (v + x2))
In Proofgold the corresponding term root is 65e393... and proposition id is 152c9f...
Proof:
Proof not loaded.
End of Section Conj_real_mul_SNo_pos__46__18
Beginning of Section Conj_real_mul_SNo_pos__47__21
L4997
Variable x : set
(*** Conj_real_mul_SNo_pos__47__21 TMF4BGCGTDo4ybAmR83Rh17t22KHayVxJLF bounty of about 25 bars ***)
L4998
Variable y : set
L4999
Variable z : set
L5000
Variable w : set
L5001
Variable u : set
L5002
Variable v : set
L5003
Variable x2 : set
L5004
Hypothesis H0 : SNo x
L5005
Hypothesis H1 : SNo y
L5006
Hypothesis H2 : SNo (x * y)
L5007
Hypothesis H3 : SNo (- (x * y))
L5008
Hypothesis H4 : (∀y2 : set, y2 ∈ ω → abs_SNo (z + - (x * y)) < eps_ y2)
L5009
Hypothesis H5 : SNo z
L5010
Hypothesis H6 : z < x * y
L5011
Hypothesis H7 : SNo w
L5012
Hypothesis H8 : SNo u
L5013
Hypothesis H9 : (z + w * u) ≤ w * y + x * u
L5014
Hypothesis H10 : SNo (w * u)
L5015
Hypothesis H11 : SNo (x * u)
L5016
Hypothesis H12 : SNo (- (x * u))
L5017
Hypothesis H13 : SNo (w * y)
L5018
Hypothesis H14 : SNo (- (w * y))
L5019
Hypothesis H15 : SNo (x + - w)
L5020
Hypothesis H16 : SNo (y + - u)
L5021
Hypothesis H17 : v ∈ ω
L5022
Hypothesis H18 : eps_ v ≤ x + - w
L5023
Hypothesis H19 : x2 ∈ ω
L5024
Hypothesis H20 : eps_ x2 ≤ y + - u
L5025
Hypothesis H22 : SNo (eps_ x2)
L5026
Theorem. (Conj_real_mul_SNo_pos__47__21)
¬ v + x2 ∈ ω
In Proofgold the corresponding term root is b7e39a... and proposition id is 070a57...
Proof:
Proof not loaded.
End of Section Conj_real_mul_SNo_pos__47__21
Beginning of Section Conj_real_mul_SNo_pos__48__19
L5032
Variable x : set
(*** Conj_real_mul_SNo_pos__48__19 TMHKL3W3kSwv2zQzL6LBPqMqE9TTEvmQT4s bounty of about 25 bars ***)
L5033
Variable y : set
L5034
Variable z : set
L5035
Variable w : set
L5036
Variable u : set
L5037
Variable v : set
L5038
Variable x2 : set
L5039
Hypothesis H0 : SNo x
L5040
Hypothesis H1 : SNo y
L5041
Hypothesis H2 : SNo (x * y)
L5042
Hypothesis H3 : SNo (- (x * y))
L5043
Hypothesis H4 : (∀y2 : set, y2 ∈ ω → abs_SNo (z + - (x * y)) < eps_ y2)
L5044
Hypothesis H5 : SNo z
L5045
Hypothesis H6 : z < x * y
L5046
Hypothesis H7 : SNo w
L5047
Hypothesis H8 : SNo u
L5048
Hypothesis H9 : (z + w * u) ≤ w * y + x * u
L5049
Hypothesis H10 : SNo (w * u)
L5050
Hypothesis H11 : SNo (x * u)
L5051
Hypothesis H12 : SNo (- (x * u))
L5052
Hypothesis H13 : SNo (w * y)
L5053
Hypothesis H14 : SNo (- (w * y))
L5054
Hypothesis H15 : SNo (x + - w)
L5055
Hypothesis H16 : SNo (y + - u)
L5056
Hypothesis H17 : v ∈ ω
L5057
Hypothesis H18 : eps_ v ≤ x + - w
L5058
Hypothesis H20 : eps_ x2 ≤ y + - u
L5059
Hypothesis H21 : SNo (eps_ v)
L5060
Theorem. (Conj_real_mul_SNo_pos__48__19)
¬ SNo (eps_ x2)
In Proofgold the corresponding term root is 995f16... and proposition id is f1eb14...
Proof:
Proof not loaded.
End of Section Conj_real_mul_SNo_pos__48__19
Beginning of Section Conj_real_mul_SNo_pos__49__4
L5066
Variable x : set
(*** Conj_real_mul_SNo_pos__49__4 TMazLgmxWH8mpmhuJVNJmzcz5TQsep9nKML bounty of about 25 bars ***)
L5067
Variable y : set
L5068
Variable z : set
L5069
Variable w : set
L5070
Variable u : set
L5071
Variable v : set
L5072
Variable x2 : set
L5073
Hypothesis H0 : SNo x
L5074
Hypothesis H1 : SNo y
L5075
Hypothesis H2 : SNo (x * y)
L5076
Hypothesis H3 : SNo (- (x * y))
L5077
Hypothesis H5 : SNo z
L5078
Hypothesis H6 : z < x * y
L5079
Hypothesis H7 : SNo w
L5080
Hypothesis H8 : SNo u
L5081
Hypothesis H9 : (z + w * u) ≤ w * y + x * u
L5082
Hypothesis H10 : SNo (w * u)
L5083
Hypothesis H11 : SNo (x * u)
L5084
Hypothesis H12 : SNo (- (x * u))
L5085
Hypothesis H13 : SNo (w * y)
L5086
Hypothesis H14 : SNo (- (w * y))
L5087
Hypothesis H15 : SNo (x + - w)
L5088
Hypothesis H16 : SNo (y + - u)
L5089
Hypothesis H17 : v ∈ ω
L5090
Hypothesis H18 : eps_ v ≤ x + - w
L5091
Hypothesis H19 : x2 ∈ ω
L5092
Hypothesis H20 : eps_ x2 ≤ y + - u
L5093
Theorem. (Conj_real_mul_SNo_pos__49__4)
¬ SNo (eps_ v)
In Proofgold the corresponding term root is 7d0b09... and proposition id is 934861...
Proof:
Proof not loaded.
End of Section Conj_real_mul_SNo_pos__49__4
Beginning of Section Conj_real_mul_SNo_pos__51__2
L5099
Variable x : set
(*** Conj_real_mul_SNo_pos__51__2 TMRrqoM1xmaAyz5ZFS9fPi5BxkBCJVW1xyq bounty of about 25 bars ***)
L5100
Variable y : set
L5101
Variable z : set
L5102
Variable w : set
L5103
Variable u : set
L5104
Hypothesis H0 : SNo x
L5105
Hypothesis H1 : SNo y
L5106
Hypothesis H3 : SNo (- (x * y))
L5107
Hypothesis H4 : (∀v : set, v ∈ SNoL x → (∀P : prop, (∀x2 : set, x2 ∈ ω → eps_ x2 ≤ x + - v → P) → P))
L5108
Hypothesis H5 : (∀v : set, v ∈ SNoL y → (∀P : prop, (∀x2 : set, x2 ∈ ω → eps_ x2 ≤ y + - v → P) → P))
L5109
Hypothesis H6 : (∀v : set, v ∈ ω → abs_SNo (z + - (x * y)) < eps_ v)
L5110
Hypothesis H7 : SNo z
L5111
Hypothesis H8 : z < x * y
L5112
Hypothesis H9 : w ∈ SNoL x
L5113
Hypothesis H10 : u ∈ SNoL y
L5114
Hypothesis H11 : SNo w
L5115
Hypothesis H12 : SNo u
L5116
Hypothesis H13 : (z + w * u) ≤ w * y + x * u
L5117
Hypothesis H14 : SNo (w * u)
L5118
Hypothesis H15 : SNo (x * u)
L5119
Hypothesis H16 : SNo (- (x * u))
L5120
Hypothesis H17 : SNo (w * y)
L5121
Hypothesis H18 : SNo (- (w * y))
L5122
Theorem. (Conj_real_mul_SNo_pos__51__2)
¬ SNo (x + - w)
In Proofgold the corresponding term root is f0dac8... and proposition id is c8ca4b...
Proof:
Proof not loaded.
End of Section Conj_real_mul_SNo_pos__51__2
Beginning of Section Conj_real_mul_SNo_pos__53__0
L5128
Variable x : set
(*** Conj_real_mul_SNo_pos__53__0 TMTyQfX87omZgNybwLvYDM8iv3BP9Mh22Hb bounty of about 25 bars ***)
L5129
Variable y : set
L5130
Variable z : set
L5131
Variable w : set
L5132
Variable u : set
L5133
Hypothesis H1 : SNo y
L5134
Hypothesis H2 : SNo (x * y)
L5135
Hypothesis H3 : SNo (- (x * y))
L5136
Hypothesis H4 : (∀v : set, v ∈ SNoL x → (∀P : prop, (∀x2 : set, x2 ∈ ω → eps_ x2 ≤ x + - v → P) → P))
L5137
Hypothesis H5 : (∀v : set, v ∈ SNoL y → (∀P : prop, (∀x2 : set, x2 ∈ ω → eps_ x2 ≤ y + - v → P) → P))
L5138
Hypothesis H6 : (∀v : set, v ∈ ω → abs_SNo (z + - (x * y)) < eps_ v)
L5139
Hypothesis H7 : SNo z
L5140
Hypothesis H8 : z < x * y
L5141
Hypothesis H9 : w ∈ SNoL x
L5142
Hypothesis H10 : u ∈ SNoL y
L5143
Hypothesis H11 : SNo w
L5144
Hypothesis H12 : SNo u
L5145
Hypothesis H13 : (z + w * u) ≤ w * y + x * u
L5146
Hypothesis H14 : SNo (w * u)
L5147
Hypothesis H15 : SNo (x * u)
L5148
Hypothesis H16 : SNo (- (x * u))
L5149
Theorem. (Conj_real_mul_SNo_pos__53__0)
¬ SNo (w * y)
In Proofgold the corresponding term root is 74efbc... and proposition id is 118cdc...
Proof:
Proof not loaded.
End of Section Conj_real_mul_SNo_pos__53__0
Beginning of Section Conj_real_mul_SNo_pos__53__1
L5155
Variable x : set
(*** Conj_real_mul_SNo_pos__53__1 TMV8HpqwnwPFwSRSkS2exHSL7CzFnnFbyZ1 bounty of about 25 bars ***)
L5156
Variable y : set
L5157
Variable z : set
L5158
Variable w : set
L5159
Variable u : set
L5160
Hypothesis H0 : SNo x
L5161
Hypothesis H2 : SNo (x * y)
L5162
Hypothesis H3 : SNo (- (x * y))
L5163
Hypothesis H4 : (∀v : set, v ∈ SNoL x → (∀P : prop, (∀x2 : set, x2 ∈ ω → eps_ x2 ≤ x + - v → P) → P))
L5164
Hypothesis H5 : (∀v : set, v ∈ SNoL y → (∀P : prop, (∀x2 : set, x2 ∈ ω → eps_ x2 ≤ y + - v → P) → P))
L5165
Hypothesis H6 : (∀v : set, v ∈ ω → abs_SNo (z + - (x * y)) < eps_ v)
L5166
Hypothesis H7 : SNo z
L5167
Hypothesis H8 : z < x * y
L5168
Hypothesis H9 : w ∈ SNoL x
L5169
Hypothesis H10 : u ∈ SNoL y
L5170
Hypothesis H11 : SNo w
L5171
Hypothesis H12 : SNo u
L5172
Hypothesis H13 : (z + w * u) ≤ w * y + x * u
L5173
Hypothesis H14 : SNo (w * u)
L5174
Hypothesis H15 : SNo (x * u)
L5175
Hypothesis H16 : SNo (- (x * u))
L5176
Theorem. (Conj_real_mul_SNo_pos__53__1)
¬ SNo (w * y)
In Proofgold the corresponding term root is 0e06ec... and proposition id is e0bff4...
Proof:
Proof not loaded.
End of Section Conj_real_mul_SNo_pos__53__1
Beginning of Section Conj_real_mul_SNo_pos__53__6
L5182
Variable x : set
(*** Conj_real_mul_SNo_pos__53__6 TMYWJyBvBrce5b1A7qcPWzwJ2GQy3mkZLwu bounty of about 25 bars ***)
L5183
Variable y : set
L5184
Variable z : set
L5185
Variable w : set
L5186
Variable u : set
L5187
Hypothesis H0 : SNo x
L5188
Hypothesis H1 : SNo y
L5189
Hypothesis H2 : SNo (x * y)
L5190
Hypothesis H3 : SNo (- (x * y))
L5191
Hypothesis H4 : (∀v : set, v ∈ SNoL x → (∀P : prop, (∀x2 : set, x2 ∈ ω → eps_ x2 ≤ x + - v → P) → P))
L5192
Hypothesis H5 : (∀v : set, v ∈ SNoL y → (∀P : prop, (∀x2 : set, x2 ∈ ω → eps_ x2 ≤ y + - v → P) → P))
L5193
Hypothesis H7 : SNo z
L5194
Hypothesis H8 : z < x * y
L5195
Hypothesis H9 : w ∈ SNoL x
L5196
Hypothesis H10 : u ∈ SNoL y
L5197
Hypothesis H11 : SNo w
L5198
Hypothesis H12 : SNo u
L5199
Hypothesis H13 : (z + w * u) ≤ w * y + x * u
L5200
Hypothesis H14 : SNo (w * u)
L5201
Hypothesis H15 : SNo (x * u)
L5202
Hypothesis H16 : SNo (- (x * u))
L5203
Theorem. (Conj_real_mul_SNo_pos__53__6)
¬ SNo (w * y)
In Proofgold the corresponding term root is 11fb52... and proposition id is 408186...
Proof:
Proof not loaded.
End of Section Conj_real_mul_SNo_pos__53__6
Beginning of Section Conj_real_mul_SNo_pos__53__10
L5209
Variable x : set
(*** Conj_real_mul_SNo_pos__53__10 TMHe6xEbMBdoEsuj6Seoa1spVSD9ZMTHY6j bounty of about 25 bars ***)
L5210
Variable y : set
L5211
Variable z : set
L5212
Variable w : set
L5213
Variable u : set
L5214
Hypothesis H0 : SNo x
L5215
Hypothesis H1 : SNo y
L5216
Hypothesis H2 : SNo (x * y)
L5217
Hypothesis H3 : SNo (- (x * y))
L5218
Hypothesis H4 : (∀v : set, v ∈ SNoL x → (∀P : prop, (∀x2 : set, x2 ∈ ω → eps_ x2 ≤ x + - v → P) → P))
L5219
Hypothesis H5 : (∀v : set, v ∈ SNoL y → (∀P : prop, (∀x2 : set, x2 ∈ ω → eps_ x2 ≤ y + - v → P) → P))
L5220
Hypothesis H6 : (∀v : set, v ∈ ω → abs_SNo (z + - (x * y)) < eps_ v)
L5221
Hypothesis H7 : SNo z
L5222
Hypothesis H8 : z < x * y
L5223
Hypothesis H9 : w ∈ SNoL x
L5224
Hypothesis H11 : SNo w
L5225
Hypothesis H12 : SNo u
L5226
Hypothesis H13 : (z + w * u) ≤ w * y + x * u
L5227
Hypothesis H14 : SNo (w * u)
L5228
Hypothesis H15 : SNo (x * u)
L5229
Hypothesis H16 : SNo (- (x * u))
L5230
Theorem. (Conj_real_mul_SNo_pos__53__10)
¬ SNo (w * y)
In Proofgold the corresponding term root is dae726... and proposition id is 746fd4...
Proof:
Proof not loaded.
End of Section Conj_real_mul_SNo_pos__53__10
Beginning of Section Conj_real_mul_SNo_pos__53__14
L5236
Variable x : set
(*** Conj_real_mul_SNo_pos__53__14 TMbdwAdYDWWmJuvDFzwMcWaHB8bFSad1jkZ bounty of about 25 bars ***)
L5237
Variable y : set
L5238
Variable z : set
L5239
Variable w : set
L5240
Variable u : set
L5241
Hypothesis H0 : SNo x
L5242
Hypothesis H1 : SNo y
L5243
Hypothesis H2 : SNo (x * y)
L5244
Hypothesis H3 : SNo (- (x * y))
L5245
Hypothesis H4 : (∀v : set, v ∈ SNoL x → (∀P : prop, (∀x2 : set, x2 ∈ ω → eps_ x2 ≤ x + - v → P) → P))
L5246
Hypothesis H5 : (∀v : set, v ∈ SNoL y → (∀P : prop, (∀x2 : set, x2 ∈ ω → eps_ x2 ≤ y + - v → P) → P))
L5247
Hypothesis H6 : (∀v : set, v ∈ ω → abs_SNo (z + - (x * y)) < eps_ v)
L5248
Hypothesis H7 : SNo z
L5249
Hypothesis H8 : z < x * y
L5250
Hypothesis H9 : w ∈ SNoL x
L5251
Hypothesis H10 : u ∈ SNoL y
L5252
Hypothesis H11 : SNo w
L5253
Hypothesis H12 : SNo u
L5254
Hypothesis H13 : (z + w * u) ≤ w * y + x * u
L5255
Hypothesis H15 : SNo (x * u)
L5256
Hypothesis H16 : SNo (- (x * u))
L5257
Theorem. (Conj_real_mul_SNo_pos__53__14)
¬ SNo (w * y)
In Proofgold the corresponding term root is 528c11... and proposition id is 69e8be...
Proof:
Proof not loaded.
End of Section Conj_real_mul_SNo_pos__53__14
Beginning of Section Conj_real_mul_SNo_pos__55__8
L5263
Variable x : set
(*** Conj_real_mul_SNo_pos__55__8 TMPwgSrgY9Am1YHBSzfx7vkTYjjkZpSGdBT bounty of about 25 bars ***)
L5264
Variable y : set
L5265
Variable z : set
L5266
Variable w : set
L5267
Variable u : set
L5268
Hypothesis H0 : SNo x
L5269
Hypothesis H1 : SNo y
L5270
Hypothesis H2 : SNo (x * y)
L5271
Hypothesis H3 : SNo (- (x * y))
L5272
Hypothesis H4 : (∀v : set, v ∈ SNoL x → (∀P : prop, (∀x2 : set, x2 ∈ ω → eps_ x2 ≤ x + - v → P) → P))
L5273
Hypothesis H5 : (∀v : set, v ∈ SNoL y → (∀P : prop, (∀x2 : set, x2 ∈ ω → eps_ x2 ≤ y + - v → P) → P))
L5274
Hypothesis H6 : (∀v : set, v ∈ ω → abs_SNo (z + - (x * y)) < eps_ v)
L5275
Hypothesis H7 : SNo z
L5276
Hypothesis H9 : w ∈ SNoL x
L5277
Hypothesis H10 : u ∈ SNoL y
L5278
Hypothesis H11 : SNo w
L5279
Hypothesis H12 : SNo u
L5280
Hypothesis H13 : (z + w * u) ≤ w * y + x * u
L5281
Hypothesis H14 : SNo (w * u)
L5282
Theorem. (Conj_real_mul_SNo_pos__55__8)
¬ SNo (x * u)
In Proofgold the corresponding term root is ba61b1... and proposition id is b403ea...
Proof:
Proof not loaded.
End of Section Conj_real_mul_SNo_pos__55__8
Beginning of Section Conj_real_mul_SNo_pos__59__5
L5288
Variable x : set
(*** Conj_real_mul_SNo_pos__59__5 TMYfp2kyiAPo757NqpD5pCy2rsEs8kj6uTQ bounty of about 25 bars ***)
L5289
Variable y : set
L5290
Variable z : set
L5291
Variable w : set
L5292
Variable u : set
L5293
Variable v : set
L5294
Hypothesis H0 : SNo x
L5295
Hypothesis H1 : SNo y
L5296
Hypothesis H2 : z ∈ ω
L5297
Hypothesis H3 : eps_ z * x < ordsucc Empty
L5298
Hypothesis H4 : eps_ z * y < ordsucc Empty
L5299
Hypothesis H6 : w + ordsucc Empty ∈ ω
L5300
Hypothesis H7 : w + ordsucc (ordsucc Empty) ∈ ω
L5301
Hypothesis H8 : u < x
L5302
Hypothesis H9 : SNo u
L5303
Hypothesis H10 : v < y
L5304
Hypothesis H11 : SNo v
L5305
Hypothesis H12 : SNo (eps_ z)
L5306
Hypothesis H13 : SNo (eps_ (w + ordsucc Empty))
L5307
Hypothesis H14 : SNo (eps_ (w + ordsucc (ordsucc Empty)))
L5308
Theorem. (Conj_real_mul_SNo_pos__59__5)
SNo (eps_ z * eps_ (w + ordsucc (ordsucc Empty))) → (u * eps_ z * eps_ (w + ordsucc (ordsucc Empty)) + (eps_ z * eps_ (w + ordsucc (ordsucc Empty))) * v + (eps_ z * eps_ (w + ordsucc (ordsucc Empty))) * eps_ z * eps_ (w + ordsucc (ordsucc Empty))) < (eps_ (w + ordsucc (ordsucc Empty)) + eps_ (w + ordsucc (ordsucc Empty))) + eps_ (w + ordsucc Empty)
In Proofgold the corresponding term root is a777ba... and proposition id is ef15af...
Proof:
Proof not loaded.
End of Section Conj_real_mul_SNo_pos__59__5
Beginning of Section Conj_real_mul_SNo_pos__69__21
L5314
Variable x : set
(*** Conj_real_mul_SNo_pos__69__21 TMLDKfg1JN8GzHWn9mxuLFpxTXYK2nwHBkL bounty of about 25 bars ***)
L5315
Variable y : set
L5316
Variable z : set
L5317
Variable w : set
L5318
Variable u : set
L5319
Variable v : set
L5320
Variable x2 : set
L5321
Variable y2 : set
L5322
Variable z2 : set
L5323
Hypothesis H0 : SNo x
L5324
Hypothesis H1 : SNo y
L5325
Hypothesis H2 : SNo (x * y)
L5326
Hypothesis H3 : SNo (- (x * y))
L5327
Hypothesis H4 : SNo (SNoCut (Repl ω (ap z)) (Repl ω (ap w)))
L5328
Hypothesis H5 : (∀w2 : set, w2 ∈ ω → SNoCut (Repl ω (ap z)) (Repl ω (ap w)) < ap w w2)
L5329
Hypothesis H6 : (∀w2 : set, w2 ∈ ω → (ap w w2 + - (eps_ w2)) < x * y)
L5330
Hypothesis H7 : u = v * y + x * x2 + - (v * x2)
L5331
Hypothesis H8 : y2 ∈ ω
L5332
Hypothesis H9 : eps_ y2 ≤ v + - x
L5333
Hypothesis H10 : z2 ∈ ω
L5334
Hypothesis H11 : eps_ z2 ≤ y + - x2
L5335
Hypothesis H12 : SNo v
L5336
Hypothesis H13 : SNo x2
L5337
Hypothesis H14 : SNo (eps_ y2)
L5338
Hypothesis H15 : SNo (eps_ z2)
L5339
Hypothesis H16 : y2 + z2 ∈ ω
L5340
Hypothesis H17 : SNo (eps_ (y2 + z2))
L5341
Hypothesis H18 : SNo (- (eps_ (y2 + z2)))
L5342
Hypothesis H19 : SNo (ap w (y2 + z2))
L5343
Hypothesis H20 : SNo (v * y)
L5344
Hypothesis H22 : SNo (- (v * x2))
L5345
Theorem. (Conj_real_mul_SNo_pos__69__21)
SNo (v * y + x * x2 + - (v * x2)) → SNoCut (Repl ω (ap z)) (Repl ω (ap w)) < u
In Proofgold the corresponding term root is 60d560... and proposition id is 9cb5f7...
Proof:
Proof not loaded.
End of Section Conj_real_mul_SNo_pos__69__21
Beginning of Section Conj_real_mul_SNo_pos__70__1
L5351
Variable x : set
(*** Conj_real_mul_SNo_pos__70__1 TMKFSRJdZG6awGoWoPHYrM3BncrHz8YTmoY bounty of about 25 bars ***)
L5352
Variable y : set
L5353
Variable z : set
L5354
Variable w : set
L5355
Variable u : set
L5356
Variable v : set
L5357
Variable x2 : set
L5358
Variable y2 : set
L5359
Variable z2 : set
L5360
Hypothesis H0 : SNo x
L5361
Hypothesis H2 : SNo (x * y)
L5362
Hypothesis H3 : SNo (- (x * y))
L5363
Hypothesis H4 : SNo (SNoCut (Repl ω (ap z)) (Repl ω (ap w)))
L5364
Hypothesis H5 : (∀w2 : set, w2 ∈ ω → SNoCut (Repl ω (ap z)) (Repl ω (ap w)) < ap w w2)
L5365
Hypothesis H6 : (∀w2 : set, w2 ∈ ω → (ap w w2 + - (eps_ w2)) < x * y)
L5366
Hypothesis H7 : u = v * y + x * x2 + - (v * x2)
L5367
Hypothesis H8 : y2 ∈ ω
L5368
Hypothesis H9 : eps_ y2 ≤ v + - x
L5369
Hypothesis H10 : z2 ∈ ω
L5370
Hypothesis H11 : eps_ z2 ≤ y + - x2
L5371
Hypothesis H12 : SNo v
L5372
Hypothesis H13 : SNo x2
L5373
Hypothesis H14 : SNo (eps_ y2)
L5374
Hypothesis H15 : SNo (eps_ z2)
L5375
Hypothesis H16 : y2 + z2 ∈ ω
L5376
Hypothesis H17 : SNo (eps_ (y2 + z2))
L5377
Hypothesis H18 : SNo (- (eps_ (y2 + z2)))
L5378
Hypothesis H19 : SNo (ap w (y2 + z2))
L5379
Hypothesis H20 : SNo (v * y)
L5380
Hypothesis H21 : SNo (x * x2)
L5381
Hypothesis H22 : SNo (v * x2)
L5382
Theorem. (Conj_real_mul_SNo_pos__70__1)
SNo (- (v * x2)) → SNoCut (Repl ω (ap z)) (Repl ω (ap w)) < u
In Proofgold the corresponding term root is 4fb8e4... and proposition id is ddf57a...
Proof:
Proof not loaded.
End of Section Conj_real_mul_SNo_pos__70__1
Beginning of Section Conj_real_mul_SNo_pos__71__3
L5388
Variable x : set
(*** Conj_real_mul_SNo_pos__71__3 TMRCGac7r8fiPRwFyjA14mZXXeyi1zdb3PB bounty of about 25 bars ***)
L5389
Variable y : set
L5390
Variable z : set
L5391
Variable w : set
L5392
Variable u : set
L5393
Variable v : set
L5394
Variable x2 : set
L5395
Variable y2 : set
L5396
Variable z2 : set
L5397
Hypothesis H0 : SNo x
L5398
Hypothesis H1 : SNo y
L5399
Hypothesis H2 : SNo (x * y)
L5400
Hypothesis H4 : SNo (SNoCut (Repl ω (ap z)) (Repl ω (ap w)))
L5401
Hypothesis H5 : (∀w2 : set, w2 ∈ ω → SNoCut (Repl ω (ap z)) (Repl ω (ap w)) < ap w w2)
L5402
Hypothesis H6 : (∀w2 : set, w2 ∈ ω → (ap w w2 + - (eps_ w2)) < x * y)
L5403
Hypothesis H7 : u = v * y + x * x2 + - (v * x2)
L5404
Hypothesis H8 : y2 ∈ ω
L5405
Hypothesis H9 : eps_ y2 ≤ v + - x
L5406
Hypothesis H10 : z2 ∈ ω
L5407
Hypothesis H11 : eps_ z2 ≤ y + - x2
L5408
Hypothesis H12 : SNo v
L5409
Hypothesis H13 : SNo x2
L5410
Hypothesis H14 : SNo (eps_ y2)
L5411
Hypothesis H15 : SNo (eps_ z2)
L5412
Hypothesis H16 : y2 + z2 ∈ ω
L5413
Hypothesis H17 : SNo (eps_ (y2 + z2))
L5414
Hypothesis H18 : SNo (- (eps_ (y2 + z2)))
L5415
Hypothesis H19 : SNo (ap w (y2 + z2))
L5416
Hypothesis H20 : SNo (v * y)
L5417
Hypothesis H21 : SNo (x * x2)
L5418
Theorem. (Conj_real_mul_SNo_pos__71__3)
SNo (v * x2) → SNoCut (Repl ω (ap z)) (Repl ω (ap w)) < u
In Proofgold the corresponding term root is 9ba4e8... and proposition id is 95399e...
Proof:
Proof not loaded.
End of Section Conj_real_mul_SNo_pos__71__3
Beginning of Section Conj_real_mul_SNo_pos__71__18
L5424
Variable x : set
(*** Conj_real_mul_SNo_pos__71__18 TMH9XTxLMp36aJmahrECEPGyxCpdSe6tYqT bounty of about 25 bars ***)
L5425
Variable y : set
L5426
Variable z : set
L5427
Variable w : set
L5428
Variable u : set
L5429
Variable v : set
L5430
Variable x2 : set
L5431
Variable y2 : set
L5432
Variable z2 : set
L5433
Hypothesis H0 : SNo x
L5434
Hypothesis H1 : SNo y
L5435
Hypothesis H2 : SNo (x * y)
L5436
Hypothesis H3 : SNo (- (x * y))
L5437
Hypothesis H4 : SNo (SNoCut (Repl ω (ap z)) (Repl ω (ap w)))
L5438
Hypothesis H5 : (∀w2 : set, w2 ∈ ω → SNoCut (Repl ω (ap z)) (Repl ω (ap w)) < ap w w2)
L5439
Hypothesis H6 : (∀w2 : set, w2 ∈ ω → (ap w w2 + - (eps_ w2)) < x * y)
L5440
Hypothesis H7 : u = v * y + x * x2 + - (v * x2)
L5441
Hypothesis H8 : y2 ∈ ω
L5442
Hypothesis H9 : eps_ y2 ≤ v + - x
L5443
Hypothesis H10 : z2 ∈ ω
L5444
Hypothesis H11 : eps_ z2 ≤ y + - x2
L5445
Hypothesis H12 : SNo v
L5446
Hypothesis H13 : SNo x2
L5447
Hypothesis H14 : SNo (eps_ y2)
L5448
Hypothesis H15 : SNo (eps_ z2)
L5449
Hypothesis H16 : y2 + z2 ∈ ω
L5450
Hypothesis H17 : SNo (eps_ (y2 + z2))
L5451
Hypothesis H19 : SNo (ap w (y2 + z2))
L5452
Hypothesis H20 : SNo (v * y)
L5453
Hypothesis H21 : SNo (x * x2)
L5454
Theorem. (Conj_real_mul_SNo_pos__71__18)
SNo (v * x2) → SNoCut (Repl ω (ap z)) (Repl ω (ap w)) < u
In Proofgold the corresponding term root is a20b5d... and proposition id is edb4ff...
Proof:
Proof not loaded.
End of Section Conj_real_mul_SNo_pos__71__18
Beginning of Section Conj_real_mul_SNo_pos__72__9
L5460
Variable x : set
(*** Conj_real_mul_SNo_pos__72__9 TMV4RUascwakY88yKdVtRf1zgxEQaH3Vf95 bounty of about 25 bars ***)
L5461
Variable y : set
L5462
Variable z : set
L5463
Variable w : set
L5464
Variable u : set
L5465
Variable v : set
L5466
Variable x2 : set
L5467
Variable y2 : set
L5468
Variable z2 : set
L5469
Hypothesis H0 : SNo x
L5470
Hypothesis H1 : SNo y
L5471
Hypothesis H2 : SNo (x * y)
L5472
Hypothesis H3 : SNo (- (x * y))
L5473
Hypothesis H4 : SNo (SNoCut (Repl ω (ap z)) (Repl ω (ap w)))
L5474
Hypothesis H5 : (∀w2 : set, w2 ∈ ω → SNoCut (Repl ω (ap z)) (Repl ω (ap w)) < ap w w2)
L5475
Hypothesis H6 : (∀w2 : set, w2 ∈ ω → (ap w w2 + - (eps_ w2)) < x * y)
L5476
Hypothesis H7 : u = v * y + x * x2 + - (v * x2)
L5477
Hypothesis H8 : y2 ∈ ω
L5478
Hypothesis H10 : z2 ∈ ω
L5479
Hypothesis H11 : eps_ z2 ≤ y + - x2
L5480
Hypothesis H12 : SNo v
L5481
Hypothesis H13 : SNo x2
L5482
Hypothesis H14 : SNo (eps_ y2)
L5483
Hypothesis H15 : SNo (eps_ z2)
L5484
Hypothesis H16 : y2 + z2 ∈ ω
L5485
Hypothesis H17 : SNo (eps_ (y2 + z2))
L5486
Hypothesis H18 : SNo (- (eps_ (y2 + z2)))
L5487
Hypothesis H19 : SNo (ap w (y2 + z2))
L5488
Hypothesis H20 : SNo (v * y)
L5489
Theorem. (Conj_real_mul_SNo_pos__72__9)
SNo (x * x2) → SNoCut (Repl ω (ap z)) (Repl ω (ap w)) < u
In Proofgold the corresponding term root is c7b505... and proposition id is fe4b73...
Proof:
Proof not loaded.
End of Section Conj_real_mul_SNo_pos__72__9
Beginning of Section Conj_real_mul_SNo_pos__72__13
L5495
Variable x : set
(*** Conj_real_mul_SNo_pos__72__13 TMKUcikKKxN15rMiQnLxKPeWqHEjqy3AAz4 bounty of about 25 bars ***)
L5496
Variable y : set
L5497
Variable z : set
L5498
Variable w : set
L5499
Variable u : set
L5500
Variable v : set
L5501
Variable x2 : set
L5502
Variable y2 : set
L5503
Variable z2 : set
L5504
Hypothesis H0 : SNo x
L5505
Hypothesis H1 : SNo y
L5506
Hypothesis H2 : SNo (x * y)
L5507
Hypothesis H3 : SNo (- (x * y))
L5508
Hypothesis H4 : SNo (SNoCut (Repl ω (ap z)) (Repl ω (ap w)))
L5509
Hypothesis H5 : (∀w2 : set, w2 ∈ ω → SNoCut (Repl ω (ap z)) (Repl ω (ap w)) < ap w w2)
L5510
Hypothesis H6 : (∀w2 : set, w2 ∈ ω → (ap w w2 + - (eps_ w2)) < x * y)
L5511
Hypothesis H7 : u = v * y + x * x2 + - (v * x2)
L5512
Hypothesis H8 : y2 ∈ ω
L5513
Hypothesis H9 : eps_ y2 ≤ v + - x
L5514
Hypothesis H10 : z2 ∈ ω
L5515
Hypothesis H11 : eps_ z2 ≤ y + - x2
L5516
Hypothesis H12 : SNo v
L5517
Hypothesis H14 : SNo (eps_ y2)
L5518
Hypothesis H15 : SNo (eps_ z2)
L5519
Hypothesis H16 : y2 + z2 ∈ ω
L5520
Hypothesis H17 : SNo (eps_ (y2 + z2))
L5521
Hypothesis H18 : SNo (- (eps_ (y2 + z2)))
L5522
Hypothesis H19 : SNo (ap w (y2 + z2))
L5523
Hypothesis H20 : SNo (v * y)
L5524
Theorem. (Conj_real_mul_SNo_pos__72__13)
SNo (x * x2) → SNoCut (Repl ω (ap z)) (Repl ω (ap w)) < u
In Proofgold the corresponding term root is 3812ab... and proposition id is 645841...
Proof:
Proof not loaded.
End of Section Conj_real_mul_SNo_pos__72__13
Beginning of Section Conj_real_mul_SNo_pos__73__16
L5530
Variable x : set
(*** Conj_real_mul_SNo_pos__73__16 TMavqhXhJZxKGNTaBveHCj3LU5rGaC1ehom bounty of about 25 bars ***)
L5531
Variable y : set
L5532
Variable z : set
L5533
Variable w : set
L5534
Variable u : set
L5535
Variable v : set
L5536
Variable x2 : set
L5537
Variable y2 : set
L5538
Variable z2 : set
L5539
Hypothesis H0 : SNo x
L5540
Hypothesis H1 : SNo y
L5541
Hypothesis H2 : SNo (x * y)
L5542
Hypothesis H3 : SNo (- (x * y))
L5543
Hypothesis H4 : SNo (SNoCut (Repl ω (ap z)) (Repl ω (ap w)))
L5544
Hypothesis H5 : (∀w2 : set, w2 ∈ ω → SNoCut (Repl ω (ap z)) (Repl ω (ap w)) < ap w w2)
L5545
Hypothesis H6 : (∀w2 : set, w2 ∈ ω → (ap w w2 + - (eps_ w2)) < x * y)
L5546
Hypothesis H7 : u = v * y + x * x2 + - (v * x2)
L5547
Hypothesis H8 : y2 ∈ ω
L5548
Hypothesis H9 : eps_ y2 ≤ v + - x
L5549
Hypothesis H10 : z2 ∈ ω
L5550
Hypothesis H11 : eps_ z2 ≤ y + - x2
L5551
Hypothesis H12 : SNo v
L5552
Hypothesis H13 : SNo x2
L5553
Hypothesis H14 : SNo (eps_ y2)
L5554
Hypothesis H15 : SNo (eps_ z2)
L5555
Hypothesis H17 : SNo (eps_ (y2 + z2))
L5556
Hypothesis H18 : SNo (- (eps_ (y2 + z2)))
L5557
Hypothesis H19 : SNo (ap w (y2 + z2))
L5558
Theorem. (Conj_real_mul_SNo_pos__73__16)
SNo (v * y) → SNoCut (Repl ω (ap z)) (Repl ω (ap w)) < u
In Proofgold the corresponding term root is 04578a... and proposition id is 2cf2fd...
Proof:
Proof not loaded.
End of Section Conj_real_mul_SNo_pos__73__16
Beginning of Section Conj_real_mul_SNo_pos__76__16
L5564
Variable x : set
(*** Conj_real_mul_SNo_pos__76__16 TMUvELkFgMHCSvioerw59HAU4nU2kKZAxNS bounty of about 25 bars ***)
L5565
Variable y : set
L5566
Variable z : set
L5567
Variable w : set
L5568
Variable u : set
L5569
Variable v : set
L5570
Variable x2 : set
L5571
Variable y2 : set
L5572
Variable z2 : set
L5573
Hypothesis H0 : SNo x
L5574
Hypothesis H1 : SNo y
L5575
Hypothesis H2 : SNo (x * y)
L5576
Hypothesis H3 : SNo (- (x * y))
L5577
Hypothesis H4 : (∀w2 : set, w2 ∈ ω → SNo (ap w w2))
L5578
Hypothesis H5 : SNo (SNoCut (Repl ω (ap z)) (Repl ω (ap w)))
L5579
Hypothesis H6 : (∀w2 : set, w2 ∈ ω → SNoCut (Repl ω (ap z)) (Repl ω (ap w)) < ap w w2)
L5580
Hypothesis H7 : (∀w2 : set, w2 ∈ ω → (ap w w2 + - (eps_ w2)) < x * y)
L5581
Hypothesis H8 : u = v * y + x * x2 + - (v * x2)
L5582
Hypothesis H9 : y2 ∈ ω
L5583
Hypothesis H10 : eps_ y2 ≤ v + - x
L5584
Hypothesis H11 : z2 ∈ ω
L5585
Hypothesis H12 : eps_ z2 ≤ y + - x2
L5586
Hypothesis H13 : SNo v
L5587
Hypothesis H14 : SNo x2
L5588
Hypothesis H15 : SNo (eps_ y2)
L5589
Hypothesis H17 : y2 + z2 ∈ ω
L5590
Theorem. (Conj_real_mul_SNo_pos__76__16)
SNo (eps_ (y2 + z2)) → SNoCut (Repl ω (ap z)) (Repl ω (ap w)) < u
In Proofgold the corresponding term root is b53ad1... and proposition id is d4b91d...
Proof:
Proof not loaded.
End of Section Conj_real_mul_SNo_pos__76__16
Beginning of Section Conj_real_mul_SNo_pos__77__10
L5596
Variable x : set
(*** Conj_real_mul_SNo_pos__77__10 TMP7spghSWDhmBudt8MhBRRP58HbokNfGwd bounty of about 25 bars ***)
L5597
Variable y : set
L5598
Variable z : set
L5599
Variable w : set
L5600
Variable u : set
L5601
Variable v : set
L5602
Variable x2 : set
L5603
Variable y2 : set
L5604
Variable z2 : set
L5605
Hypothesis H0 : SNo x
L5606
Hypothesis H1 : SNo y
L5607
Hypothesis H2 : SNo (x * y)
L5608
Hypothesis H3 : SNo (- (x * y))
L5609
Hypothesis H4 : (∀w2 : set, w2 ∈ ω → SNo (ap w w2))
L5610
Hypothesis H5 : SNo (SNoCut (Repl ω (ap z)) (Repl ω (ap w)))
L5611
Hypothesis H6 : (∀w2 : set, w2 ∈ ω → SNoCut (Repl ω (ap z)) (Repl ω (ap w)) < ap w w2)
L5612
Hypothesis H7 : (∀w2 : set, w2 ∈ ω → (ap w w2 + - (eps_ w2)) < x * y)
L5613
Hypothesis H8 : u = v * y + x * x2 + - (v * x2)
L5614
Hypothesis H9 : y2 ∈ ω
L5615
Hypothesis H11 : z2 ∈ ω
L5616
Hypothesis H12 : eps_ z2 ≤ y + - x2
L5617
Hypothesis H13 : SNo v
L5618
Hypothesis H14 : SNo x2
L5619
Hypothesis H15 : SNo (eps_ y2)
L5620
Hypothesis H16 : SNo (eps_ z2)
L5621
Theorem. (Conj_real_mul_SNo_pos__77__10)
y2 + z2 ∈ ω → SNoCut (Repl ω (ap z)) (Repl ω (ap w)) < u
In Proofgold the corresponding term root is 318429... and proposition id is 084e1d...
Proof:
Proof not loaded.
End of Section Conj_real_mul_SNo_pos__77__10
Beginning of Section Conj_real_mul_SNo_pos__84__2
L5627
Variable x : set
(*** Conj_real_mul_SNo_pos__84__2 TMYaoVDtuKAyxHHR5vbiDH8NvLiSPNcsriK bounty of about 25 bars ***)
L5628
Variable y : set
L5629
Variable z : set
L5630
Variable w : set
L5631
Variable u : set
L5632
Variable v : set
L5633
Variable x2 : set
L5634
Variable y2 : set
L5635
Variable z2 : set
L5636
Hypothesis H0 : SNo x
L5637
Hypothesis H1 : SNo y
L5638
Hypothesis H3 : SNo (- (x * y))
L5639
Hypothesis H4 : SNo (SNoCut (Repl ω (ap z)) (Repl ω (ap w)))
L5640
Hypothesis H5 : (∀w2 : set, w2 ∈ ω → SNoCut (Repl ω (ap z)) (Repl ω (ap w)) < ap w w2)
L5641
Hypothesis H6 : (∀w2 : set, w2 ∈ ω → (ap w w2 + - (eps_ w2)) < x * y)
L5642
Hypothesis H7 : u = v * y + x * x2 + - (v * x2)
L5643
Hypothesis H8 : y2 ∈ ω
L5644
Hypothesis H9 : eps_ y2 ≤ x + - v
L5645
Hypothesis H10 : z2 ∈ ω
L5646
Hypothesis H11 : eps_ z2 ≤ x2 + - y
L5647
Hypothesis H12 : SNo v
L5648
Hypothesis H13 : SNo x2
L5649
Hypothesis H14 : SNo (eps_ y2)
L5650
Hypothesis H15 : SNo (eps_ z2)
L5651
Hypothesis H16 : y2 + z2 ∈ ω
L5652
Hypothesis H17 : SNo (eps_ (y2 + z2))
L5653
Hypothesis H18 : SNo (- (eps_ (y2 + z2)))
L5654
Hypothesis H19 : SNo (ap w (y2 + z2))
L5655
Theorem. (Conj_real_mul_SNo_pos__84__2)
SNo (v * y) → SNoCut (Repl ω (ap z)) (Repl ω (ap w)) < u
In Proofgold the corresponding term root is d15734... and proposition id is 4084d2...
Proof:
Proof not loaded.
End of Section Conj_real_mul_SNo_pos__84__2
Beginning of Section Conj_real_mul_SNo_pos__85__6
L5661
Variable x : set
(*** Conj_real_mul_SNo_pos__85__6 TMMT46Q6jcs4CzRLXZGgdpoxmrVM5bPdM7A bounty of about 25 bars ***)
L5662
Variable y : set
L5663
Variable z : set
L5664
Variable w : set
L5665
Variable u : set
L5666
Variable v : set
L5667
Variable x2 : set
L5668
Variable y2 : set
L5669
Variable z2 : set
L5670
Hypothesis H0 : SNo x
L5671
Hypothesis H1 : SNo y
L5672
Hypothesis H2 : SNo (x * y)
L5673
Hypothesis H3 : SNo (- (x * y))
L5674
Hypothesis H4 : (∀w2 : set, w2 ∈ ω → SNo (ap w w2))
L5675
Hypothesis H5 : SNo (SNoCut (Repl ω (ap z)) (Repl ω (ap w)))
L5676
Hypothesis H7 : (∀w2 : set, w2 ∈ ω → (ap w w2 + - (eps_ w2)) < x * y)
L5677
Hypothesis H8 : u = v * y + x * x2 + - (v * x2)
L5678
Hypothesis H9 : y2 ∈ ω
L5679
Hypothesis H10 : eps_ y2 ≤ x + - v
L5680
Hypothesis H11 : z2 ∈ ω
L5681
Hypothesis H12 : eps_ z2 ≤ x2 + - y
L5682
Hypothesis H13 : SNo v
L5683
Hypothesis H14 : SNo x2
L5684
Hypothesis H15 : SNo (eps_ y2)
L5685
Hypothesis H16 : SNo (eps_ z2)
L5686
Hypothesis H17 : y2 + z2 ∈ ω
L5687
Hypothesis H18 : SNo (eps_ (y2 + z2))
L5688
Hypothesis H19 : SNo (- (eps_ (y2 + z2)))
L5689
Theorem. (Conj_real_mul_SNo_pos__85__6)
SNo (ap w (y2 + z2)) → SNoCut (Repl ω (ap z)) (Repl ω (ap w)) < u
In Proofgold the corresponding term root is 51fd37... and proposition id is 9f2b01...
Proof:
Proof not loaded.
End of Section Conj_real_mul_SNo_pos__85__6
Beginning of Section Conj_real_mul_SNo_pos__85__14
L5695
Variable x : set
(*** Conj_real_mul_SNo_pos__85__14 TMWdzBZ64sitNpnkdBGJJJPf7zRkVQH3AWM bounty of about 25 bars ***)
L5696
Variable y : set
L5697
Variable z : set
L5698
Variable w : set
L5699
Variable u : set
L5700
Variable v : set
L5701
Variable x2 : set
L5702
Variable y2 : set
L5703
Variable z2 : set
L5704
Hypothesis H0 : SNo x
L5705
Hypothesis H1 : SNo y
L5706
Hypothesis H2 : SNo (x * y)
L5707
Hypothesis H3 : SNo (- (x * y))
L5708
Hypothesis H4 : (∀w2 : set, w2 ∈ ω → SNo (ap w w2))
L5709
Hypothesis H5 : SNo (SNoCut (Repl ω (ap z)) (Repl ω (ap w)))
L5710
Hypothesis H6 : (∀w2 : set, w2 ∈ ω → SNoCut (Repl ω (ap z)) (Repl ω (ap w)) < ap w w2)
L5711
Hypothesis H7 : (∀w2 : set, w2 ∈ ω → (ap w w2 + - (eps_ w2)) < x * y)
L5712
Hypothesis H8 : u = v * y + x * x2 + - (v * x2)
L5713
Hypothesis H9 : y2 ∈ ω
L5714
Hypothesis H10 : eps_ y2 ≤ x + - v
L5715
Hypothesis H11 : z2 ∈ ω
L5716
Hypothesis H12 : eps_ z2 ≤ x2 + - y
L5717
Hypothesis H13 : SNo v
L5718
Hypothesis H15 : SNo (eps_ y2)
L5719
Hypothesis H16 : SNo (eps_ z2)
L5720
Hypothesis H17 : y2 + z2 ∈ ω
L5721
Hypothesis H18 : SNo (eps_ (y2 + z2))
L5722
Hypothesis H19 : SNo (- (eps_ (y2 + z2)))
L5723
Theorem. (Conj_real_mul_SNo_pos__85__14)
SNo (ap w (y2 + z2)) → SNoCut (Repl ω (ap z)) (Repl ω (ap w)) < u
In Proofgold the corresponding term root is 0fa5cc... and proposition id is 296bf0...
Proof:
Proof not loaded.
End of Section Conj_real_mul_SNo_pos__85__14
Beginning of Section Conj_real_mul_SNo_pos__85__19
L5729
Variable x : set
(*** Conj_real_mul_SNo_pos__85__19 TMUXjYoxk8YmZQodrkRoJZ9wDduJgayZTrD bounty of about 25 bars ***)
L5730
Variable y : set
L5731
Variable z : set
L5732
Variable w : set
L5733
Variable u : set
L5734
Variable v : set
L5735
Variable x2 : set
L5736
Variable y2 : set
L5737
Variable z2 : set
L5738
Hypothesis H0 : SNo x
L5739
Hypothesis H1 : SNo y
L5740
Hypothesis H2 : SNo (x * y)
L5741
Hypothesis H3 : SNo (- (x * y))
L5742
Hypothesis H4 : (∀w2 : set, w2 ∈ ω → SNo (ap w w2))
L5743
Hypothesis H5 : SNo (SNoCut (Repl ω (ap z)) (Repl ω (ap w)))
L5744
Hypothesis H6 : (∀w2 : set, w2 ∈ ω → SNoCut (Repl ω (ap z)) (Repl ω (ap w)) < ap w w2)
L5745
Hypothesis H7 : (∀w2 : set, w2 ∈ ω → (ap w w2 + - (eps_ w2)) < x * y)
L5746
Hypothesis H8 : u = v * y + x * x2 + - (v * x2)
L5747
Hypothesis H9 : y2 ∈ ω
L5748
Hypothesis H10 : eps_ y2 ≤ x + - v
L5749
Hypothesis H11 : z2 ∈ ω
L5750
Hypothesis H12 : eps_ z2 ≤ x2 + - y
L5751
Hypothesis H13 : SNo v
L5752
Hypothesis H14 : SNo x2
L5753
Hypothesis H15 : SNo (eps_ y2)
L5754
Hypothesis H16 : SNo (eps_ z2)
L5755
Hypothesis H17 : y2 + z2 ∈ ω
L5756
Hypothesis H18 : SNo (eps_ (y2 + z2))
L5757
Theorem. (Conj_real_mul_SNo_pos__85__19)
SNo (ap w (y2 + z2)) → SNoCut (Repl ω (ap z)) (Repl ω (ap w)) < u
In Proofgold the corresponding term root is 228ea1... and proposition id is 4ff90e...
Proof:
Proof not loaded.
End of Section Conj_real_mul_SNo_pos__85__19
Beginning of Section Conj_real_mul_SNo_pos__86__4
L5763
Variable x : set
(*** Conj_real_mul_SNo_pos__86__4 TMNyw3HmEqwWf7fEoWNCgzrsamFfPV2Sqy8 bounty of about 25 bars ***)
L5764
Variable y : set
L5765
Variable z : set
L5766
Variable w : set
L5767
Variable u : set
L5768
Variable v : set
L5769
Variable x2 : set
L5770
Variable y2 : set
L5771
Variable z2 : set
L5772
Hypothesis H0 : SNo x
L5773
Hypothesis H1 : SNo y
L5774
Hypothesis H2 : SNo (x * y)
L5775
Hypothesis H3 : SNo (- (x * y))
L5776
Hypothesis H5 : SNo (SNoCut (Repl ω (ap z)) (Repl ω (ap w)))
L5777
Hypothesis H6 : (∀w2 : set, w2 ∈ ω → SNoCut (Repl ω (ap z)) (Repl ω (ap w)) < ap w w2)
L5778
Hypothesis H7 : (∀w2 : set, w2 ∈ ω → (ap w w2 + - (eps_ w2)) < x * y)
L5779
Hypothesis H8 : u = v * y + x * x2 + - (v * x2)
L5780
Hypothesis H9 : y2 ∈ ω
L5781
Hypothesis H10 : eps_ y2 ≤ x + - v
L5782
Hypothesis H11 : z2 ∈ ω
L5783
Hypothesis H12 : eps_ z2 ≤ x2 + - y
L5784
Hypothesis H13 : SNo v
L5785
Hypothesis H14 : SNo x2
L5786
Hypothesis H15 : SNo (eps_ y2)
L5787
Hypothesis H16 : SNo (eps_ z2)
L5788
Hypothesis H17 : y2 + z2 ∈ ω
L5789
Hypothesis H18 : SNo (eps_ (y2 + z2))
L5790
Theorem. (Conj_real_mul_SNo_pos__86__4)
SNo (- (eps_ (y2 + z2))) → SNoCut (Repl ω (ap z)) (Repl ω (ap w)) < u
In Proofgold the corresponding term root is b20495... and proposition id is 16d2b5...
Proof:
Proof not loaded.
End of Section Conj_real_mul_SNo_pos__86__4
Beginning of Section Conj_real_mul_SNo_pos__86__18
L5796
Variable x : set
(*** Conj_real_mul_SNo_pos__86__18 TMdSBk7x2HshTHBCnLsRavjer91rHjqXLSg bounty of about 25 bars ***)
L5797
Variable y : set
L5798
Variable z : set
L5799
Variable w : set
L5800
Variable u : set
L5801
Variable v : set
L5802
Variable x2 : set
L5803
Variable y2 : set
L5804
Variable z2 : set
L5805
Hypothesis H0 : SNo x
L5806
Hypothesis H1 : SNo y
L5807
Hypothesis H2 : SNo (x * y)
L5808
Hypothesis H3 : SNo (- (x * y))
L5809
Hypothesis H4 : (∀w2 : set, w2 ∈ ω → SNo (ap w w2))
L5810
Hypothesis H5 : SNo (SNoCut (Repl ω (ap z)) (Repl ω (ap w)))
L5811
Hypothesis H6 : (∀w2 : set, w2 ∈ ω → SNoCut (Repl ω (ap z)) (Repl ω (ap w)) < ap w w2)
L5812
Hypothesis H7 : (∀w2 : set, w2 ∈ ω → (ap w w2 + - (eps_ w2)) < x * y)
L5813
Hypothesis H8 : u = v * y + x * x2 + - (v * x2)
L5814
Hypothesis H9 : y2 ∈ ω
L5815
Hypothesis H10 : eps_ y2 ≤ x + - v
L5816
Hypothesis H11 : z2 ∈ ω
L5817
Hypothesis H12 : eps_ z2 ≤ x2 + - y
L5818
Hypothesis H13 : SNo v
L5819
Hypothesis H14 : SNo x2
L5820
Hypothesis H15 : SNo (eps_ y2)
L5821
Hypothesis H16 : SNo (eps_ z2)
L5822
Hypothesis H17 : y2 + z2 ∈ ω
L5823
Theorem. (Conj_real_mul_SNo_pos__86__18)
SNo (- (eps_ (y2 + z2))) → SNoCut (Repl ω (ap z)) (Repl ω (ap w)) < u
In Proofgold the corresponding term root is 45902a... and proposition id is 4be90e...
Proof:
Proof not loaded.
End of Section Conj_real_mul_SNo_pos__86__18
Beginning of Section Conj_real_mul_SNo_pos__87__14
L5829
Variable x : set
(*** Conj_real_mul_SNo_pos__87__14 TMLrgfFoPgJS9Pnupz55Z2qw5zScDM4P4Ps bounty of about 25 bars ***)
L5830
Variable y : set
L5831
Variable z : set
L5832
Variable w : set
L5833
Variable u : set
L5834
Variable v : set
L5835
Variable x2 : set
L5836
Variable y2 : set
L5837
Variable z2 : set
L5838
Hypothesis H0 : SNo x
L5839
Hypothesis H1 : SNo y
L5840
Hypothesis H2 : SNo (x * y)
L5841
Hypothesis H3 : SNo (- (x * y))
L5842
Hypothesis H4 : (∀w2 : set, w2 ∈ ω → SNo (ap w w2))
L5843
Hypothesis H5 : SNo (SNoCut (Repl ω (ap z)) (Repl ω (ap w)))
L5844
Hypothesis H6 : (∀w2 : set, w2 ∈ ω → SNoCut (Repl ω (ap z)) (Repl ω (ap w)) < ap w w2)
L5845
Hypothesis H7 : (∀w2 : set, w2 ∈ ω → (ap w w2 + - (eps_ w2)) < x * y)
L5846
Hypothesis H8 : u = v * y + x * x2 + - (v * x2)
L5847
Hypothesis H9 : y2 ∈ ω
L5848
Hypothesis H10 : eps_ y2 ≤ x + - v
L5849
Hypothesis H11 : z2 ∈ ω
L5850
Hypothesis H12 : eps_ z2 ≤ x2 + - y
L5851
Hypothesis H13 : SNo v
L5852
Hypothesis H15 : SNo (eps_ y2)
L5853
Hypothesis H16 : SNo (eps_ z2)
L5854
Hypothesis H17 : y2 + z2 ∈ ω
L5855
Theorem. (Conj_real_mul_SNo_pos__87__14)
SNo (eps_ (y2 + z2)) → SNoCut (Repl ω (ap z)) (Repl ω (ap w)) < u
In Proofgold the corresponding term root is 26fa43... and proposition id is 1ac53c...
Proof:
Proof not loaded.
End of Section Conj_real_mul_SNo_pos__87__14
Beginning of Section Conj_real_mul_SNo_pos__88__10
L5861
Variable x : set
(*** Conj_real_mul_SNo_pos__88__10 TMF62MDyp1m1NKt3LjpM27aipbovhL1NfmY bounty of about 25 bars ***)
L5862
Variable y : set
L5863
Variable z : set
L5864
Variable w : set
L5865
Variable u : set
L5866
Variable v : set
L5867
Variable x2 : set
L5868
Variable y2 : set
L5869
Variable z2 : set
L5870
Hypothesis H0 : SNo x
L5871
Hypothesis H1 : SNo y
L5872
Hypothesis H2 : SNo (x * y)
L5873
Hypothesis H3 : SNo (- (x * y))
L5874
Hypothesis H4 : (∀w2 : set, w2 ∈ ω → SNo (ap w w2))
L5875
Hypothesis H5 : SNo (SNoCut (Repl ω (ap z)) (Repl ω (ap w)))
L5876
Hypothesis H6 : (∀w2 : set, w2 ∈ ω → SNoCut (Repl ω (ap z)) (Repl ω (ap w)) < ap w w2)
L5877
Hypothesis H7 : (∀w2 : set, w2 ∈ ω → (ap w w2 + - (eps_ w2)) < x * y)
L5878
Hypothesis H8 : u = v * y + x * x2 + - (v * x2)
L5879
Hypothesis H9 : y2 ∈ ω
L5880
Hypothesis H11 : z2 ∈ ω
L5881
Hypothesis H12 : eps_ z2 ≤ x2 + - y
L5882
Hypothesis H13 : SNo v
L5883
Hypothesis H14 : SNo x2
L5884
Hypothesis H15 : SNo (eps_ y2)
L5885
Hypothesis H16 : SNo (eps_ z2)
L5886
Theorem. (Conj_real_mul_SNo_pos__88__10)
y2 + z2 ∈ ω → SNoCut (Repl ω (ap z)) (Repl ω (ap w)) < u
In Proofgold the corresponding term root is 491784... and proposition id is c9c72a...
Proof:
Proof not loaded.
End of Section Conj_real_mul_SNo_pos__88__10
Beginning of Section Conj_real_mul_SNo_pos__90__14
L5892
Variable x : set
(*** Conj_real_mul_SNo_pos__90__14 TMbudEvWFMNHLGb8uRp7rAVcxjE2G7XbRLq bounty of about 25 bars ***)
L5893
Variable y : set
L5894
Variable z : set
L5895
Variable w : set
L5896
Variable u : set
L5897
Variable v : set
L5898
Variable x2 : set
L5899
Variable y2 : set
L5900
Variable z2 : set
L5901
Hypothesis H0 : SNo x
L5902
Hypothesis H1 : SNo y
L5903
Hypothesis H2 : SNo (x * y)
L5904
Hypothesis H3 : SNo (- (x * y))
L5905
Hypothesis H4 : (∀w2 : set, w2 ∈ ω → SNo (ap w w2))
L5906
Hypothesis H5 : SNo (SNoCut (Repl ω (ap z)) (Repl ω (ap w)))
L5907
Hypothesis H6 : (∀w2 : set, w2 ∈ ω → SNoCut (Repl ω (ap z)) (Repl ω (ap w)) < ap w w2)
L5908
Hypothesis H7 : (∀w2 : set, w2 ∈ ω → (ap w w2 + - (eps_ w2)) < x * y)
L5909
Hypothesis H8 : u = v * y + x * x2 + - (v * x2)
L5910
Hypothesis H9 : y2 ∈ ω
L5911
Hypothesis H10 : eps_ y2 ≤ x + - v
L5912
Hypothesis H11 : z2 ∈ ω
L5913
Hypothesis H12 : eps_ z2 ≤ x2 + - y
L5914
Hypothesis H13 : SNo v
L5915
Theorem. (Conj_real_mul_SNo_pos__90__14)
SNo (eps_ y2) → SNoCut (Repl ω (ap z)) (Repl ω (ap w)) < u
In Proofgold the corresponding term root is d24498... and proposition id is f186ed...
Proof:
Proof not loaded.
End of Section Conj_real_mul_SNo_pos__90__14
Beginning of Section Conj_real_mul_SNo_pos__93__1
L5921
Variable x : set
(*** Conj_real_mul_SNo_pos__93__1 TMMNuwvWGseTceibVcc5PYNTszjHsSkAKGp bounty of about 25 bars ***)
L5922
Variable y : set
L5923
Variable z : set
L5924
Variable w : set
L5925
Variable u : set
L5926
Variable v : set
L5927
Variable x2 : set
L5928
Variable y2 : set
L5929
Variable z2 : set
L5930
Hypothesis H0 : SNo x
L5931
Hypothesis H2 : SNo (x * y)
L5932
Hypothesis H3 : SNo (SNoCut (Repl ω (ap z)) (Repl ω (ap w)))
L5933
Hypothesis H4 : (∀w2 : set, w2 ∈ ω → ap z w2 < SNoCut (Repl ω (ap z)) (Repl ω (ap w)))
L5934
Hypothesis H5 : (∀w2 : set, w2 ∈ ω → x * y < ap z w2 + eps_ w2)
L5935
Hypothesis H6 : u = v * y + x * x2 + - (v * x2)
L5936
Hypothesis H7 : y2 ∈ ω
L5937
Hypothesis H8 : eps_ y2 ≤ v + - x
L5938
Hypothesis H9 : z2 ∈ ω
L5939
Hypothesis H10 : eps_ z2 ≤ x2 + - y
L5940
Hypothesis H11 : SNo v
L5941
Hypothesis H12 : SNo x2
L5942
Hypothesis H13 : SNo (eps_ y2)
L5943
Hypothesis H14 : SNo (eps_ z2)
L5944
Hypothesis H15 : y2 + z2 ∈ ω
L5945
Hypothesis H16 : SNo (eps_ (y2 + z2))
L5946
Hypothesis H17 : SNo (ap z (y2 + z2))
L5947
Hypothesis H18 : SNo (v * y)
L5948
Hypothesis H19 : SNo (x * x2)
L5949
Theorem. (Conj_real_mul_SNo_pos__93__1)
SNo (v * x2) → u < SNoCut (Repl ω (ap z)) (Repl ω (ap w))
In Proofgold the corresponding term root is e188f5... and proposition id is dde1c2...
Proof:
Proof not loaded.
End of Section Conj_real_mul_SNo_pos__93__1
Beginning of Section Conj_real_mul_SNo_pos__93__16
L5955
Variable x : set
(*** Conj_real_mul_SNo_pos__93__16 TMRo6vy1XXhLQfqwtP4pijfyb9abm8kmUd5 bounty of about 25 bars ***)
L5956
Variable y : set
L5957
Variable z : set
L5958
Variable w : set
L5959
Variable u : set
L5960
Variable v : set
L5961
Variable x2 : set
L5962
Variable y2 : set
L5963
Variable z2 : set
L5964
Hypothesis H0 : SNo x
L5965
Hypothesis H1 : SNo y
L5966
Hypothesis H2 : SNo (x * y)
L5967
Hypothesis H3 : SNo (SNoCut (Repl ω (ap z)) (Repl ω (ap w)))
L5968
Hypothesis H4 : (∀w2 : set, w2 ∈ ω → ap z w2 < SNoCut (Repl ω (ap z)) (Repl ω (ap w)))
L5969
Hypothesis H5 : (∀w2 : set, w2 ∈ ω → x * y < ap z w2 + eps_ w2)
L5970
Hypothesis H6 : u = v * y + x * x2 + - (v * x2)
L5971
Hypothesis H7 : y2 ∈ ω
L5972
Hypothesis H8 : eps_ y2 ≤ v + - x
L5973
Hypothesis H9 : z2 ∈ ω
L5974
Hypothesis H10 : eps_ z2 ≤ x2 + - y
L5975
Hypothesis H11 : SNo v
L5976
Hypothesis H12 : SNo x2
L5977
Hypothesis H13 : SNo (eps_ y2)
L5978
Hypothesis H14 : SNo (eps_ z2)
L5979
Hypothesis H15 : y2 + z2 ∈ ω
L5980
Hypothesis H17 : SNo (ap z (y2 + z2))
L5981
Hypothesis H18 : SNo (v * y)
L5982
Hypothesis H19 : SNo (x * x2)
L5983
Theorem. (Conj_real_mul_SNo_pos__93__16)
SNo (v * x2) → u < SNoCut (Repl ω (ap z)) (Repl ω (ap w))
In Proofgold the corresponding term root is 530585... and proposition id is 64fefe...
Proof:
Proof not loaded.
End of Section Conj_real_mul_SNo_pos__93__16
Beginning of Section Conj_real_mul_SNo_pos__94__6
L5989
Variable x : set
(*** Conj_real_mul_SNo_pos__94__6 TMbgTCgYKSERHSQs7YuweXcpMN1N1LbrkZz bounty of about 25 bars ***)
L5990
Variable y : set
L5991
Variable z : set
L5992
Variable w : set
L5993
Variable u : set
L5994
Variable v : set
L5995
Variable x2 : set
L5996
Variable y2 : set
L5997
Variable z2 : set
L5998
Hypothesis H0 : SNo x
L5999
Hypothesis H1 : SNo y
L6000
Hypothesis H2 : SNo (x * y)
L6001
Hypothesis H3 : SNo (SNoCut (Repl ω (ap z)) (Repl ω (ap w)))
L6002
Hypothesis H4 : (∀w2 : set, w2 ∈ ω → ap z w2 < SNoCut (Repl ω (ap z)) (Repl ω (ap w)))
L6003
Hypothesis H5 : (∀w2 : set, w2 ∈ ω → x * y < ap z w2 + eps_ w2)
L6004
Hypothesis H7 : y2 ∈ ω
L6005
Hypothesis H8 : eps_ y2 ≤ v + - x
L6006
Hypothesis H9 : z2 ∈ ω
L6007
Hypothesis H10 : eps_ z2 ≤ x2 + - y
L6008
Hypothesis H11 : SNo v
L6009
Hypothesis H12 : SNo x2
L6010
Hypothesis H13 : SNo (eps_ y2)
L6011
Hypothesis H14 : SNo (eps_ z2)
L6012
Hypothesis H15 : y2 + z2 ∈ ω
L6013
Hypothesis H16 : SNo (eps_ (y2 + z2))
L6014
Hypothesis H17 : SNo (ap z (y2 + z2))
L6015
Hypothesis H18 : SNo (v * y)
L6016
Theorem. (Conj_real_mul_SNo_pos__94__6)
SNo (x * x2) → u < SNoCut (Repl ω (ap z)) (Repl ω (ap w))
In Proofgold the corresponding term root is 2dab0c... and proposition id is 9c2b5c...
Proof:
Proof not loaded.
End of Section Conj_real_mul_SNo_pos__94__6
Beginning of Section Conj_real_mul_SNo_pos__94__7
L6022
Variable x : set
(*** Conj_real_mul_SNo_pos__94__7 TMX1hBM2erXLDPuyfnHYdtDLGXX5yTPik7G bounty of about 25 bars ***)
L6023
Variable y : set
L6024
Variable z : set
L6025
Variable w : set
L6026
Variable u : set
L6027
Variable v : set
L6028
Variable x2 : set
L6029
Variable y2 : set
L6030
Variable z2 : set
L6031
Hypothesis H0 : SNo x
L6032
Hypothesis H1 : SNo y
L6033
Hypothesis H2 : SNo (x * y)
L6034
Hypothesis H3 : SNo (SNoCut (Repl ω (ap z)) (Repl ω (ap w)))
L6035
Hypothesis H4 : (∀w2 : set, w2 ∈ ω → ap z w2 < SNoCut (Repl ω (ap z)) (Repl ω (ap w)))
L6036
Hypothesis H5 : (∀w2 : set, w2 ∈ ω → x * y < ap z w2 + eps_ w2)
L6037
Hypothesis H6 : u = v * y + x * x2 + - (v * x2)
L6038
Hypothesis H8 : eps_ y2 ≤ v + - x
L6039
Hypothesis H9 : z2 ∈ ω
L6040
Hypothesis H10 : eps_ z2 ≤ x2 + - y
L6041
Hypothesis H11 : SNo v
L6042
Hypothesis H12 : SNo x2
L6043
Hypothesis H13 : SNo (eps_ y2)
L6044
Hypothesis H14 : SNo (eps_ z2)
L6045
Hypothesis H15 : y2 + z2 ∈ ω
L6046
Hypothesis H16 : SNo (eps_ (y2 + z2))
L6047
Hypothesis H17 : SNo (ap z (y2 + z2))
L6048
Hypothesis H18 : SNo (v * y)
L6049
Theorem. (Conj_real_mul_SNo_pos__94__7)
SNo (x * x2) → u < SNoCut (Repl ω (ap z)) (Repl ω (ap w))
In Proofgold the corresponding term root is 7e7d8a... and proposition id is 67e2da...
Proof:
Proof not loaded.
End of Section Conj_real_mul_SNo_pos__94__7
Beginning of Section Conj_real_mul_SNo_pos__95__14
L6055
Variable x : set
(*** Conj_real_mul_SNo_pos__95__14 TMPz8f46YXM6m3SqrdpjdANzGLmRD2ccRtH 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
Variable x2 : set
L6062
Variable y2 : set
L6063
Variable z2 : set
L6064
Hypothesis H0 : SNo x
L6065
Hypothesis H1 : SNo y
L6066
Hypothesis H2 : SNo (x * y)
L6067
Hypothesis H3 : SNo (SNoCut (Repl ω (ap z)) (Repl ω (ap w)))
L6068
Hypothesis H4 : (∀w2 : set, w2 ∈ ω → ap z w2 < SNoCut (Repl ω (ap z)) (Repl ω (ap w)))
L6069
Hypothesis H5 : (∀w2 : set, w2 ∈ ω → x * y < ap z w2 + eps_ w2)
L6070
Hypothesis H6 : u = v * y + x * x2 + - (v * x2)
L6071
Hypothesis H7 : y2 ∈ ω
L6072
Hypothesis H8 : eps_ y2 ≤ v + - x
L6073
Hypothesis H9 : z2 ∈ ω
L6074
Hypothesis H10 : eps_ z2 ≤ x2 + - y
L6075
Hypothesis H11 : SNo v
L6076
Hypothesis H12 : SNo x2
L6077
Hypothesis H13 : SNo (eps_ y2)
L6078
Hypothesis H15 : y2 + z2 ∈ ω
L6079
Hypothesis H16 : SNo (eps_ (y2 + z2))
L6080
Hypothesis H17 : SNo (ap z (y2 + z2))
L6081
Theorem. (Conj_real_mul_SNo_pos__95__14)
SNo (v * y) → u < SNoCut (Repl ω (ap z)) (Repl ω (ap w))
In Proofgold the corresponding term root is 231356... and proposition id is c0d4a1...
Proof:
Proof not loaded.
End of Section Conj_real_mul_SNo_pos__95__14
Beginning of Section Conj_real_mul_SNo_pos__100__9
L6087
Variable x : set
(*** Conj_real_mul_SNo_pos__100__9 TMPSTEx3vtjUBY15NtokfywDSJbGNdNDHqo bounty of about 25 bars ***)
L6088
Variable y : set
L6089
Variable z : set
L6090
Variable w : set
L6091
Variable u : set
L6092
Variable v : set
L6093
Variable x2 : set
L6094
Variable y2 : set
L6095
Variable z2 : set
L6096
Hypothesis H0 : SNo x
L6097
Hypothesis H1 : SNo y
L6098
Hypothesis H2 : SNo (x * y)
L6099
Hypothesis H3 : (∀w2 : set, w2 ∈ ω → SNo (ap z w2))
L6100
Hypothesis H4 : SNo (SNoCut (Repl ω (ap z)) (Repl ω (ap w)))
L6101
Hypothesis H5 : (∀w2 : set, w2 ∈ ω → ap z w2 < SNoCut (Repl ω (ap z)) (Repl ω (ap w)))
L6102
Hypothesis H6 : (∀w2 : set, w2 ∈ ω → x * y < ap z w2 + eps_ w2)
L6103
Hypothesis H7 : u = v * y + x * x2 + - (v * x2)
L6104
Hypothesis H8 : y2 ∈ ω
L6105
Hypothesis H10 : z2 ∈ ω
L6106
Hypothesis H11 : eps_ z2 ≤ x2 + - y
L6107
Hypothesis H12 : SNo v
L6108
Hypothesis H13 : SNo x2
L6109
Theorem. (Conj_real_mul_SNo_pos__100__9)
SNo (eps_ y2) → u < SNoCut (Repl ω (ap z)) (Repl ω (ap w))
In Proofgold the corresponding term root is bbc80f... and proposition id is 5a4e0c...
Proof:
Proof not loaded.
End of Section Conj_real_mul_SNo_pos__100__9
Beginning of Section Conj_real_mul_SNo_pos__100__13
L6115
Variable x : set
(*** Conj_real_mul_SNo_pos__100__13 TMGywPDTVZsCpNgmTZKcUAScjC6e3qJ3fnc bounty of about 25 bars ***)
L6116
Variable y : set
L6117
Variable z : set
L6118
Variable w : set
L6119
Variable u : set
L6120
Variable v : set
L6121
Variable x2 : set
L6122
Variable y2 : set
L6123
Variable z2 : set
L6124
Hypothesis H0 : SNo x
L6125
Hypothesis H1 : SNo y
L6126
Hypothesis H2 : SNo (x * y)
L6127
Hypothesis H3 : (∀w2 : set, w2 ∈ ω → SNo (ap z w2))
L6128
Hypothesis H4 : SNo (SNoCut (Repl ω (ap z)) (Repl ω (ap w)))
L6129
Hypothesis H5 : (∀w2 : set, w2 ∈ ω → ap z w2 < SNoCut (Repl ω (ap z)) (Repl ω (ap w)))
L6130
Hypothesis H6 : (∀w2 : set, w2 ∈ ω → x * y < ap z w2 + eps_ w2)
L6131
Hypothesis H7 : u = v * y + x * x2 + - (v * x2)
L6132
Hypothesis H8 : y2 ∈ ω
L6133
Hypothesis H9 : eps_ y2 ≤ v + - x
L6134
Hypothesis H10 : z2 ∈ ω
L6135
Hypothesis H11 : eps_ z2 ≤ x2 + - y
L6136
Hypothesis H12 : SNo v
L6137
Theorem. (Conj_real_mul_SNo_pos__100__13)
SNo (eps_ y2) → u < SNoCut (Repl ω (ap z)) (Repl ω (ap w))
In Proofgold the corresponding term root is dbbc83... and proposition id is beab75...
Proof:
Proof not loaded.
End of Section Conj_real_mul_SNo_pos__100__13
Beginning of Section Conj_real_mul_SNo_pos__101__1
L6143
Variable x : set
(*** Conj_real_mul_SNo_pos__101__1 TMLeS97XXykwEmtZ6mGUKsmAYVeZ6weoQco bounty of about 25 bars ***)
L6144
Variable y : set
L6145
Variable z : set
L6146
Variable w : set
L6147
Variable u : set
L6148
Variable v : set
L6149
Variable x2 : set
L6150
Variable y2 : set
L6151
Variable z2 : set
L6152
Hypothesis H0 : SNo x
L6153
Hypothesis H2 : SNo (x * y)
L6154
Hypothesis H3 : SNo (SNoCut (Repl ω (ap z)) (Repl ω (ap w)))
L6155
Hypothesis H4 : (∀w2 : set, w2 ∈ ω → ap z w2 < SNoCut (Repl ω (ap z)) (Repl ω (ap w)))
L6156
Hypothesis H5 : (∀w2 : set, w2 ∈ ω → x * y < ap z w2 + eps_ w2)
L6157
Hypothesis H6 : u = v * y + x * x2 + - (v * x2)
L6158
Hypothesis H7 : y2 ∈ ω
L6159
Hypothesis H8 : eps_ y2 ≤ x + - v
L6160
Hypothesis H9 : z2 ∈ ω
L6161
Hypothesis H10 : eps_ z2 ≤ y + - x2
L6162
Hypothesis H11 : SNo v
L6163
Hypothesis H12 : SNo x2
L6164
Hypothesis H13 : SNo (eps_ y2)
L6165
Hypothesis H14 : SNo (eps_ z2)
L6166
Hypothesis H15 : y2 + z2 ∈ ω
L6167
Hypothesis H16 : SNo (eps_ (y2 + z2))
L6168
Hypothesis H17 : SNo (ap z (y2 + z2))
L6169
Hypothesis H18 : SNo (v * y)
L6170
Hypothesis H19 : SNo (x * x2)
L6171
Hypothesis H20 : SNo (- (v * x2))
L6172
Theorem. (Conj_real_mul_SNo_pos__101__1)
SNo (v * y + x * x2 + - (v * x2)) → u < SNoCut (Repl ω (ap z)) (Repl ω (ap w))
In Proofgold the corresponding term root is c1664b... and proposition id is dc5367...
Proof:
Proof not loaded.
End of Section Conj_real_mul_SNo_pos__101__1
Beginning of Section Conj_real_mul_SNo_pos__101__20
L6178
Variable x : set
(*** Conj_real_mul_SNo_pos__101__20 TMMbmK2RVyBo3LH5MrTdFiQ485b4986p29S bounty of about 25 bars ***)
L6179
Variable y : set
L6180
Variable z : set
L6181
Variable w : set
L6182
Variable u : set
L6183
Variable v : set
L6184
Variable x2 : set
L6185
Variable y2 : set
L6186
Variable z2 : set
L6187
Hypothesis H0 : SNo x
L6188
Hypothesis H1 : SNo y
L6189
Hypothesis H2 : SNo (x * y)
L6190
Hypothesis H3 : SNo (SNoCut (Repl ω (ap z)) (Repl ω (ap w)))
L6191
Hypothesis H4 : (∀w2 : set, w2 ∈ ω → ap z w2 < SNoCut (Repl ω (ap z)) (Repl ω (ap w)))
L6192
Hypothesis H5 : (∀w2 : set, w2 ∈ ω → x * y < ap z w2 + eps_ w2)
L6193
Hypothesis H6 : u = v * y + x * x2 + - (v * x2)
L6194
Hypothesis H7 : y2 ∈ ω
L6195
Hypothesis H8 : eps_ y2 ≤ x + - v
L6196
Hypothesis H9 : z2 ∈ ω
L6197
Hypothesis H10 : eps_ z2 ≤ y + - x2
L6198
Hypothesis H11 : SNo v
L6199
Hypothesis H12 : SNo x2
L6200
Hypothesis H13 : SNo (eps_ y2)
L6201
Hypothesis H14 : SNo (eps_ z2)
L6202
Hypothesis H15 : y2 + z2 ∈ ω
L6203
Hypothesis H16 : SNo (eps_ (y2 + z2))
L6204
Hypothesis H17 : SNo (ap z (y2 + z2))
L6205
Hypothesis H18 : SNo (v * y)
L6206
Hypothesis H19 : SNo (x * x2)
L6207
Theorem. (Conj_real_mul_SNo_pos__101__20)
SNo (v * y + x * x2 + - (v * x2)) → u < SNoCut (Repl ω (ap z)) (Repl ω (ap w))
In Proofgold the corresponding term root is 2f117b... and proposition id is ab4069...
Proof:
Proof not loaded.
End of Section Conj_real_mul_SNo_pos__101__20
Beginning of Section Conj_real_mul_SNo_pos__104__4
L6213
Variable x : set
(*** Conj_real_mul_SNo_pos__104__4 TMP8NEigTJwwkZDiNpJqy4DqWUqo9ELxy9N bounty of about 25 bars ***)
L6214
Variable y : set
L6215
Variable z : set
L6216
Variable w : set
L6217
Variable u : set
L6218
Variable v : set
L6219
Variable x2 : set
L6220
Variable y2 : set
L6221
Variable z2 : set
L6222
Hypothesis H0 : SNo x
L6223
Hypothesis H1 : SNo y
L6224
Hypothesis H2 : SNo (x * y)
L6225
Hypothesis H3 : SNo (SNoCut (Repl ω (ap z)) (Repl ω (ap w)))
L6226
Hypothesis H5 : (∀w2 : set, w2 ∈ ω → x * y < ap z w2 + eps_ w2)
L6227
Hypothesis H6 : u = v * y + x * x2 + - (v * x2)
L6228
Hypothesis H7 : y2 ∈ ω
L6229
Hypothesis H8 : eps_ y2 ≤ x + - v
L6230
Hypothesis H9 : z2 ∈ ω
L6231
Hypothesis H10 : eps_ z2 ≤ y + - x2
L6232
Hypothesis H11 : SNo v
L6233
Hypothesis H12 : SNo x2
L6234
Hypothesis H13 : SNo (eps_ y2)
L6235
Hypothesis H14 : SNo (eps_ z2)
L6236
Hypothesis H15 : y2 + z2 ∈ ω
L6237
Hypothesis H16 : SNo (eps_ (y2 + z2))
L6238
Hypothesis H17 : SNo (ap z (y2 + z2))
L6239
Hypothesis H18 : SNo (v * y)
L6240
Theorem. (Conj_real_mul_SNo_pos__104__4)
SNo (x * x2) → u < SNoCut (Repl ω (ap z)) (Repl ω (ap w))
In Proofgold the corresponding term root is 9682e3... and proposition id is bed07e...
Proof:
Proof not loaded.
End of Section Conj_real_mul_SNo_pos__104__4
Beginning of Section Conj_real_mul_SNo_pos__105__6
L6246
Variable x : set
(*** Conj_real_mul_SNo_pos__105__6 TMQLicaHDBE8BFp4NK5G6zJ9etv5dy8ndyH bounty of about 25 bars ***)
L6247
Variable y : set
L6248
Variable z : set
L6249
Variable w : set
L6250
Variable u : set
L6251
Variable v : set
L6252
Variable x2 : set
L6253
Variable y2 : set
L6254
Variable z2 : set
L6255
Hypothesis H0 : SNo x
L6256
Hypothesis H1 : SNo y
L6257
Hypothesis H2 : SNo (x * y)
L6258
Hypothesis H3 : SNo (SNoCut (Repl ω (ap z)) (Repl ω (ap w)))
L6259
Hypothesis H4 : (∀w2 : set, w2 ∈ ω → ap z w2 < SNoCut (Repl ω (ap z)) (Repl ω (ap w)))
L6260
Hypothesis H5 : (∀w2 : set, w2 ∈ ω → x * y < ap z w2 + eps_ w2)
L6261
Hypothesis H7 : y2 ∈ ω
L6262
Hypothesis H8 : eps_ y2 ≤ x + - v
L6263
Hypothesis H9 : z2 ∈ ω
L6264
Hypothesis H10 : eps_ z2 ≤ y + - x2
L6265
Hypothesis H11 : SNo v
L6266
Hypothesis H12 : SNo x2
L6267
Hypothesis H13 : SNo (eps_ y2)
L6268
Hypothesis H14 : SNo (eps_ z2)
L6269
Hypothesis H15 : y2 + z2 ∈ ω
L6270
Hypothesis H16 : SNo (eps_ (y2 + z2))
L6271
Hypothesis H17 : SNo (ap z (y2 + z2))
L6272
Theorem. (Conj_real_mul_SNo_pos__105__6)
SNo (v * y) → u < SNoCut (Repl ω (ap z)) (Repl ω (ap w))
In Proofgold the corresponding term root is 66ffb1... and proposition id is c86735...
Proof:
Proof not loaded.
End of Section Conj_real_mul_SNo_pos__105__6
Beginning of Section Conj_real_mul_SNo_pos__106__17
L6278
Variable x : set
(*** Conj_real_mul_SNo_pos__106__17 TMbMLeidfTMuiPd34tmK9QmnsMoXhdNpDPe bounty of about 25 bars ***)
L6279
Variable y : set
L6280
Variable z : set
L6281
Variable w : set
L6282
Variable u : set
L6283
Variable v : set
L6284
Variable x2 : set
L6285
Variable y2 : set
L6286
Variable z2 : set
L6287
Hypothesis H0 : SNo x
L6288
Hypothesis H1 : SNo y
L6289
Hypothesis H2 : SNo (x * y)
L6290
Hypothesis H3 : (∀w2 : set, w2 ∈ ω → SNo (ap z w2))
L6291
Hypothesis H4 : SNo (SNoCut (Repl ω (ap z)) (Repl ω (ap w)))
L6292
Hypothesis H5 : (∀w2 : set, w2 ∈ ω → ap z w2 < SNoCut (Repl ω (ap z)) (Repl ω (ap w)))
L6293
Hypothesis H6 : (∀w2 : set, w2 ∈ ω → x * y < ap z w2 + eps_ w2)
L6294
Hypothesis H7 : u = v * y + x * x2 + - (v * x2)
L6295
Hypothesis H8 : y2 ∈ ω
L6296
Hypothesis H9 : eps_ y2 ≤ x + - v
L6297
Hypothesis H10 : z2 ∈ ω
L6298
Hypothesis H11 : eps_ z2 ≤ y + - x2
L6299
Hypothesis H12 : SNo v
L6300
Hypothesis H13 : SNo x2
L6301
Hypothesis H14 : SNo (eps_ y2)
L6302
Hypothesis H15 : SNo (eps_ z2)
L6303
Hypothesis H16 : y2 + z2 ∈ ω
L6304
Theorem. (Conj_real_mul_SNo_pos__106__17)
SNo (ap z (y2 + z2)) → u < SNoCut (Repl ω (ap z)) (Repl ω (ap w))
In Proofgold the corresponding term root is b09c7d... and proposition id is c40903...
Proof:
Proof not loaded.
End of Section Conj_real_mul_SNo_pos__106__17
Beginning of Section Conj_real_mul_SNo_pos__107__16
L6310
Variable x : set
(*** Conj_real_mul_SNo_pos__107__16 TMMx8VRrAt4HzBnaKYjQCL6SMsruuCpUeLT bounty of about 25 bars ***)
L6311
Variable y : set
L6312
Variable z : set
L6313
Variable w : set
L6314
Variable u : set
L6315
Variable v : set
L6316
Variable x2 : set
L6317
Variable y2 : set
L6318
Variable z2 : set
L6319
Hypothesis H0 : SNo x
L6320
Hypothesis H1 : SNo y
L6321
Hypothesis H2 : SNo (x * y)
L6322
Hypothesis H3 : (∀w2 : set, w2 ∈ ω → SNo (ap z w2))
L6323
Hypothesis H4 : SNo (SNoCut (Repl ω (ap z)) (Repl ω (ap w)))
L6324
Hypothesis H5 : (∀w2 : set, w2 ∈ ω → ap z w2 < SNoCut (Repl ω (ap z)) (Repl ω (ap w)))
L6325
Hypothesis H6 : (∀w2 : set, w2 ∈ ω → x * y < ap z w2 + eps_ w2)
L6326
Hypothesis H7 : u = v * y + x * x2 + - (v * x2)
L6327
Hypothesis H8 : y2 ∈ ω
L6328
Hypothesis H9 : eps_ y2 ≤ x + - v
L6329
Hypothesis H10 : z2 ∈ ω
L6330
Hypothesis H11 : eps_ z2 ≤ y + - x2
L6331
Hypothesis H12 : SNo v
L6332
Hypothesis H13 : SNo x2
L6333
Hypothesis H14 : SNo (eps_ y2)
L6334
Hypothesis H15 : SNo (eps_ z2)
L6335
Theorem. (Conj_real_mul_SNo_pos__107__16)
SNo (eps_ (y2 + z2)) → u < SNoCut (Repl ω (ap z)) (Repl ω (ap w))
In Proofgold the corresponding term root is f7b5e0... and proposition id is 5657cf...
Proof:
Proof not loaded.
End of Section Conj_real_mul_SNo_pos__107__16
Beginning of Section Conj_real_mul_SNo_pos__108__6
L6341
Variable x : set
(*** Conj_real_mul_SNo_pos__108__6 TMM5toNWPBBZSzZ7iamMVJoohpLoNNbKT4E bounty of about 25 bars ***)
L6342
Variable y : set
L6343
Variable z : set
L6344
Variable w : set
L6345
Variable u : set
L6346
Variable v : set
L6347
Variable x2 : set
L6348
Variable y2 : set
L6349
Variable z2 : set
L6350
Hypothesis H0 : SNo x
L6351
Hypothesis H1 : SNo y
L6352
Hypothesis H2 : SNo (x * y)
L6353
Hypothesis H3 : (∀w2 : set, w2 ∈ ω → SNo (ap z w2))
L6354
Hypothesis H4 : SNo (SNoCut (Repl ω (ap z)) (Repl ω (ap w)))
L6355
Hypothesis H5 : (∀w2 : set, w2 ∈ ω → ap z w2 < SNoCut (Repl ω (ap z)) (Repl ω (ap w)))
L6356
Hypothesis H7 : u = v * y + x * x2 + - (v * x2)
L6357
Hypothesis H8 : y2 ∈ ω
L6358
Hypothesis H9 : eps_ y2 ≤ x + - v
L6359
Hypothesis H10 : z2 ∈ ω
L6360
Hypothesis H11 : eps_ z2 ≤ y + - x2
L6361
Hypothesis H12 : SNo v
L6362
Hypothesis H13 : SNo x2
L6363
Hypothesis H14 : SNo (eps_ y2)
L6364
Hypothesis H15 : SNo (eps_ z2)
L6365
Theorem. (Conj_real_mul_SNo_pos__108__6)
y2 + z2 ∈ ω → u < SNoCut (Repl ω (ap z)) (Repl ω (ap w))
In Proofgold the corresponding term root is 1a80f6... and proposition id is 6eb4b9...
Proof:
Proof not loaded.
End of Section Conj_real_mul_SNo_pos__108__6
Beginning of Section Conj_real_mul_SNo_pos__108__11
L6371
Variable x : set
(*** Conj_real_mul_SNo_pos__108__11 TMPR34o5X3U7h6suDWD1unFoVouhasQFHoP bounty of about 25 bars ***)
L6372
Variable y : set
L6373
Variable z : set
L6374
Variable w : set
L6375
Variable u : set
L6376
Variable v : set
L6377
Variable x2 : set
L6378
Variable y2 : set
L6379
Variable z2 : set
L6380
Hypothesis H0 : SNo x
L6381
Hypothesis H1 : SNo y
L6382
Hypothesis H2 : SNo (x * y)
L6383
Hypothesis H3 : (∀w2 : set, w2 ∈ ω → SNo (ap z w2))
L6384
Hypothesis H4 : SNo (SNoCut (Repl ω (ap z)) (Repl ω (ap w)))
L6385
Hypothesis H5 : (∀w2 : set, w2 ∈ ω → ap z w2 < SNoCut (Repl ω (ap z)) (Repl ω (ap w)))
L6386
Hypothesis H6 : (∀w2 : set, w2 ∈ ω → x * y < ap z w2 + eps_ w2)
L6387
Hypothesis H7 : u = v * y + x * x2 + - (v * x2)
L6388
Hypothesis H8 : y2 ∈ ω
L6389
Hypothesis H9 : eps_ y2 ≤ x + - v
L6390
Hypothesis H10 : z2 ∈ ω
L6391
Hypothesis H12 : SNo v
L6392
Hypothesis H13 : SNo x2
L6393
Hypothesis H14 : SNo (eps_ y2)
L6394
Hypothesis H15 : SNo (eps_ z2)
L6395
Theorem. (Conj_real_mul_SNo_pos__108__11)
y2 + z2 ∈ ω → u < SNoCut (Repl ω (ap z)) (Repl ω (ap w))
In Proofgold the corresponding term root is 37c360... and proposition id is 79ea4b...
Proof:
Proof not loaded.
End of Section Conj_real_mul_SNo_pos__108__11
Beginning of Section Conj_real_mul_SNo_pos__109__3
L6401
Variable x : set
(*** Conj_real_mul_SNo_pos__109__3 TMFyecQjJ8sdb9a8tSQTcWxkPj33XtVwQZ3 bounty of about 25 bars ***)
L6402
Variable y : set
L6403
Variable z : set
L6404
Variable w : set
L6405
Variable u : set
L6406
Variable v : set
L6407
Variable x2 : set
L6408
Variable y2 : set
L6409
Variable z2 : set
L6410
Hypothesis H0 : SNo x
L6411
Hypothesis H1 : SNo y
L6412
Hypothesis H2 : SNo (x * y)
L6413
Hypothesis H4 : SNo (SNoCut (Repl ω (ap z)) (Repl ω (ap w)))
L6414
Hypothesis H5 : (∀w2 : set, w2 ∈ ω → ap z w2 < SNoCut (Repl ω (ap z)) (Repl ω (ap w)))
L6415
Hypothesis H6 : (∀w2 : set, w2 ∈ ω → x * y < ap z w2 + eps_ w2)
L6416
Hypothesis H7 : u = v * y + x * x2 + - (v * x2)
L6417
Hypothesis H8 : y2 ∈ ω
L6418
Hypothesis H9 : eps_ y2 ≤ x + - v
L6419
Hypothesis H10 : z2 ∈ ω
L6420
Hypothesis H11 : eps_ z2 ≤ y + - x2
L6421
Hypothesis H12 : SNo v
L6422
Hypothesis H13 : SNo x2
L6423
Hypothesis H14 : SNo (eps_ y2)
L6424
Theorem. (Conj_real_mul_SNo_pos__109__3)
SNo (eps_ z2) → u < SNoCut (Repl ω (ap z)) (Repl ω (ap w))
In Proofgold the corresponding term root is cdcc91... and proposition id is 1f0603...
Proof:
Proof not loaded.
End of Section Conj_real_mul_SNo_pos__109__3
Beginning of Section Conj_real_mul_SNo_pos__111__23
L6430
Variable x : set
(*** Conj_real_mul_SNo_pos__111__23 TMUyZuNkPQJbLAyEgpPPdTBz9WPYU6YJFQi bounty of about 25 bars ***)
L6431
Variable y : set
L6432
Variable z : set
L6433
Variable w : set
L6434
Variable u : set
L6435
Variable v : set
L6436
Hypothesis H0 : ¬ x * y ∈ real
L6437
Hypothesis H1 : SNo x
L6438
Hypothesis H2 : SNo y
L6439
Hypothesis H3 : SNo (x * y)
L6440
Hypothesis H4 : SNo (- (x * y))
L6441
Hypothesis H5 : (∀x2 : set, x2 ∈ SNoL x → (∀P : prop, (∀y2 : set, y2 ∈ ω → eps_ y2 ≤ x + - x2 → P) → P))
L6442
Hypothesis H6 : (∀x2 : set, x2 ∈ SNoR x → (∀P : prop, (∀y2 : set, y2 ∈ ω → eps_ y2 ≤ x2 + - x → P) → P))
L6443
Hypothesis H7 : (∀x2 : set, x2 ∈ SNoL y → (∀P : prop, (∀y2 : set, y2 ∈ ω → eps_ y2 ≤ y + - x2 → P) → P))
L6444
Hypothesis H8 : (∀x2 : set, x2 ∈ SNoR y → (∀P : prop, (∀y2 : set, y2 ∈ ω → eps_ y2 ≤ x2 + - y → P) → P))
L6445
Hypothesis H9 : SNoCutP z w
L6446
Hypothesis H10 : (∀x2 : set, x2 ∈ z → (∀P : prop, (∀y2 : set, y2 ∈ SNoL x → (∀z2 : set, z2 ∈ SNoL y → x2 = y2 * y + x * z2 + - (y2 * z2) → P)) → (∀y2 : set, y2 ∈ SNoR x → (∀z2 : set, z2 ∈ SNoR y → x2 = y2 * y + x * z2 + - (y2 * z2) → P)) → P))
L6447
Hypothesis H11 : (∀x2 : set, x2 ∈ w → (∀P : prop, (∀y2 : set, y2 ∈ SNoL x → (∀z2 : set, z2 ∈ SNoR y → x2 = y2 * y + x * z2 + - (y2 * z2) → P)) → (∀y2 : set, y2 ∈ SNoR x → (∀z2 : set, z2 ∈ SNoL y → x2 = y2 * y + x * z2 + - (y2 * z2) → P)) → P))
L6448
Hypothesis H12 : x * y = SNoCut z w
L6449
Hypothesis H13 : u ∈ setexp (SNoS_ ω) ω
L6450
Hypothesis H14 : v ∈ setexp (SNoS_ ω) ω
L6451
Hypothesis H15 : (∀x2 : set, x2 ∈ ω → SNo (ap u x2))
L6452
Hypothesis H16 : (∀x2 : set, x2 ∈ ω → SNo (ap v x2))
L6453
Hypothesis H17 : (∀x2 : set, x2 ∈ ω → ap u x2 < x * y)
L6454
Hypothesis H18 : (∀x2 : set, x2 ∈ ω → x * y < ap v x2)
L6455
Hypothesis H19 : SNoCutP (Repl ω (ap u)) (Repl ω (ap v))
L6456
Hypothesis H20 : SNo (SNoCut (Repl ω (ap u)) (Repl ω (ap v)))
L6457
Hypothesis H21 : (∀x2 : set, x2 ∈ ω → ap u x2 < SNoCut (Repl ω (ap u)) (Repl ω (ap v)))
L6458
Hypothesis H22 : (∀x2 : set, x2 ∈ ω → SNoCut (Repl ω (ap u)) (Repl ω (ap v)) < ap v x2)
L6459
Hypothesis H24 : (∀x2 : set, x2 ∈ ω → (∀y2 : set, y2 ∈ x2 → ap u y2 < ap u x2))
L6460
Hypothesis H25 : (∀x2 : set, x2 ∈ ω → (ap v x2 + - (eps_ x2)) < x * y)
L6461
Hypothesis H26 : (∀x2 : set, x2 ∈ ω → (∀y2 : set, y2 ∈ x2 → ap v x2 < ap v y2))
L6462
Theorem. (Conj_real_mul_SNo_pos__111__23)
x * y ≠ SNoCut (Repl ω (ap u)) (Repl ω (ap v))
In Proofgold the corresponding term root is 382102... and proposition id is 03b9a7...
Proof:
Proof not loaded.
End of Section Conj_real_mul_SNo_pos__111__23
Beginning of Section Conj_real_mul_SNo_pos__112__11
L6468
Variable x : set
(*** Conj_real_mul_SNo_pos__112__11 TMX2HaetPie6ZpHwoVVmuanNJHBv7TbmcXo bounty of about 25 bars ***)
L6469
Variable y : set
L6470
Variable z : set
L6471
Variable w : set
L6472
Variable u : set
L6473
Variable v : set
L6474
Hypothesis H0 : ¬ x * y ∈ real
L6475
Hypothesis H1 : SNo x
L6476
Hypothesis H2 : SNo y
L6477
Hypothesis H3 : SNo (x * y)
L6478
Hypothesis H4 : SNo (- (x * y))
L6479
Hypothesis H5 : (∀x2 : set, x2 ∈ SNoL x → (∀P : prop, (∀y2 : set, y2 ∈ ω → eps_ y2 ≤ x + - x2 → P) → P))
L6480
Hypothesis H6 : (∀x2 : set, x2 ∈ SNoR x → (∀P : prop, (∀y2 : set, y2 ∈ ω → eps_ y2 ≤ x2 + - x → P) → P))
L6481
Hypothesis H7 : (∀x2 : set, x2 ∈ SNoL y → (∀P : prop, (∀y2 : set, y2 ∈ ω → eps_ y2 ≤ y + - x2 → P) → P))
L6482
Hypothesis H8 : (∀x2 : set, x2 ∈ SNoR y → (∀P : prop, (∀y2 : set, y2 ∈ ω → eps_ y2 ≤ x2 + - y → P) → P))
L6483
Hypothesis H9 : SNoCutP z w
L6484
Hypothesis H10 : (∀x2 : set, x2 ∈ z → (∀P : prop, (∀y2 : set, y2 ∈ SNoL x → (∀z2 : set, z2 ∈ SNoL y → x2 = y2 * y + x * z2 + - (y2 * z2) → P)) → (∀y2 : set, y2 ∈ SNoR x → (∀z2 : set, z2 ∈ SNoR y → x2 = y2 * y + x * z2 + - (y2 * z2) → P)) → P))
L6485
Hypothesis H12 : x * y = SNoCut z w
L6486
Hypothesis H13 : u ∈ setexp (SNoS_ ω) ω
L6487
Hypothesis H14 : v ∈ setexp (SNoS_ ω) ω
L6488
Hypothesis H15 : (∀x2 : set, x2 ∈ ω → (ap v x2 + - (eps_ x2)) < x * y ∧ x * y < ap v x2 ∧ (∀y2 : set, y2 ∈ x2 → ap v x2 < ap v y2))
L6489
Hypothesis H16 : (∀x2 : set, x2 ∈ ω → SNo (ap u x2))
L6490
Hypothesis H17 : (∀x2 : set, x2 ∈ ω → SNo (ap v x2))
L6491
Hypothesis H18 : (∀x2 : set, x2 ∈ ω → ap u x2 < x * y)
L6492
Hypothesis H19 : (∀x2 : set, x2 ∈ ω → x * y < ap v x2)
L6493
Hypothesis H20 : SNoCutP (Repl ω (ap u)) (Repl ω (ap v))
L6494
Hypothesis H21 : SNo (SNoCut (Repl ω (ap u)) (Repl ω (ap v)))
L6495
Hypothesis H22 : (∀x2 : set, x2 ∈ ω → ap u x2 < SNoCut (Repl ω (ap u)) (Repl ω (ap v)))
L6496
Hypothesis H23 : (∀x2 : set, x2 ∈ ω → SNoCut (Repl ω (ap u)) (Repl ω (ap v)) < ap v x2)
L6497
Hypothesis H24 : (∀x2 : set, x2 ∈ ω → x * y < ap u x2 + eps_ x2)
L6498
Hypothesis H25 : (∀x2 : set, x2 ∈ ω → (∀y2 : set, y2 ∈ x2 → ap u y2 < ap u x2))
L6499
Hypothesis H26 : (∀x2 : set, x2 ∈ ω → (ap v x2 + - (eps_ x2)) < x * y)
L6500
Theorem. (Conj_real_mul_SNo_pos__112__11)
¬ (∀x2 : set, x2 ∈ ω → (∀y2 : set, y2 ∈ x2 → ap v x2 < ap v y2))
In Proofgold the corresponding term root is 4e42b7... and proposition id is db124b...
Proof:
Proof not loaded.
End of Section Conj_real_mul_SNo_pos__112__11
Beginning of Section Conj_real_mul_SNo_pos__112__22
L6506
Variable x : set
(*** Conj_real_mul_SNo_pos__112__22 TMWSSwAqXy5XqUzCwkGp4CgvBjtyoNU6pxi bounty of about 25 bars ***)
L6507
Variable y : set
L6508
Variable z : set
L6509
Variable w : set
L6510
Variable u : set
L6511
Variable v : set
L6512
Hypothesis H0 : ¬ x * y ∈ real
L6513
Hypothesis H1 : SNo x
L6514
Hypothesis H2 : SNo y
L6515
Hypothesis H3 : SNo (x * y)
L6516
Hypothesis H4 : SNo (- (x * y))
L6517
Hypothesis H5 : (∀x2 : set, x2 ∈ SNoL x → (∀P : prop, (∀y2 : set, y2 ∈ ω → eps_ y2 ≤ x + - x2 → P) → P))
L6518
Hypothesis H6 : (∀x2 : set, x2 ∈ SNoR x → (∀P : prop, (∀y2 : set, y2 ∈ ω → eps_ y2 ≤ x2 + - x → P) → P))
L6519
Hypothesis H7 : (∀x2 : set, x2 ∈ SNoL y → (∀P : prop, (∀y2 : set, y2 ∈ ω → eps_ y2 ≤ y + - x2 → P) → P))
L6520
Hypothesis H8 : (∀x2 : set, x2 ∈ SNoR y → (∀P : prop, (∀y2 : set, y2 ∈ ω → eps_ y2 ≤ x2 + - y → P) → P))
L6521
Hypothesis H9 : SNoCutP z w
L6522
Hypothesis H10 : (∀x2 : set, x2 ∈ z → (∀P : prop, (∀y2 : set, y2 ∈ SNoL x → (∀z2 : set, z2 ∈ SNoL y → x2 = y2 * y + x * z2 + - (y2 * z2) → P)) → (∀y2 : set, y2 ∈ SNoR x → (∀z2 : set, z2 ∈ SNoR y → x2 = y2 * y + x * z2 + - (y2 * z2) → P)) → P))
L6523
Hypothesis H11 : (∀x2 : set, x2 ∈ w → (∀P : prop, (∀y2 : set, y2 ∈ SNoL x → (∀z2 : set, z2 ∈ SNoR y → x2 = y2 * y + x * z2 + - (y2 * z2) → P)) → (∀y2 : set, y2 ∈ SNoR x → (∀z2 : set, z2 ∈ SNoL y → x2 = y2 * y + x * z2 + - (y2 * z2) → P)) → P))
L6524
Hypothesis H12 : x * y = SNoCut z w
L6525
Hypothesis H13 : u ∈ setexp (SNoS_ ω) ω
L6526
Hypothesis H14 : v ∈ setexp (SNoS_ ω) ω
L6527
Hypothesis H15 : (∀x2 : set, x2 ∈ ω → (ap v x2 + - (eps_ x2)) < x * y ∧ x * y < ap v x2 ∧ (∀y2 : set, y2 ∈ x2 → ap v x2 < ap v y2))
L6528
Hypothesis H16 : (∀x2 : set, x2 ∈ ω → SNo (ap u x2))
L6529
Hypothesis H17 : (∀x2 : set, x2 ∈ ω → SNo (ap v x2))
L6530
Hypothesis H18 : (∀x2 : set, x2 ∈ ω → ap u x2 < x * y)
L6531
Hypothesis H19 : (∀x2 : set, x2 ∈ ω → x * y < ap v x2)
L6532
Hypothesis H20 : SNoCutP (Repl ω (ap u)) (Repl ω (ap v))
L6533
Hypothesis H21 : SNo (SNoCut (Repl ω (ap u)) (Repl ω (ap v)))
L6534
Hypothesis H23 : (∀x2 : set, x2 ∈ ω → SNoCut (Repl ω (ap u)) (Repl ω (ap v)) < ap v x2)
L6535
Hypothesis H24 : (∀x2 : set, x2 ∈ ω → x * y < ap u x2 + eps_ x2)
L6536
Hypothesis H25 : (∀x2 : set, x2 ∈ ω → (∀y2 : set, y2 ∈ x2 → ap u y2 < ap u x2))
L6537
Hypothesis H26 : (∀x2 : set, x2 ∈ ω → (ap v x2 + - (eps_ x2)) < x * y)
L6538
Theorem. (Conj_real_mul_SNo_pos__112__22)
¬ (∀x2 : set, x2 ∈ ω → (∀y2 : set, y2 ∈ x2 → ap v x2 < ap v y2))
In Proofgold the corresponding term root is 2d0197... and proposition id is a4e8ee...
Proof:
Proof not loaded.
End of Section Conj_real_mul_SNo_pos__112__22
Beginning of Section Conj_real_mul_SNo_pos__113__6
L6544
Variable x : set
(*** Conj_real_mul_SNo_pos__113__6 TMcTCm79ELB739kanE9hTnqu3YKGSECxqm6 bounty of about 25 bars ***)
L6545
Variable y : set
L6546
Variable z : set
L6547
Variable w : set
L6548
Variable u : set
L6549
Variable v : set
L6550
Hypothesis H0 : ¬ x * y ∈ real
L6551
Hypothesis H1 : SNo x
L6552
Hypothesis H2 : SNo y
L6553
Hypothesis H3 : SNo (x * y)
L6554
Hypothesis H4 : SNo (- (x * y))
L6555
Hypothesis H5 : (∀x2 : set, x2 ∈ SNoL x → (∀P : prop, (∀y2 : set, y2 ∈ ω → eps_ y2 ≤ x + - x2 → P) → P))
L6556
Hypothesis H7 : (∀x2 : set, x2 ∈ SNoL y → (∀P : prop, (∀y2 : set, y2 ∈ ω → eps_ y2 ≤ y + - x2 → P) → P))
L6557
Hypothesis H8 : (∀x2 : set, x2 ∈ SNoR y → (∀P : prop, (∀y2 : set, y2 ∈ ω → eps_ y2 ≤ x2 + - y → P) → P))
L6558
Hypothesis H9 : SNoCutP z w
L6559
Hypothesis H10 : (∀x2 : set, x2 ∈ z → (∀P : prop, (∀y2 : set, y2 ∈ SNoL x → (∀z2 : set, z2 ∈ SNoL y → x2 = y2 * y + x * z2 + - (y2 * z2) → P)) → (∀y2 : set, y2 ∈ SNoR x → (∀z2 : set, z2 ∈ SNoR y → x2 = y2 * y + x * z2 + - (y2 * z2) → P)) → P))
L6560
Hypothesis H11 : (∀x2 : set, x2 ∈ w → (∀P : prop, (∀y2 : set, y2 ∈ SNoL x → (∀z2 : set, z2 ∈ SNoR y → x2 = y2 * y + x * z2 + - (y2 * z2) → P)) → (∀y2 : set, y2 ∈ SNoR x → (∀z2 : set, z2 ∈ SNoL y → x2 = y2 * y + x * z2 + - (y2 * z2) → P)) → P))
L6561
Hypothesis H12 : x * y = SNoCut z w
L6562
Hypothesis H13 : u ∈ setexp (SNoS_ ω) ω
L6563
Hypothesis H14 : v ∈ setexp (SNoS_ ω) ω
L6564
Hypothesis H15 : (∀x2 : set, x2 ∈ ω → (ap v x2 + - (eps_ x2)) < x * y ∧ x * y < ap v x2 ∧ (∀y2 : set, y2 ∈ x2 → ap v x2 < ap v y2))
L6565
Hypothesis H16 : (∀x2 : set, x2 ∈ ω → SNo (ap u x2))
L6566
Hypothesis H17 : (∀x2 : set, x2 ∈ ω → SNo (ap v x2))
L6567
Hypothesis H18 : (∀x2 : set, x2 ∈ ω → ap u x2 < x * y)
L6568
Hypothesis H19 : (∀x2 : set, x2 ∈ ω → x * y < ap v x2)
L6569
Hypothesis H20 : SNoCutP (Repl ω (ap u)) (Repl ω (ap v))
L6570
Hypothesis H21 : SNo (SNoCut (Repl ω (ap u)) (Repl ω (ap v)))
L6571
Hypothesis H22 : (∀x2 : set, x2 ∈ ω → ap u x2 < SNoCut (Repl ω (ap u)) (Repl ω (ap v)))
L6572
Hypothesis H23 : (∀x2 : set, x2 ∈ ω → SNoCut (Repl ω (ap u)) (Repl ω (ap v)) < ap v x2)
L6573
Hypothesis H24 : (∀x2 : set, x2 ∈ ω → x * y < ap u x2 + eps_ x2)
L6574
Hypothesis H25 : (∀x2 : set, x2 ∈ ω → (∀y2 : set, y2 ∈ x2 → ap u y2 < ap u x2))
L6575
Theorem. (Conj_real_mul_SNo_pos__113__6)
¬ (∀x2 : set, x2 ∈ ω → (ap v x2 + - (eps_ x2)) < x * y)
In Proofgold the corresponding term root is e1bf76... and proposition id is a20466...
Proof:
Proof not loaded.
End of Section Conj_real_mul_SNo_pos__113__6
Beginning of Section Conj_real_mul_SNo_pos__114__6
L6581
Variable x : set
(*** Conj_real_mul_SNo_pos__114__6 TMVRFeBQhjzdKomgrnvSS3iwN1MctH1Pjgk bounty of about 25 bars ***)
L6582
Variable y : set
L6583
Variable z : set
L6584
Variable w : set
L6585
Variable u : set
L6586
Variable v : set
L6587
Hypothesis H0 : ¬ x * y ∈ real
L6588
Hypothesis H1 : SNo x
L6589
Hypothesis H2 : SNo y
L6590
Hypothesis H3 : SNo (x * y)
L6591
Hypothesis H4 : SNo (- (x * y))
L6592
Hypothesis H5 : (∀x2 : set, x2 ∈ SNoL x → (∀P : prop, (∀y2 : set, y2 ∈ ω → eps_ y2 ≤ x + - x2 → P) → P))
L6593
Hypothesis H7 : (∀x2 : set, x2 ∈ SNoL y → (∀P : prop, (∀y2 : set, y2 ∈ ω → eps_ y2 ≤ y + - x2 → P) → P))
L6594
Hypothesis H8 : (∀x2 : set, x2 ∈ SNoR y → (∀P : prop, (∀y2 : set, y2 ∈ ω → eps_ y2 ≤ x2 + - y → P) → P))
L6595
Hypothesis H9 : SNoCutP z w
L6596
Hypothesis H10 : (∀x2 : set, x2 ∈ z → (∀P : prop, (∀y2 : set, y2 ∈ SNoL x → (∀z2 : set, z2 ∈ SNoL y → x2 = y2 * y + x * z2 + - (y2 * z2) → P)) → (∀y2 : set, y2 ∈ SNoR x → (∀z2 : set, z2 ∈ SNoR y → x2 = y2 * y + x * z2 + - (y2 * z2) → P)) → P))
L6597
Hypothesis H11 : (∀x2 : set, x2 ∈ w → (∀P : prop, (∀y2 : set, y2 ∈ SNoL x → (∀z2 : set, z2 ∈ SNoR y → x2 = y2 * y + x * z2 + - (y2 * z2) → P)) → (∀y2 : set, y2 ∈ SNoR x → (∀z2 : set, z2 ∈ SNoL y → x2 = y2 * y + x * z2 + - (y2 * z2) → P)) → P))
L6598
Hypothesis H12 : x * y = SNoCut z w
L6599
Hypothesis H13 : u ∈ setexp (SNoS_ ω) ω
L6600
Hypothesis H14 : (∀x2 : set, x2 ∈ ω → ap u x2 < x * y ∧ x * y < ap u x2 + eps_ x2 ∧ (∀y2 : set, y2 ∈ x2 → ap u y2 < ap u x2))
L6601
Hypothesis H15 : v ∈ setexp (SNoS_ ω) ω
L6602
Hypothesis H16 : (∀x2 : set, x2 ∈ ω → (ap v x2 + - (eps_ x2)) < x * y ∧ x * y < ap v x2 ∧ (∀y2 : set, y2 ∈ x2 → ap v x2 < ap v y2))
L6603
Hypothesis H17 : (∀x2 : set, x2 ∈ ω → SNo (ap u x2))
L6604
Hypothesis H18 : (∀x2 : set, x2 ∈ ω → SNo (ap v x2))
L6605
Hypothesis H19 : (∀x2 : set, x2 ∈ ω → ap u x2 < x * y)
L6606
Hypothesis H20 : (∀x2 : set, x2 ∈ ω → x * y < ap v x2)
L6607
Hypothesis H21 : SNoCutP (Repl ω (ap u)) (Repl ω (ap v))
L6608
Hypothesis H22 : SNo (SNoCut (Repl ω (ap u)) (Repl ω (ap v)))
L6609
Hypothesis H23 : (∀x2 : set, x2 ∈ ω → ap u x2 < SNoCut (Repl ω (ap u)) (Repl ω (ap v)))
L6610
Hypothesis H24 : (∀x2 : set, x2 ∈ ω → SNoCut (Repl ω (ap u)) (Repl ω (ap v)) < ap v x2)
L6611
Hypothesis H25 : (∀x2 : set, x2 ∈ ω → x * y < ap u x2 + eps_ x2)
L6612
Theorem. (Conj_real_mul_SNo_pos__114__6)
¬ (∀x2 : set, x2 ∈ ω → (∀y2 : set, y2 ∈ x2 → ap u y2 < ap u x2))
In Proofgold the corresponding term root is 714be6... and proposition id is bff0dd...
Proof:
Proof not loaded.
End of Section Conj_real_mul_SNo_pos__114__6
Beginning of Section Conj_real_mul_SNo_pos__114__21
L6618
Variable x : set
(*** Conj_real_mul_SNo_pos__114__21 TMEonVXa2cMo835BTYV884Rt5YdxnuYp1Mv bounty of about 25 bars ***)
L6619
Variable y : set
L6620
Variable z : set
L6621
Variable w : set
L6622
Variable u : set
L6623
Variable v : set
L6624
Hypothesis H0 : ¬ x * y ∈ real
L6625
Hypothesis H1 : SNo x
L6626
Hypothesis H2 : SNo y
L6627
Hypothesis H3 : SNo (x * y)
L6628
Hypothesis H4 : SNo (- (x * y))
L6629
Hypothesis H5 : (∀x2 : set, x2 ∈ SNoL x → (∀P : prop, (∀y2 : set, y2 ∈ ω → eps_ y2 ≤ x + - x2 → P) → P))
L6630
Hypothesis H6 : (∀x2 : set, x2 ∈ SNoR x → (∀P : prop, (∀y2 : set, y2 ∈ ω → eps_ y2 ≤ x2 + - x → P) → P))
L6631
Hypothesis H7 : (∀x2 : set, x2 ∈ SNoL y → (∀P : prop, (∀y2 : set, y2 ∈ ω → eps_ y2 ≤ y + - x2 → P) → P))
L6632
Hypothesis H8 : (∀x2 : set, x2 ∈ SNoR y → (∀P : prop, (∀y2 : set, y2 ∈ ω → eps_ y2 ≤ x2 + - y → P) → P))
L6633
Hypothesis H9 : SNoCutP z w
L6634
Hypothesis H10 : (∀x2 : set, x2 ∈ z → (∀P : prop, (∀y2 : set, y2 ∈ SNoL x → (∀z2 : set, z2 ∈ SNoL y → x2 = y2 * y + x * z2 + - (y2 * z2) → P)) → (∀y2 : set, y2 ∈ SNoR x → (∀z2 : set, z2 ∈ SNoR y → x2 = y2 * y + x * z2 + - (y2 * z2) → P)) → P))
L6635
Hypothesis H11 : (∀x2 : set, x2 ∈ w → (∀P : prop, (∀y2 : set, y2 ∈ SNoL x → (∀z2 : set, z2 ∈ SNoR y → x2 = y2 * y + x * z2 + - (y2 * z2) → P)) → (∀y2 : set, y2 ∈ SNoR x → (∀z2 : set, z2 ∈ SNoL y → x2 = y2 * y + x * z2 + - (y2 * z2) → P)) → P))
L6636
Hypothesis H12 : x * y = SNoCut z w
L6637
Hypothesis H13 : u ∈ setexp (SNoS_ ω) ω
L6638
Hypothesis H14 : (∀x2 : set, x2 ∈ ω → ap u x2 < x * y ∧ x * y < ap u x2 + eps_ x2 ∧ (∀y2 : set, y2 ∈ x2 → ap u y2 < ap u x2))
L6639
Hypothesis H15 : v ∈ setexp (SNoS_ ω) ω
L6640
Hypothesis H16 : (∀x2 : set, x2 ∈ ω → (ap v x2 + - (eps_ x2)) < x * y ∧ x * y < ap v x2 ∧ (∀y2 : set, y2 ∈ x2 → ap v x2 < ap v y2))
L6641
Hypothesis H17 : (∀x2 : set, x2 ∈ ω → SNo (ap u x2))
L6642
Hypothesis H18 : (∀x2 : set, x2 ∈ ω → SNo (ap v x2))
L6643
Hypothesis H19 : (∀x2 : set, x2 ∈ ω → ap u x2 < x * y)
L6644
Hypothesis H20 : (∀x2 : set, x2 ∈ ω → x * y < ap v x2)
L6645
Hypothesis H22 : SNo (SNoCut (Repl ω (ap u)) (Repl ω (ap v)))
L6646
Hypothesis H23 : (∀x2 : set, x2 ∈ ω → ap u x2 < SNoCut (Repl ω (ap u)) (Repl ω (ap v)))
L6647
Hypothesis H24 : (∀x2 : set, x2 ∈ ω → SNoCut (Repl ω (ap u)) (Repl ω (ap v)) < ap v x2)
L6648
Hypothesis H25 : (∀x2 : set, x2 ∈ ω → x * y < ap u x2 + eps_ x2)
L6649
Theorem. (Conj_real_mul_SNo_pos__114__21)
¬ (∀x2 : set, x2 ∈ ω → (∀y2 : set, y2 ∈ x2 → ap u y2 < ap u x2))
In Proofgold the corresponding term root is 12b077... and proposition id is 9bcca1...
Proof:
Proof not loaded.
End of Section Conj_real_mul_SNo_pos__114__21
Beginning of Section Conj_real_mul_SNo_pos__114__25
L6655
Variable x : set
(*** Conj_real_mul_SNo_pos__114__25 TMLijcXx5rgBoVx7vcQLai5nJ2fa8xnkmbZ bounty of about 25 bars ***)
L6656
Variable y : set
L6657
Variable z : set
L6658
Variable w : set
L6659
Variable u : set
L6660
Variable v : set
L6661
Hypothesis H0 : ¬ x * y ∈ real
L6662
Hypothesis H1 : SNo x
L6663
Hypothesis H2 : SNo y
L6664
Hypothesis H3 : SNo (x * y)
L6665
Hypothesis H4 : SNo (- (x * y))
L6666
Hypothesis H5 : (∀x2 : set, x2 ∈ SNoL x → (∀P : prop, (∀y2 : set, y2 ∈ ω → eps_ y2 ≤ x + - x2 → P) → P))
L6667
Hypothesis H6 : (∀x2 : set, x2 ∈ SNoR x → (∀P : prop, (∀y2 : set, y2 ∈ ω → eps_ y2 ≤ x2 + - x → P) → P))
L6668
Hypothesis H7 : (∀x2 : set, x2 ∈ SNoL y → (∀P : prop, (∀y2 : set, y2 ∈ ω → eps_ y2 ≤ y + - x2 → P) → P))
L6669
Hypothesis H8 : (∀x2 : set, x2 ∈ SNoR y → (∀P : prop, (∀y2 : set, y2 ∈ ω → eps_ y2 ≤ x2 + - y → P) → P))
L6670
Hypothesis H9 : SNoCutP z w
L6671
Hypothesis H10 : (∀x2 : set, x2 ∈ z → (∀P : prop, (∀y2 : set, y2 ∈ SNoL x → (∀z2 : set, z2 ∈ SNoL y → x2 = y2 * y + x * z2 + - (y2 * z2) → P)) → (∀y2 : set, y2 ∈ SNoR x → (∀z2 : set, z2 ∈ SNoR y → x2 = y2 * y + x * z2 + - (y2 * z2) → P)) → P))
L6672
Hypothesis H11 : (∀x2 : set, x2 ∈ w → (∀P : prop, (∀y2 : set, y2 ∈ SNoL x → (∀z2 : set, z2 ∈ SNoR y → x2 = y2 * y + x * z2 + - (y2 * z2) → P)) → (∀y2 : set, y2 ∈ SNoR x → (∀z2 : set, z2 ∈ SNoL y → x2 = y2 * y + x * z2 + - (y2 * z2) → P)) → P))
L6673
Hypothesis H12 : x * y = SNoCut z w
L6674
Hypothesis H13 : u ∈ setexp (SNoS_ ω) ω
L6675
Hypothesis H14 : (∀x2 : set, x2 ∈ ω → ap u x2 < x * y ∧ x * y < ap u x2 + eps_ x2 ∧ (∀y2 : set, y2 ∈ x2 → ap u y2 < ap u x2))
L6676
Hypothesis H15 : v ∈ setexp (SNoS_ ω) ω
L6677
Hypothesis H16 : (∀x2 : set, x2 ∈ ω → (ap v x2 + - (eps_ x2)) < x * y ∧ x * y < ap v x2 ∧ (∀y2 : set, y2 ∈ x2 → ap v x2 < ap v y2))
L6678
Hypothesis H17 : (∀x2 : set, x2 ∈ ω → SNo (ap u x2))
L6679
Hypothesis H18 : (∀x2 : set, x2 ∈ ω → SNo (ap v x2))
L6680
Hypothesis H19 : (∀x2 : set, x2 ∈ ω → ap u x2 < x * y)
L6681
Hypothesis H20 : (∀x2 : set, x2 ∈ ω → x * y < ap v x2)
L6682
Hypothesis H21 : SNoCutP (Repl ω (ap u)) (Repl ω (ap v))
L6683
Hypothesis H22 : SNo (SNoCut (Repl ω (ap u)) (Repl ω (ap v)))
L6684
Hypothesis H23 : (∀x2 : set, x2 ∈ ω → ap u x2 < SNoCut (Repl ω (ap u)) (Repl ω (ap v)))
L6685
Hypothesis H24 : (∀x2 : set, x2 ∈ ω → SNoCut (Repl ω (ap u)) (Repl ω (ap v)) < ap v x2)
L6686
Theorem. (Conj_real_mul_SNo_pos__114__25)
¬ (∀x2 : set, x2 ∈ ω → (∀y2 : set, y2 ∈ x2 → ap u y2 < ap u x2))
In Proofgold the corresponding term root is 0a3aa1... and proposition id is fe8cf3...
Proof:
Proof not loaded.
End of Section Conj_real_mul_SNo_pos__114__25
Beginning of Section Conj_real_mul_SNo_pos__115__11
L6692
Variable x : set
(*** Conj_real_mul_SNo_pos__115__11 TMLq6VCy3gfhjaj9j7PcL141BCrbndUpCxc bounty of about 25 bars ***)
L6693
Variable y : set
L6694
Variable z : set
L6695
Variable w : set
L6696
Variable u : set
L6697
Variable v : set
L6698
Hypothesis H0 : ¬ x * y ∈ real
L6699
Hypothesis H1 : SNo x
L6700
Hypothesis H2 : SNo y
L6701
Hypothesis H3 : SNo (x * y)
L6702
Hypothesis H4 : SNo (- (x * y))
L6703
Hypothesis H5 : (∀x2 : set, x2 ∈ SNoL x → (∀P : prop, (∀y2 : set, y2 ∈ ω → eps_ y2 ≤ x + - x2 → P) → P))
L6704
Hypothesis H6 : (∀x2 : set, x2 ∈ SNoR x → (∀P : prop, (∀y2 : set, y2 ∈ ω → eps_ y2 ≤ x2 + - x → P) → P))
L6705
Hypothesis H7 : (∀x2 : set, x2 ∈ SNoL y → (∀P : prop, (∀y2 : set, y2 ∈ ω → eps_ y2 ≤ y + - x2 → P) → P))
L6706
Hypothesis H8 : (∀x2 : set, x2 ∈ SNoR y → (∀P : prop, (∀y2 : set, y2 ∈ ω → eps_ y2 ≤ x2 + - y → P) → P))
L6707
Hypothesis H9 : SNoCutP z w
L6708
Hypothesis H10 : (∀x2 : set, x2 ∈ z → (∀P : prop, (∀y2 : set, y2 ∈ SNoL x → (∀z2 : set, z2 ∈ SNoL y → x2 = y2 * y + x * z2 + - (y2 * z2) → P)) → (∀y2 : set, y2 ∈ SNoR x → (∀z2 : set, z2 ∈ SNoR y → x2 = y2 * y + x * z2 + - (y2 * z2) → P)) → P))
L6709
Hypothesis H12 : x * y = SNoCut z w
L6710
Hypothesis H13 : u ∈ setexp (SNoS_ ω) ω
L6711
Hypothesis H14 : (∀x2 : set, x2 ∈ ω → ap u x2 < x * y ∧ x * y < ap u x2 + eps_ x2 ∧ (∀y2 : set, y2 ∈ x2 → ap u y2 < ap u x2))
L6712
Hypothesis H15 : v ∈ setexp (SNoS_ ω) ω
L6713
Hypothesis H16 : (∀x2 : set, x2 ∈ ω → (ap v x2 + - (eps_ x2)) < x * y ∧ x * y < ap v x2 ∧ (∀y2 : set, y2 ∈ x2 → ap v x2 < ap v y2))
L6714
Hypothesis H17 : (∀x2 : set, x2 ∈ ω → SNo (ap u x2))
L6715
Hypothesis H18 : (∀x2 : set, x2 ∈ ω → SNo (ap v x2))
L6716
Hypothesis H19 : (∀x2 : set, x2 ∈ ω → ap u x2 < x * y)
L6717
Hypothesis H20 : (∀x2 : set, x2 ∈ ω → x * y < ap v x2)
L6718
Hypothesis H21 : SNoCutP (Repl ω (ap u)) (Repl ω (ap v))
L6719
Hypothesis H22 : SNo (SNoCut (Repl ω (ap u)) (Repl ω (ap v)))
L6720
Hypothesis H23 : (∀x2 : set, x2 ∈ ω → ap u x2 < SNoCut (Repl ω (ap u)) (Repl ω (ap v)))
L6721
Hypothesis H24 : (∀x2 : set, x2 ∈ ω → SNoCut (Repl ω (ap u)) (Repl ω (ap v)) < ap v x2)
L6722
Theorem. (Conj_real_mul_SNo_pos__115__11)
¬ (∀x2 : set, x2 ∈ ω → x * y < ap u x2 + eps_ x2)
In Proofgold the corresponding term root is 51dd80... and proposition id is 0861ff...
Proof:
Proof not loaded.
End of Section Conj_real_mul_SNo_pos__115__11
Beginning of Section Conj_real_mul_SNo_pos__115__21
L6728
Variable x : set
(*** Conj_real_mul_SNo_pos__115__21 TMJagLpgWeRdeipF5jne5tuzY5f4qRiBGJ3 bounty of about 25 bars ***)
L6729
Variable y : set
L6730
Variable z : set
L6731
Variable w : set
L6732
Variable u : set
L6733
Variable v : set
L6734
Hypothesis H0 : ¬ x * y ∈ real
L6735
Hypothesis H1 : SNo x
L6736
Hypothesis H2 : SNo y
L6737
Hypothesis H3 : SNo (x * y)
L6738
Hypothesis H4 : SNo (- (x * y))
L6739
Hypothesis H5 : (∀x2 : set, x2 ∈ SNoL x → (∀P : prop, (∀y2 : set, y2 ∈ ω → eps_ y2 ≤ x + - x2 → P) → P))
L6740
Hypothesis H6 : (∀x2 : set, x2 ∈ SNoR x → (∀P : prop, (∀y2 : set, y2 ∈ ω → eps_ y2 ≤ x2 + - x → P) → P))
L6741
Hypothesis H7 : (∀x2 : set, x2 ∈ SNoL y → (∀P : prop, (∀y2 : set, y2 ∈ ω → eps_ y2 ≤ y + - x2 → P) → P))
L6742
Hypothesis H8 : (∀x2 : set, x2 ∈ SNoR y → (∀P : prop, (∀y2 : set, y2 ∈ ω → eps_ y2 ≤ x2 + - y → P) → P))
L6743
Hypothesis H9 : SNoCutP z w
L6744
Hypothesis H10 : (∀x2 : set, x2 ∈ z → (∀P : prop, (∀y2 : set, y2 ∈ SNoL x → (∀z2 : set, z2 ∈ SNoL y → x2 = y2 * y + x * z2 + - (y2 * z2) → P)) → (∀y2 : set, y2 ∈ SNoR x → (∀z2 : set, z2 ∈ SNoR y → x2 = y2 * y + x * z2 + - (y2 * z2) → P)) → P))
L6745
Hypothesis H11 : (∀x2 : set, x2 ∈ w → (∀P : prop, (∀y2 : set, y2 ∈ SNoL x → (∀z2 : set, z2 ∈ SNoR y → x2 = y2 * y + x * z2 + - (y2 * z2) → P)) → (∀y2 : set, y2 ∈ SNoR x → (∀z2 : set, z2 ∈ SNoL y → x2 = y2 * y + x * z2 + - (y2 * z2) → P)) → P))
L6746
Hypothesis H12 : x * y = SNoCut z w
L6747
Hypothesis H13 : u ∈ setexp (SNoS_ ω) ω
L6748
Hypothesis H14 : (∀x2 : set, x2 ∈ ω → ap u x2 < x * y ∧ x * y < ap u x2 + eps_ x2 ∧ (∀y2 : set, y2 ∈ x2 → ap u y2 < ap u x2))
L6749
Hypothesis H15 : v ∈ setexp (SNoS_ ω) ω
L6750
Hypothesis H16 : (∀x2 : set, x2 ∈ ω → (ap v x2 + - (eps_ x2)) < x * y ∧ x * y < ap v x2 ∧ (∀y2 : set, y2 ∈ x2 → ap v x2 < ap v y2))
L6751
Hypothesis H17 : (∀x2 : set, x2 ∈ ω → SNo (ap u x2))
L6752
Hypothesis H18 : (∀x2 : set, x2 ∈ ω → SNo (ap v x2))
L6753
Hypothesis H19 : (∀x2 : set, x2 ∈ ω → ap u x2 < x * y)
L6754
Hypothesis H20 : (∀x2 : set, x2 ∈ ω → x * y < ap v x2)
L6755
Hypothesis H22 : SNo (SNoCut (Repl ω (ap u)) (Repl ω (ap v)))
L6756
Hypothesis H23 : (∀x2 : set, x2 ∈ ω → ap u x2 < SNoCut (Repl ω (ap u)) (Repl ω (ap v)))
L6757
Hypothesis H24 : (∀x2 : set, x2 ∈ ω → SNoCut (Repl ω (ap u)) (Repl ω (ap v)) < ap v x2)
L6758
Theorem. (Conj_real_mul_SNo_pos__115__21)
¬ (∀x2 : set, x2 ∈ ω → x * y < ap u x2 + eps_ x2)
In Proofgold the corresponding term root is 6cfe97... and proposition id is 4faa66...
Proof:
Proof not loaded.
End of Section Conj_real_mul_SNo_pos__115__21
Beginning of Section Conj_real_mul_SNo_pos__116__0
L6764
Variable x : set
(*** Conj_real_mul_SNo_pos__116__0 TMWLgpD24AqoEUGfa1kfgrRZg7UzT5ssvzA bounty of about 25 bars ***)
L6765
Variable y : set
L6766
Variable z : set
L6767
Variable w : set
L6768
Variable u : set
L6769
Variable v : set
L6770
Hypothesis H1 : SNo x
L6771
Hypothesis H2 : SNo y
L6772
Hypothesis H3 : SNo (x * y)
L6773
Hypothesis H4 : SNo (- (x * y))
L6774
Hypothesis H5 : (∀x2 : set, x2 ∈ SNoL x → (∀P : prop, (∀y2 : set, y2 ∈ ω → eps_ y2 ≤ x + - x2 → P) → P))
L6775
Hypothesis H6 : (∀x2 : set, x2 ∈ SNoR x → (∀P : prop, (∀y2 : set, y2 ∈ ω → eps_ y2 ≤ x2 + - x → P) → P))
L6776
Hypothesis H7 : (∀x2 : set, x2 ∈ SNoL y → (∀P : prop, (∀y2 : set, y2 ∈ ω → eps_ y2 ≤ y + - x2 → P) → P))
L6777
Hypothesis H8 : (∀x2 : set, x2 ∈ SNoR y → (∀P : prop, (∀y2 : set, y2 ∈ ω → eps_ y2 ≤ x2 + - y → P) → P))
L6778
Hypothesis H9 : SNoCutP z w
L6779
Hypothesis H10 : (∀x2 : set, x2 ∈ z → (∀P : prop, (∀y2 : set, y2 ∈ SNoL x → (∀z2 : set, z2 ∈ SNoL y → x2 = y2 * y + x * z2 + - (y2 * z2) → P)) → (∀y2 : set, y2 ∈ SNoR x → (∀z2 : set, z2 ∈ SNoR y → x2 = y2 * y + x * z2 + - (y2 * z2) → P)) → P))
L6780
Hypothesis H11 : (∀x2 : set, x2 ∈ w → (∀P : prop, (∀y2 : set, y2 ∈ SNoL x → (∀z2 : set, z2 ∈ SNoR y → x2 = y2 * y + x * z2 + - (y2 * z2) → P)) → (∀y2 : set, y2 ∈ SNoR x → (∀z2 : set, z2 ∈ SNoL y → x2 = y2 * y + x * z2 + - (y2 * z2) → P)) → P))
L6781
Hypothesis H12 : x * y = SNoCut z w
L6782
Hypothesis H13 : u ∈ setexp (SNoS_ ω) ω
L6783
Hypothesis H14 : (∀x2 : set, x2 ∈ ω → ap u x2 < x * y ∧ x * y < ap u x2 + eps_ x2 ∧ (∀y2 : set, y2 ∈ x2 → ap u y2 < ap u x2))
L6784
Hypothesis H15 : v ∈ setexp (SNoS_ ω) ω
L6785
Hypothesis H16 : (∀x2 : set, x2 ∈ ω → (ap v x2 + - (eps_ x2)) < x * y ∧ x * y < ap v x2 ∧ (∀y2 : set, y2 ∈ x2 → ap v x2 < ap v y2))
L6786
Hypothesis H17 : (∀x2 : set, x2 ∈ ω → SNo (ap u x2))
L6787
Hypothesis H18 : (∀x2 : set, x2 ∈ ω → SNo (ap v x2))
L6788
Hypothesis H19 : (∀x2 : set, x2 ∈ ω → ap u x2 < x * y)
L6789
Hypothesis H20 : (∀x2 : set, x2 ∈ ω → x * y < ap v x2)
L6790
Theorem. (Conj_real_mul_SNo_pos__116__0)
¬ (∀x2 : set, x2 ∈ ω → (∀y2 : set, y2 ∈ ω → ap u x2 < ap v y2))
In Proofgold the corresponding term root is e29646... and proposition id is 96bddb...
Proof:
Proof not loaded.
End of Section Conj_real_mul_SNo_pos__116__0
Beginning of Section Conj_real_mul_SNo_pos__116__5
L6796
Variable x : set
(*** Conj_real_mul_SNo_pos__116__5 TMQW1q6rqbDTqAyefnGkcRYpuqZpCgascYs bounty of about 25 bars ***)
L6797
Variable y : set
L6798
Variable z : set
L6799
Variable w : set
L6800
Variable u : set
L6801
Variable v : set
L6802
Hypothesis H0 : ¬ x * y ∈ real
L6803
Hypothesis H1 : SNo x
L6804
Hypothesis H2 : SNo y
L6805
Hypothesis H3 : SNo (x * y)
L6806
Hypothesis H4 : SNo (- (x * y))
L6807
Hypothesis H6 : (∀x2 : set, x2 ∈ SNoR x → (∀P : prop, (∀y2 : set, y2 ∈ ω → eps_ y2 ≤ x2 + - x → P) → P))
L6808
Hypothesis H7 : (∀x2 : set, x2 ∈ SNoL y → (∀P : prop, (∀y2 : set, y2 ∈ ω → eps_ y2 ≤ y + - x2 → P) → P))
L6809
Hypothesis H8 : (∀x2 : set, x2 ∈ SNoR y → (∀P : prop, (∀y2 : set, y2 ∈ ω → eps_ y2 ≤ x2 + - y → P) → P))
L6810
Hypothesis H9 : SNoCutP z w
L6811
Hypothesis H10 : (∀x2 : set, x2 ∈ z → (∀P : prop, (∀y2 : set, y2 ∈ SNoL x → (∀z2 : set, z2 ∈ SNoL y → x2 = y2 * y + x * z2 + - (y2 * z2) → P)) → (∀y2 : set, y2 ∈ SNoR x → (∀z2 : set, z2 ∈ SNoR y → x2 = y2 * y + x * z2 + - (y2 * z2) → P)) → P))
L6812
Hypothesis H11 : (∀x2 : set, x2 ∈ w → (∀P : prop, (∀y2 : set, y2 ∈ SNoL x → (∀z2 : set, z2 ∈ SNoR y → x2 = y2 * y + x * z2 + - (y2 * z2) → P)) → (∀y2 : set, y2 ∈ SNoR x → (∀z2 : set, z2 ∈ SNoL y → x2 = y2 * y + x * z2 + - (y2 * z2) → P)) → P))
L6813
Hypothesis H12 : x * y = SNoCut z w
L6814
Hypothesis H13 : u ∈ setexp (SNoS_ ω) ω
L6815
Hypothesis H14 : (∀x2 : set, x2 ∈ ω → ap u x2 < x * y ∧ x * y < ap u x2 + eps_ x2 ∧ (∀y2 : set, y2 ∈ x2 → ap u y2 < ap u x2))
L6816
Hypothesis H15 : v ∈ setexp (SNoS_ ω) ω
L6817
Hypothesis H16 : (∀x2 : set, x2 ∈ ω → (ap v x2 + - (eps_ x2)) < x * y ∧ x * y < ap v x2 ∧ (∀y2 : set, y2 ∈ x2 → ap v x2 < ap v y2))
L6818
Hypothesis H17 : (∀x2 : set, x2 ∈ ω → SNo (ap u x2))
L6819
Hypothesis H18 : (∀x2 : set, x2 ∈ ω → SNo (ap v x2))
L6820
Hypothesis H19 : (∀x2 : set, x2 ∈ ω → ap u x2 < x * y)
L6821
Hypothesis H20 : (∀x2 : set, x2 ∈ ω → x * y < ap v x2)
L6822
Theorem. (Conj_real_mul_SNo_pos__116__5)
¬ (∀x2 : set, x2 ∈ ω → (∀y2 : set, y2 ∈ ω → ap u x2 < ap v y2))
In Proofgold the corresponding term root is 65851b... and proposition id is 2c2c74...
Proof:
Proof not loaded.
End of Section Conj_real_mul_SNo_pos__116__5
Beginning of Section Conj_real_mul_SNo_pos__117__0
L6828
Variable x : set
(*** Conj_real_mul_SNo_pos__117__0 TMVXXT3ELwQTZq5ayKEGjUzaKdmxLH8YbV1 bounty of about 25 bars ***)
L6829
Variable y : set
L6830
Variable z : set
L6831
Variable w : set
L6832
Variable u : set
L6833
Variable v : set
L6834
Hypothesis H1 : SNo x
L6835
Hypothesis H2 : SNo y
L6836
Hypothesis H3 : SNo (x * y)
L6837
Hypothesis H4 : SNo (- (x * y))
L6838
Hypothesis H5 : (∀x2 : set, x2 ∈ SNoL x → (∀P : prop, (∀y2 : set, y2 ∈ ω → eps_ y2 ≤ x + - x2 → P) → P))
L6839
Hypothesis H6 : (∀x2 : set, x2 ∈ SNoR x → (∀P : prop, (∀y2 : set, y2 ∈ ω → eps_ y2 ≤ x2 + - x → P) → P))
L6840
Hypothesis H7 : (∀x2 : set, x2 ∈ SNoL y → (∀P : prop, (∀y2 : set, y2 ∈ ω → eps_ y2 ≤ y + - x2 → P) → P))
L6841
Hypothesis H8 : (∀x2 : set, x2 ∈ SNoR y → (∀P : prop, (∀y2 : set, y2 ∈ ω → eps_ y2 ≤ x2 + - y → P) → P))
L6842
Hypothesis H9 : SNoCutP z w
L6843
Hypothesis H10 : (∀x2 : set, x2 ∈ z → (∀P : prop, (∀y2 : set, y2 ∈ SNoL x → (∀z2 : set, z2 ∈ SNoL y → x2 = y2 * y + x * z2 + - (y2 * z2) → P)) → (∀y2 : set, y2 ∈ SNoR x → (∀z2 : set, z2 ∈ SNoR y → x2 = y2 * y + x * z2 + - (y2 * z2) → P)) → P))
L6844
Hypothesis H11 : (∀x2 : set, x2 ∈ w → (∀P : prop, (∀y2 : set, y2 ∈ SNoL x → (∀z2 : set, z2 ∈ SNoR y → x2 = y2 * y + x * z2 + - (y2 * z2) → P)) → (∀y2 : set, y2 ∈ SNoR x → (∀z2 : set, z2 ∈ SNoL y → x2 = y2 * y + x * z2 + - (y2 * z2) → P)) → P))
L6845
Hypothesis H12 : x * y = SNoCut z w
L6846
Hypothesis H13 : u ∈ setexp (SNoS_ ω) ω
L6847
Hypothesis H14 : (∀x2 : set, x2 ∈ ω → ap u x2 < x * y ∧ x * y < ap u x2 + eps_ x2 ∧ (∀y2 : set, y2 ∈ x2 → ap u y2 < ap u x2))
L6848
Hypothesis H15 : v ∈ setexp (SNoS_ ω) ω
L6849
Hypothesis H16 : (∀x2 : set, x2 ∈ ω → (ap v x2 + - (eps_ x2)) < x * y ∧ x * y < ap v x2 ∧ (∀y2 : set, y2 ∈ x2 → ap v x2 < ap v y2))
L6850
Hypothesis H17 : (∀x2 : set, x2 ∈ ω → SNo (ap u x2))
L6851
Hypothesis H18 : (∀x2 : set, x2 ∈ ω → SNo (ap v x2))
L6852
Hypothesis H19 : (∀x2 : set, x2 ∈ ω → ap u x2 < x * y)
L6853
Theorem. (Conj_real_mul_SNo_pos__117__0)
¬ (∀x2 : set, x2 ∈ ω → x * y < ap v x2)
In Proofgold the corresponding term root is 4c4700... and proposition id is 025088...
Proof:
Proof not loaded.
End of Section Conj_real_mul_SNo_pos__117__0
Beginning of Section Conj_real_mul_SNo_pos__117__15
L6859
Variable x : set
(*** Conj_real_mul_SNo_pos__117__15 TMZoNHtsQScAkXRUB74sAfGbsNZCkqC9eRY bounty of about 25 bars ***)
L6860
Variable y : set
L6861
Variable z : set
L6862
Variable w : set
L6863
Variable u : set
L6864
Variable v : set
L6865
Hypothesis H0 : ¬ x * y ∈ real
L6866
Hypothesis H1 : SNo x
L6867
Hypothesis H2 : SNo y
L6868
Hypothesis H3 : SNo (x * y)
L6869
Hypothesis H4 : SNo (- (x * y))
L6870
Hypothesis H5 : (∀x2 : set, x2 ∈ SNoL x → (∀P : prop, (∀y2 : set, y2 ∈ ω → eps_ y2 ≤ x + - x2 → P) → P))
L6871
Hypothesis H6 : (∀x2 : set, x2 ∈ SNoR x → (∀P : prop, (∀y2 : set, y2 ∈ ω → eps_ y2 ≤ x2 + - x → P) → P))
L6872
Hypothesis H7 : (∀x2 : set, x2 ∈ SNoL y → (∀P : prop, (∀y2 : set, y2 ∈ ω → eps_ y2 ≤ y + - x2 → P) → P))
L6873
Hypothesis H8 : (∀x2 : set, x2 ∈ SNoR y → (∀P : prop, (∀y2 : set, y2 ∈ ω → eps_ y2 ≤ x2 + - y → P) → P))
L6874
Hypothesis H9 : SNoCutP z w
L6875
Hypothesis H10 : (∀x2 : set, x2 ∈ z → (∀P : prop, (∀y2 : set, y2 ∈ SNoL x → (∀z2 : set, z2 ∈ SNoL y → x2 = y2 * y + x * z2 + - (y2 * z2) → P)) → (∀y2 : set, y2 ∈ SNoR x → (∀z2 : set, z2 ∈ SNoR y → x2 = y2 * y + x * z2 + - (y2 * z2) → P)) → P))
L6876
Hypothesis H11 : (∀x2 : set, x2 ∈ w → (∀P : prop, (∀y2 : set, y2 ∈ SNoL x → (∀z2 : set, z2 ∈ SNoR y → x2 = y2 * y + x * z2 + - (y2 * z2) → P)) → (∀y2 : set, y2 ∈ SNoR x → (∀z2 : set, z2 ∈ SNoL y → x2 = y2 * y + x * z2 + - (y2 * z2) → P)) → P))
L6877
Hypothesis H12 : x * y = SNoCut z w
L6878
Hypothesis H13 : u ∈ setexp (SNoS_ ω) ω
L6879
Hypothesis H14 : (∀x2 : set, x2 ∈ ω → ap u x2 < x * y ∧ x * y < ap u x2 + eps_ x2 ∧ (∀y2 : set, y2 ∈ x2 → ap u y2 < ap u x2))
L6880
Hypothesis H16 : (∀x2 : set, x2 ∈ ω → (ap v x2 + - (eps_ x2)) < x * y ∧ x * y < ap v x2 ∧ (∀y2 : set, y2 ∈ x2 → ap v x2 < ap v y2))
L6881
Hypothesis H17 : (∀x2 : set, x2 ∈ ω → SNo (ap u x2))
L6882
Hypothesis H18 : (∀x2 : set, x2 ∈ ω → SNo (ap v x2))
L6883
Hypothesis H19 : (∀x2 : set, x2 ∈ ω → ap u x2 < x * y)
L6884
Theorem. (Conj_real_mul_SNo_pos__117__15)
¬ (∀x2 : set, x2 ∈ ω → x * y < ap v x2)
In Proofgold the corresponding term root is d0c683... and proposition id is 9557da...
Proof:
Proof not loaded.
End of Section Conj_real_mul_SNo_pos__117__15
Beginning of Section Conj_real_mul_SNo_pos__118__18
L6890
Variable x : set
(*** Conj_real_mul_SNo_pos__118__18 TMdEAgnLyH1pY7FMoKQ6j2XUeKksZr1zxYz bounty of about 25 bars ***)
L6891
Variable y : set
L6892
Variable z : set
L6893
Variable w : set
L6894
Variable u : set
L6895
Variable v : set
L6896
Hypothesis H0 : ¬ x * y ∈ real
L6897
Hypothesis H1 : SNo x
L6898
Hypothesis H2 : SNo y
L6899
Hypothesis H3 : SNo (x * y)
L6900
Hypothesis H4 : SNo (- (x * y))
L6901
Hypothesis H5 : (∀x2 : set, x2 ∈ SNoL x → (∀P : prop, (∀y2 : set, y2 ∈ ω → eps_ y2 ≤ x + - x2 → P) → P))
L6902
Hypothesis H6 : (∀x2 : set, x2 ∈ SNoR x → (∀P : prop, (∀y2 : set, y2 ∈ ω → eps_ y2 ≤ x2 + - x → P) → P))
L6903
Hypothesis H7 : (∀x2 : set, x2 ∈ SNoL y → (∀P : prop, (∀y2 : set, y2 ∈ ω → eps_ y2 ≤ y + - x2 → P) → P))
L6904
Hypothesis H8 : (∀x2 : set, x2 ∈ SNoR y → (∀P : prop, (∀y2 : set, y2 ∈ ω → eps_ y2 ≤ x2 + - y → P) → P))
L6905
Hypothesis H9 : SNoCutP z w
L6906
Hypothesis H10 : (∀x2 : set, x2 ∈ z → (∀P : prop, (∀y2 : set, y2 ∈ SNoL x → (∀z2 : set, z2 ∈ SNoL y → x2 = y2 * y + x * z2 + - (y2 * z2) → P)) → (∀y2 : set, y2 ∈ SNoR x → (∀z2 : set, z2 ∈ SNoR y → x2 = y2 * y + x * z2 + - (y2 * z2) → P)) → P))
L6907
Hypothesis H11 : (∀x2 : set, x2 ∈ w → (∀P : prop, (∀y2 : set, y2 ∈ SNoL x → (∀z2 : set, z2 ∈ SNoR y → x2 = y2 * y + x * z2 + - (y2 * z2) → P)) → (∀y2 : set, y2 ∈ SNoR x → (∀z2 : set, z2 ∈ SNoL y → x2 = y2 * y + x * z2 + - (y2 * z2) → P)) → P))
L6908
Hypothesis H12 : x * y = SNoCut z w
L6909
Hypothesis H13 : u ∈ setexp (SNoS_ ω) ω
L6910
Hypothesis H14 : (∀x2 : set, x2 ∈ ω → ap u x2 < x * y ∧ x * y < ap u x2 + eps_ x2 ∧ (∀y2 : set, y2 ∈ x2 → ap u y2 < ap u x2))
L6911
Hypothesis H15 : v ∈ setexp (SNoS_ ω) ω
L6912
Hypothesis H16 : (∀x2 : set, x2 ∈ ω → (ap v x2 + - (eps_ x2)) < x * y ∧ x * y < ap v x2 ∧ (∀y2 : set, y2 ∈ x2 → ap v x2 < ap v y2))
L6913
Hypothesis H17 : (∀x2 : set, x2 ∈ ω → SNo (ap u x2))
L6914
Theorem. (Conj_real_mul_SNo_pos__118__18)
¬ (∀x2 : set, x2 ∈ ω → ap u x2 < x * y)
In Proofgold the corresponding term root is 8349a9... and proposition id is f059fc...
Proof:
Proof not loaded.
End of Section Conj_real_mul_SNo_pos__118__18
Beginning of Section Conj_real_mul_SNo_pos__123__21
L6920
Variable x : set
(*** Conj_real_mul_SNo_pos__123__21 TMTCSz5NCrQuzQJSN975pypcTqBqoqAP2Kr bounty of about 25 bars ***)
L6921
Variable y : set
L6922
Variable z : set
L6923
Variable w : set
L6924
Hypothesis H0 : Empty < x
L6925
Hypothesis H1 : Empty < y
L6926
Hypothesis H2 : ¬ x * y ∈ real
L6927
Hypothesis H3 : SNo x
L6928
Hypothesis H4 : x ∈ SNoS_ (ordsucc ω)
L6929
Hypothesis H5 : x < ω
L6930
Hypothesis H6 : SNo y
L6931
Hypothesis H7 : y ∈ SNoS_ (ordsucc ω)
L6932
Hypothesis H8 : y < ω
L6933
Hypothesis H9 : (∀u : set, u ∈ ω → (∀P : prop, (∀v : set, v ∈ SNoS_ ω → Empty < v → v < x → x < v + eps_ u → P) → P))
L6934
Hypothesis H10 : (∀u : set, u ∈ ω → (∀P : prop, (∀v : set, v ∈ SNoS_ ω → Empty < v → v < y → y < v + eps_ u → P) → P))
L6935
Hypothesis H11 : SNo (x * y)
L6936
Hypothesis H12 : SNo (- (x * y))
L6937
Hypothesis H13 : (∀u : set, SNo u → SNoLev u ∈ ω → SNoLev u ∈ SNoLev (x * y))
L6938
Hypothesis H14 : (∀u : set, u ∈ SNoL x → (∀P : prop, (∀v : set, v ∈ ω → eps_ v ≤ x + - u → P) → P))
L6939
Hypothesis H15 : (∀u : set, u ∈ SNoR x → (∀P : prop, (∀v : set, v ∈ ω → eps_ v ≤ u + - x → P) → P))
L6940
Hypothesis H16 : (∀u : set, u ∈ SNoL y → (∀P : prop, (∀v : set, v ∈ ω → eps_ v ≤ y + - u → P) → P))
L6941
Hypothesis H17 : (∀u : set, u ∈ SNoR y → (∀P : prop, (∀v : set, v ∈ ω → eps_ v ≤ u + - y → P) → P))
L6942
Hypothesis H18 : SNoCutP z w
L6943
Hypothesis H19 : (∀u : set, u ∈ z → (∀P : prop, (∀v : set, v ∈ SNoL x → (∀x2 : set, x2 ∈ SNoL y → u = v * y + x * x2 + - (v * x2) → P)) → (∀v : set, v ∈ SNoR x → (∀x2 : set, x2 ∈ SNoR y → u = v * y + x * x2 + - (v * x2) → P)) → P))
L6944
Hypothesis H20 : (∀u : set, u ∈ w → (∀P : prop, (∀v : set, v ∈ SNoL x → (∀x2 : set, x2 ∈ SNoR y → u = v * y + x * x2 + - (v * x2) → P)) → (∀v : set, v ∈ SNoR x → (∀x2 : set, x2 ∈ SNoL y → u = v * y + x * x2 + - (v * x2) → P)) → P))
L6945
Theorem. (Conj_real_mul_SNo_pos__123__21)
¬ (∀u : set, u ∈ SNoS_ ω → (∀v : set, v ∈ ω → abs_SNo (u + - (x * y)) < eps_ v) → u = x * y)
In Proofgold the corresponding term root is 0cd9d0... and proposition id is f8b6d7...
Proof:
Proof not loaded.
End of Section Conj_real_mul_SNo_pos__123__21
Beginning of Section Conj_real_mul_SNo_pos__127__3
L6951
Variable x : set
(*** Conj_real_mul_SNo_pos__127__3 TMYNkZb3QP4bEZayAUe4ZNhandvPjKMtd5g bounty of about 25 bars ***)
L6952
Variable y : set
L6953
Hypothesis H0 : Empty < x
L6954
Hypothesis H1 : Empty < y
L6955
Hypothesis H2 : ¬ x * y ∈ real
L6956
Hypothesis H4 : x ∈ SNoS_ (ordsucc ω)
L6957
Hypothesis H5 : x < ω
L6958
Hypothesis H6 : (∀z : set, z ∈ SNoS_ ω → (∀w : set, w ∈ ω → abs_SNo (z + - x) < eps_ w) → z = x)
L6959
Hypothesis H7 : SNo y
L6960
Hypothesis H8 : y ∈ SNoS_ (ordsucc ω)
L6961
Hypothesis H9 : y < ω
L6962
Hypothesis H10 : (∀z : set, z ∈ SNoS_ ω → (∀w : set, w ∈ ω → abs_SNo (z + - y) < eps_ w) → z = y)
L6963
Hypothesis H11 : (∀z : set, z ∈ ω → (∀P : prop, (∀w : set, w ∈ SNoS_ ω → Empty < w → w < x → x < w + eps_ z → P) → P))
L6964
Hypothesis H12 : (∀z : set, z ∈ ω → (∀P : prop, (∀w : set, w ∈ SNoS_ ω → Empty < w → w < y → y < w + eps_ z → P) → P))
L6965
Hypothesis H13 : SNo (x * y)
L6966
Hypothesis H14 : SNo (- (x * y))
L6967
Hypothesis H15 : (∀z : set, SNo z → SNoLev z ∈ ω → SNoLev z ∈ SNoLev (x * y))
L6968
Hypothesis H16 : Subq (SNoL x) (SNoS_ ω)
L6969
Hypothesis H17 : Subq (SNoR x) (SNoS_ ω)
L6970
Hypothesis H18 : Subq (SNoL y) (SNoS_ ω)
L6971
Hypothesis H19 : Subq (SNoR y) (SNoS_ ω)
L6972
Theorem. (Conj_real_mul_SNo_pos__127__3)
¬ (∀z : set, z ∈ SNoL x → (∀P : prop, (∀w : set, w ∈ ω → eps_ w ≤ x + - z → P) → P))
In Proofgold the corresponding term root is ac502a... and proposition id is f5bbd9...
Proof:
Proof not loaded.
End of Section Conj_real_mul_SNo_pos__127__3
Beginning of Section Conj_real_mul_SNo_pos__129__4
L6978
Variable x : set
(*** Conj_real_mul_SNo_pos__129__4 TMQTB8232tywc8kYk8Pg4whEct1TpnX5Tje bounty of about 25 bars ***)
L6979
Variable y : set
L6980
Hypothesis H0 : Empty < x
L6981
Hypothesis H1 : Empty < y
L6982
Hypothesis H2 : ¬ x * y ∈ real
L6983
Hypothesis H3 : SNo x
L6984
Hypothesis H5 : x < ω
L6985
Hypothesis H6 : (∀z : set, z ∈ SNoS_ ω → (∀w : set, w ∈ ω → abs_SNo (z + - x) < eps_ w) → z = x)
L6986
Hypothesis H7 : SNo y
L6987
Hypothesis H8 : SNoLev y ∈ ordsucc ω
L6988
Hypothesis H9 : y ∈ SNoS_ (ordsucc ω)
L6989
Hypothesis H10 : y < ω
L6990
Hypothesis H11 : (∀z : set, z ∈ SNoS_ ω → (∀w : set, w ∈ ω → abs_SNo (z + - y) < eps_ w) → z = y)
L6991
Hypothesis H12 : (∀z : set, z ∈ ω → (∀P : prop, (∀w : set, w ∈ SNoS_ ω → Empty < w → w < x → x < w + eps_ z → P) → P))
L6992
Hypothesis H13 : (∀z : set, z ∈ ω → (∀P : prop, (∀w : set, w ∈ SNoS_ ω → Empty < w → w < y → y < w + eps_ z → P) → P))
L6993
Hypothesis H14 : SNo (x * y)
L6994
Hypothesis H15 : SNo (- (x * y))
L6995
Hypothesis H16 : (∀z : set, SNo z → SNoLev z ∈ ω → SNoLev z ∈ SNoLev (x * y))
L6996
Hypothesis H17 : Subq (SNoL x) (SNoS_ ω)
L6997
Hypothesis H18 : Subq (SNoR x) (SNoS_ ω)
L6998
Theorem. (Conj_real_mul_SNo_pos__129__4)
¬ Subq (SNoL y) (SNoS_ ω)
In Proofgold the corresponding term root is 3801ad... and proposition id is dcd874...
Proof:
Proof not loaded.
End of Section Conj_real_mul_SNo_pos__129__4
Beginning of Section Conj_real_mul_SNo_pos__130__0
L7004
Variable x : set
(*** Conj_real_mul_SNo_pos__130__0 TMRRwPzS6KKfBzsw5XaoSKd8ao7ZN7Z2HFN bounty of about 25 bars ***)
L7005
Variable y : set
L7006
Hypothesis H1 : Empty < y
L7007
Hypothesis H2 : ¬ x * y ∈ real
L7008
Hypothesis H3 : SNo x
L7009
Hypothesis H4 : SNoLev x ∈ ordsucc ω
L7010
Hypothesis H5 : x ∈ SNoS_ (ordsucc ω)
L7011
Hypothesis H6 : x < ω
L7012
Hypothesis H7 : (∀z : set, z ∈ SNoS_ ω → (∀w : set, w ∈ ω → abs_SNo (z + - x) < eps_ w) → z = x)
L7013
Hypothesis H8 : SNo y
L7014
Hypothesis H9 : SNoLev y ∈ ordsucc ω
L7015
Hypothesis H10 : y ∈ SNoS_ (ordsucc ω)
L7016
Hypothesis H11 : y < ω
L7017
Hypothesis H12 : (∀z : set, z ∈ SNoS_ ω → (∀w : set, w ∈ ω → abs_SNo (z + - y) < eps_ w) → z = y)
L7018
Hypothesis H13 : (∀z : set, z ∈ ω → (∀P : prop, (∀w : set, w ∈ SNoS_ ω → Empty < w → w < x → x < w + eps_ z → P) → P))
L7019
Hypothesis H14 : (∀z : set, z ∈ ω → (∀P : prop, (∀w : set, w ∈ SNoS_ ω → Empty < w → w < y → y < w + eps_ z → P) → P))
L7020
Hypothesis H15 : SNo (x * y)
L7021
Hypothesis H16 : SNo (- (x * y))
L7022
Hypothesis H17 : (∀z : set, SNo z → SNoLev z ∈ ω → SNoLev z ∈ SNoLev (x * y))
L7023
Hypothesis H18 : Subq (SNoL x) (SNoS_ ω)
L7024
Theorem. (Conj_real_mul_SNo_pos__130__0)
¬ Subq (SNoR x) (SNoS_ ω)
In Proofgold the corresponding term root is 80facd... and proposition id is ee4978...
Proof:
Proof not loaded.
End of Section Conj_real_mul_SNo_pos__130__0
Beginning of Section Conj_real_mul_SNo_pos__132__4
L7030
Variable x : set
(*** Conj_real_mul_SNo_pos__132__4 TMPEKF2qMeDC3XrA3wWyAvwFdzZTHJEV73F bounty of about 25 bars ***)
L7031
Variable y : set
L7032
Hypothesis H0 : Empty < x
L7033
Hypothesis H1 : Empty < y
L7034
Hypothesis H2 : ¬ x * y ∈ real
L7035
Hypothesis H3 : SNo x
L7036
Hypothesis H5 : x ∈ SNoS_ (ordsucc ω)
L7037
Hypothesis H6 : x < ω
L7038
Hypothesis H7 : (∀z : set, z ∈ SNoS_ ω → (∀w : set, w ∈ ω → abs_SNo (z + - x) < eps_ w) → z = x)
L7039
Hypothesis H8 : SNo y
L7040
Hypothesis H9 : SNoLev y ∈ ordsucc ω
L7041
Hypothesis H10 : y ∈ SNoS_ (ordsucc ω)
L7042
Hypothesis H11 : y < ω
L7043
Hypothesis H12 : (∀z : set, z ∈ SNoS_ ω → (∀w : set, w ∈ ω → abs_SNo (z + - y) < eps_ w) → z = y)
L7044
Hypothesis H13 : (∀z : set, z ∈ ω → (∀P : prop, (∀w : set, w ∈ SNoS_ ω → Empty < w → w < x → x < w + eps_ z → P) → P))
L7045
Hypothesis H14 : (∀z : set, z ∈ ω → (∀P : prop, (∀w : set, w ∈ SNoS_ ω → Empty < w → w < y → y < w + eps_ z → P) → P))
L7046
Hypothesis H15 : SNo (x * y)
L7047
Hypothesis H16 : SNo (- (x * y))
L7048
Hypothesis H17 : nIn (SNoLev (x * y)) ω
L7049
Theorem. (Conj_real_mul_SNo_pos__132__4)
¬ (∀z : set, SNo z → SNoLev z ∈ ω → SNoLev z ∈ SNoLev (x * y))
In Proofgold the corresponding term root is 46b8fa... and proposition id is 9e3c58...
Proof:
Proof not loaded.
End of Section Conj_real_mul_SNo_pos__132__4
Beginning of Section Conj_real_mul_SNo_pos__133__12
L7055
Variable x : set
(*** Conj_real_mul_SNo_pos__133__12 TMFdfgHoVkM7aZUJZJ9pGVAiMiQdyGgJHdz bounty of about 25 bars ***)
L7056
Variable y : set
L7057
Hypothesis H0 : Empty < x
L7058
Hypothesis H1 : Empty < y
L7059
Hypothesis H2 : ¬ x * y ∈ real
L7060
Hypothesis H3 : SNo x
L7061
Hypothesis H4 : SNoLev x ∈ ordsucc ω
L7062
Hypothesis H5 : x ∈ SNoS_ (ordsucc ω)
L7063
Hypothesis H6 : x < ω
L7064
Hypothesis H7 : (∀z : set, z ∈ SNoS_ ω → (∀w : set, w ∈ ω → abs_SNo (z + - x) < eps_ w) → z = x)
L7065
Hypothesis H8 : SNo y
L7066
Hypothesis H9 : SNoLev y ∈ ordsucc ω
L7067
Hypothesis H10 : y ∈ SNoS_ (ordsucc ω)
L7068
Hypothesis H11 : y < ω
L7069
Hypothesis H13 : (∀z : set, z ∈ ω → (∀P : prop, (∀w : set, w ∈ SNoS_ ω → Empty < w → w < x → x < w + eps_ z → P) → P))
L7070
Hypothesis H14 : (∀z : set, z ∈ ω → (∀P : prop, (∀w : set, w ∈ SNoS_ ω → Empty < w → w < y → y < w + eps_ z → P) → P))
L7071
Hypothesis H15 : SNo (x * y)
L7072
Hypothesis H16 : SNo (- (x * y))
L7073
Theorem. (Conj_real_mul_SNo_pos__133__12)
¬ nIn (SNoLev (x * y)) ω
In Proofgold the corresponding term root is f311c9... and proposition id is e346a2...
Proof:
Proof not loaded.
End of Section Conj_real_mul_SNo_pos__133__12
Beginning of Section Conj_real_mul_SNo_pos__135__10
L7079
Variable x : set
(*** Conj_real_mul_SNo_pos__135__10 TMJyD52P98j7bD8o1aoVGQbrBp35vzHcoWt bounty of about 25 bars ***)
L7080
Variable y : set
L7081
Hypothesis H0 : Empty < x
L7082
Hypothesis H1 : Empty < y
L7083
Hypothesis H2 : ¬ x * y ∈ real
L7084
Hypothesis H3 : SNo x
L7085
Hypothesis H4 : SNoLev x ∈ ordsucc ω
L7086
Hypothesis H5 : x ∈ SNoS_ (ordsucc ω)
L7087
Hypothesis H6 : x < ω
L7088
Hypothesis H7 : (∀z : set, z ∈ SNoS_ ω → (∀w : set, w ∈ ω → abs_SNo (z + - x) < eps_ w) → z = x)
L7089
Hypothesis H8 : SNo y
L7090
Hypothesis H9 : SNoLev y ∈ ordsucc ω
L7091
Hypothesis H11 : y < ω
L7092
Hypothesis H12 : (∀z : set, z ∈ SNoS_ ω → (∀w : set, w ∈ ω → abs_SNo (z + - y) < eps_ w) → z = y)
L7093
Hypothesis H13 : (∀z : set, z ∈ ω → (∀P : prop, (∀w : set, w ∈ SNoS_ ω → Empty < w → w < x → x < w + eps_ z → P) → P))
L7094
Hypothesis H14 : (∀z : set, z ∈ ω → (∀P : prop, (∀w : set, w ∈ SNoS_ ω → Empty < w → w < y → y < w + eps_ z → P) → P))
L7095
Theorem. (Conj_real_mul_SNo_pos__135__10)
¬ SNo (x * y)
In Proofgold the corresponding term root is eedbd3... and proposition id is e4ae81...
Proof:
Proof not loaded.
End of Section Conj_real_mul_SNo_pos__135__10
Beginning of Section Conj_real_mul_SNo_pos__136__12
L7101
Variable x : set
(*** Conj_real_mul_SNo_pos__136__12 TMS5PZE3TZPN4ND1ujYfpfUWCYedEWQFXeq bounty of about 25 bars ***)
L7102
Variable y : set
L7103
Hypothesis H0 : Empty < x
L7104
Hypothesis H1 : Empty < y
L7105
Hypothesis H2 : ¬ x * y ∈ real
L7106
Hypothesis H3 : SNo x
L7107
Hypothesis H4 : SNoLev x ∈ ordsucc ω
L7108
Hypothesis H5 : x ∈ SNoS_ (ordsucc ω)
L7109
Hypothesis H6 : x < ω
L7110
Hypothesis H7 : (∀z : set, z ∈ SNoS_ ω → (∀w : set, w ∈ ω → abs_SNo (z + - x) < eps_ w) → z = x)
L7111
Hypothesis H8 : SNo y
L7112
Hypothesis H9 : SNoLev y ∈ ordsucc ω
L7113
Hypothesis H10 : y ∈ SNoS_ (ordsucc ω)
L7114
Hypothesis H11 : y < ω
L7115
Hypothesis H13 : (∀z : set, z ∈ ω → (∃w : set, w ∈ SNoS_ ω ∧ (w < y ∧ y < w + eps_ z)))
L7116
Hypothesis H14 : (∀z : set, z ∈ ω → (∀P : prop, (∀w : set, w ∈ SNoS_ ω → Empty < w → w < x → x < w + eps_ z → P) → P))
L7117
Theorem. (Conj_real_mul_SNo_pos__136__12)
¬ (∀z : set, z ∈ ω → (∀P : prop, (∀w : set, w ∈ SNoS_ ω → Empty < w → w < y → y < w + eps_ z → P) → P))
In Proofgold the corresponding term root is 11e638... and proposition id is 602eae...
Proof:
Proof not loaded.
End of Section Conj_real_mul_SNo_pos__136__12
Beginning of Section Conj_real_mul_SNo_pos__137__4
L7123
Variable x : set
(*** Conj_real_mul_SNo_pos__137__4 TMbbdYEKJCjEjJTwRC5ytepJEaBK8R6mgxW bounty of about 25 bars ***)
L7124
Variable y : set
L7125
Hypothesis H0 : Empty < x
L7126
Hypothesis H1 : Empty < y
L7127
Hypothesis H2 : ¬ x * y ∈ real
L7128
Hypothesis H3 : SNo x
L7129
Hypothesis H5 : x ∈ SNoS_ (ordsucc ω)
L7130
Hypothesis H6 : x < ω
L7131
Hypothesis H7 : (∀z : set, z ∈ SNoS_ ω → (∀w : set, w ∈ ω → abs_SNo (z + - x) < eps_ w) → z = x)
L7132
Hypothesis H8 : (∀z : set, z ∈ ω → (∃w : set, w ∈ SNoS_ ω ∧ (w < x ∧ x < w + eps_ z)))
L7133
Hypothesis H9 : SNo y
L7134
Hypothesis H10 : SNoLev y ∈ ordsucc ω
L7135
Hypothesis H11 : y ∈ SNoS_ (ordsucc ω)
L7136
Hypothesis H12 : y < ω
L7137
Hypothesis H13 : (∀z : set, z ∈ SNoS_ ω → (∀w : set, w ∈ ω → abs_SNo (z + - y) < eps_ w) → z = y)
L7138
Hypothesis H14 : (∀z : set, z ∈ ω → (∃w : set, w ∈ SNoS_ ω ∧ (w < y ∧ y < w + eps_ z)))
L7139
Theorem. (Conj_real_mul_SNo_pos__137__4)
¬ (∀z : set, z ∈ ω → (∀P : prop, (∀w : set, w ∈ SNoS_ ω → Empty < w → w < x → x < w + eps_ z → P) → P))
In Proofgold the corresponding term root is 883f60... and proposition id is 06a7d7...
Proof:
Proof not loaded.
End of Section Conj_real_mul_SNo_pos__137__4
Beginning of Section Conj_abs_SNo_intvl_bd__1__1
L7145
Variable x : set
(*** Conj_abs_SNo_intvl_bd__1__1 TMaPQSoyBWSFwADtt1AdDWQ33syELGW1SDa bounty of about 25 bars ***)
L7146
Variable y : set
L7147
Variable z : set
L7148
Hypothesis H0 : SNo x
L7149
Hypothesis H2 : SNo z
L7150
Hypothesis H3 : y < x + z
L7151
Hypothesis H4 : Empty ≤ y + - x
L7152
Theorem. (Conj_abs_SNo_intvl_bd__1__1)
abs_SNo (y + - x) = y + - x → abs_SNo (y + - x) < z
In Proofgold the corresponding term root is ca5865... and proposition id is a6e8c4...
Proof:
Proof not loaded.
End of Section Conj_abs_SNo_intvl_bd__1__1
Beginning of Section Conj_pos_small_real_recip_ex__6__9
L7158
Variable x : set
(*** Conj_pos_small_real_recip_ex__6__9 TMPFhGY4vjtoJ52pG4WQyXYdYaHUNp5YYGB bounty of about 25 bars ***)
L7159
Variable y : set
L7160
Variable z : set
L7161
Hypothesis H0 : x < ordsucc Empty
L7162
Hypothesis H1 : SNo x
L7163
Hypothesis H2 : SNo (SNoCut (Sep (SNoS_ ω) (λw : set ⇒ x * w < ordsucc Empty)) (Sep (SNoS_ ω) (λw : set ⇒ ordsucc Empty < x * w)))
L7164
Hypothesis H3 : (∀w : set, w ∈ Sep (SNoS_ ω) (λu : set ⇒ ordsucc Empty < x * u) → SNoCut (Sep (SNoS_ ω) (λu : set ⇒ x * u < ordsucc Empty)) (Sep (SNoS_ ω) (λu : set ⇒ ordsucc Empty < x * u)) < w)
L7165
Hypothesis H4 : y ∈ SNoS_ ω
L7166
Hypothesis H5 : (∀w : set, w ∈ ω → abs_SNo (y + - (SNoCut (Sep (SNoS_ ω) (λu : set ⇒ x * u < ordsucc Empty)) (Sep (SNoS_ ω) (λu : set ⇒ ordsucc Empty < x * u)))) < eps_ w)
L7167
Hypothesis H6 : SNo y
L7168
Hypothesis H7 : SNo (x * y)
L7169
Hypothesis H8 : SNoCut (Sep (SNoS_ ω) (λw : set ⇒ x * w < ordsucc Empty)) (Sep (SNoS_ ω) (λw : set ⇒ ordsucc Empty < x * w)) < y
L7170
Hypothesis H10 : ordsucc Empty ≤ x * y + - (eps_ z)
L7171
Theorem. (Conj_pos_small_real_recip_ex__6__9)
¬ SNoCut (Sep (SNoS_ ω) (λw : set ⇒ x * w < ordsucc Empty)) (Sep (SNoS_ ω) (λw : set ⇒ ordsucc Empty < x * w)) < y + - (eps_ z)
In Proofgold the corresponding term root is ec1dec... and proposition id is 1de34a...
Proof:
Proof not loaded.
End of Section Conj_pos_small_real_recip_ex__6__9
Beginning of Section Conj_pos_small_real_recip_ex__8__0
L7177
Variable x : set
(*** Conj_pos_small_real_recip_ex__8__0 TMRvBrLEdB8cJdjdMEKvs4LRugzKhp57Mof bounty of about 25 bars ***)
L7178
Variable y : set
L7179
Hypothesis H1 : ¬ (∃z : set, z ∈ real ∧ x * z = ordsucc Empty)
L7180
Hypothesis H2 : SNo x
L7181
Hypothesis H3 : SNo (SNoCut (Sep (SNoS_ ω) (λz : set ⇒ x * z < ordsucc Empty)) (Sep (SNoS_ ω) (λz : set ⇒ ordsucc Empty < x * z)))
L7182
Hypothesis H4 : (∀z : set, z ∈ Sep (SNoS_ ω) (λw : set ⇒ x * w < ordsucc Empty) → z < SNoCut (Sep (SNoS_ ω) (λw : set ⇒ x * w < ordsucc Empty)) (Sep (SNoS_ ω) (λw : set ⇒ ordsucc Empty < x * w)))
L7183
Hypothesis H5 : (∀z : set, z ∈ Sep (SNoS_ ω) (λw : set ⇒ ordsucc Empty < x * w) → SNoCut (Sep (SNoS_ ω) (λw : set ⇒ x * w < ordsucc Empty)) (Sep (SNoS_ ω) (λw : set ⇒ ordsucc Empty < x * w)) < z)
L7184
Hypothesis H6 : y ∈ SNoS_ ω
L7185
Hypothesis H7 : (∀z : set, z ∈ ω → abs_SNo (y + - (SNoCut (Sep (SNoS_ ω) (λw : set ⇒ x * w < ordsucc Empty)) (Sep (SNoS_ ω) (λw : set ⇒ ordsucc Empty < x * w)))) < eps_ z)
L7186
Hypothesis H8 : SNo y
L7187
Hypothesis H9 : y ∈ real
L7188
Hypothesis H10 : SNo (x * y)
L7189
Hypothesis H11 : (∀z : set, z ∈ SNoS_ ω → (∀w : set, w ∈ ω → abs_SNo (z + - (x * y)) < eps_ w) → z = x * y)
L7190
Hypothesis H12 : x * y < ordsucc Empty → (∃z : set, z ∈ ω ∧ (x * y + eps_ z) ≤ ordsucc Empty)
L7191
Theorem. (Conj_pos_small_real_recip_ex__8__0)
(ordsucc Empty < x * y → (∃z : set, z ∈ ω ∧ ordsucc Empty ≤ x * y + - (eps_ z))) → y = SNoCut (Sep (SNoS_ ω) (λz : set ⇒ x * z < ordsucc Empty)) (Sep (SNoS_ ω) (λz : set ⇒ ordsucc Empty < x * z))
In Proofgold the corresponding term root is 57b867... and proposition id is b65cca...
Proof:
Proof not loaded.
End of Section Conj_pos_small_real_recip_ex__8__0
Beginning of Section Conj_pos_small_real_recip_ex__9__3
L7197
Variable x : set
(*** Conj_pos_small_real_recip_ex__9__3 TMUC6SVsJ4pj4b6h72e8rth32dqURg96cm2 bounty of about 25 bars ***)
L7198
Variable y : set
L7199
Hypothesis H0 : x < ordsucc Empty
L7200
Hypothesis H1 : ¬ (∃z : set, z ∈ real ∧ x * z = ordsucc Empty)
L7201
Hypothesis H2 : SNo x
L7202
Hypothesis H4 : (∀z : set, z ∈ Sep (SNoS_ ω) (λw : set ⇒ x * w < ordsucc Empty) → z < SNoCut (Sep (SNoS_ ω) (λw : set ⇒ x * w < ordsucc Empty)) (Sep (SNoS_ ω) (λw : set ⇒ ordsucc Empty < x * w)))
L7203
Hypothesis H5 : (∀z : set, z ∈ Sep (SNoS_ ω) (λw : set ⇒ ordsucc Empty < x * w) → SNoCut (Sep (SNoS_ ω) (λw : set ⇒ x * w < ordsucc Empty)) (Sep (SNoS_ ω) (λw : set ⇒ ordsucc Empty < x * w)) < z)
L7204
Hypothesis H6 : y ∈ SNoS_ ω
L7205
Hypothesis H7 : (∀z : set, z ∈ ω → abs_SNo (y + - (SNoCut (Sep (SNoS_ ω) (λw : set ⇒ x * w < ordsucc Empty)) (Sep (SNoS_ ω) (λw : set ⇒ ordsucc Empty < x * w)))) < eps_ z)
L7206
Hypothesis H8 : SNo y
L7207
Hypothesis H9 : y ∈ real
L7208
Hypothesis H10 : SNo (x * y)
L7209
Hypothesis H11 : (∀z : set, z ∈ SNoS_ ω → (∀w : set, w ∈ ω → abs_SNo (z + - (x * y)) < eps_ w) → z = x * y)
L7210
Hypothesis H12 : SNo (- (x * y))
L7211
Theorem. (Conj_pos_small_real_recip_ex__9__3)
(x * y < ordsucc Empty → (∃z : set, z ∈ ω ∧ (x * y + eps_ z) ≤ ordsucc Empty)) → y = SNoCut (Sep (SNoS_ ω) (λz : set ⇒ x * z < ordsucc Empty)) (Sep (SNoS_ ω) (λz : set ⇒ ordsucc Empty < x * z))
In Proofgold the corresponding term root is 46835a... and proposition id is d24167...
Proof:
Proof not loaded.
End of Section Conj_pos_small_real_recip_ex__9__3
Beginning of Section Conj_pos_small_real_recip_ex__11__1
L7217
Variable x : set
(*** Conj_pos_small_real_recip_ex__11__1 TMMyT9UihYiCAz1imqwARNcY4xDFPpK2wyS bounty of about 25 bars ***)
L7218
Variable y : set
L7219
Hypothesis H0 : x ∈ real
L7220
Hypothesis H2 : ¬ (∃z : set, z ∈ real ∧ x * z = ordsucc Empty)
L7221
Hypothesis H3 : SNo x
L7222
Hypothesis H4 : SNo (SNoCut (Sep (SNoS_ ω) (λz : set ⇒ x * z < ordsucc Empty)) (Sep (SNoS_ ω) (λz : set ⇒ ordsucc Empty < x * z)))
L7223
Hypothesis H5 : (∀z : set, z ∈ Sep (SNoS_ ω) (λw : set ⇒ x * w < ordsucc Empty) → z < SNoCut (Sep (SNoS_ ω) (λw : set ⇒ x * w < ordsucc Empty)) (Sep (SNoS_ ω) (λw : set ⇒ ordsucc Empty < x * w)))
L7224
Hypothesis H6 : (∀z : set, z ∈ Sep (SNoS_ ω) (λw : set ⇒ ordsucc Empty < x * w) → SNoCut (Sep (SNoS_ ω) (λw : set ⇒ x * w < ordsucc Empty)) (Sep (SNoS_ ω) (λw : set ⇒ ordsucc Empty < x * w)) < z)
L7225
Hypothesis H7 : y ∈ SNoS_ ω
L7226
Hypothesis H8 : (∀z : set, z ∈ ω → abs_SNo (y + - (SNoCut (Sep (SNoS_ ω) (λw : set ⇒ x * w < ordsucc Empty)) (Sep (SNoS_ ω) (λw : set ⇒ ordsucc Empty < x * w)))) < eps_ z)
L7227
Hypothesis H9 : SNo y
L7228
Hypothesis H10 : y ∈ real
L7229
Theorem. (Conj_pos_small_real_recip_ex__11__1)
x * y ∈ real → y = SNoCut (Sep (SNoS_ ω) (λz : set ⇒ x * z < ordsucc Empty)) (Sep (SNoS_ ω) (λz : set ⇒ ordsucc Empty < x * z))
In Proofgold the corresponding term root is 531d50... and proposition id is edbff8...
Proof:
Proof not loaded.
End of Section Conj_pos_small_real_recip_ex__11__1
Beginning of Section Conj_pos_small_real_recip_ex__11__2
L7235
Variable x : set
(*** Conj_pos_small_real_recip_ex__11__2 TMbRkHPN6tCX8U7x1vPQss3jfzXWm36frrA bounty of about 25 bars ***)
L7236
Variable y : set
L7237
Hypothesis H0 : x ∈ real
L7238
Hypothesis H1 : x < ordsucc Empty
L7239
Hypothesis H3 : SNo x
L7240
Hypothesis H4 : SNo (SNoCut (Sep (SNoS_ ω) (λz : set ⇒ x * z < ordsucc Empty)) (Sep (SNoS_ ω) (λz : set ⇒ ordsucc Empty < x * z)))
L7241
Hypothesis H5 : (∀z : set, z ∈ Sep (SNoS_ ω) (λw : set ⇒ x * w < ordsucc Empty) → z < SNoCut (Sep (SNoS_ ω) (λw : set ⇒ x * w < ordsucc Empty)) (Sep (SNoS_ ω) (λw : set ⇒ ordsucc Empty < x * w)))
L7242
Hypothesis H6 : (∀z : set, z ∈ Sep (SNoS_ ω) (λw : set ⇒ ordsucc Empty < x * w) → SNoCut (Sep (SNoS_ ω) (λw : set ⇒ x * w < ordsucc Empty)) (Sep (SNoS_ ω) (λw : set ⇒ ordsucc Empty < x * w)) < z)
L7243
Hypothesis H7 : y ∈ SNoS_ ω
L7244
Hypothesis H8 : (∀z : set, z ∈ ω → abs_SNo (y + - (SNoCut (Sep (SNoS_ ω) (λw : set ⇒ x * w < ordsucc Empty)) (Sep (SNoS_ ω) (λw : set ⇒ ordsucc Empty < x * w)))) < eps_ z)
L7245
Hypothesis H9 : SNo y
L7246
Hypothesis H10 : y ∈ real
L7247
Theorem. (Conj_pos_small_real_recip_ex__11__2)
x * y ∈ real → y = SNoCut (Sep (SNoS_ ω) (λz : set ⇒ x * z < ordsucc Empty)) (Sep (SNoS_ ω) (λz : set ⇒ ordsucc Empty < x * z))
In Proofgold the corresponding term root is 49e26f... and proposition id is 3b81fc...
Proof:
Proof not loaded.
End of Section Conj_pos_small_real_recip_ex__11__2
Beginning of Section Conj_pos_small_real_recip_ex__11__7
L7253
Variable x : set
(*** Conj_pos_small_real_recip_ex__11__7 TMYt1mpvVAUNeAz8K7uEQ76yn4oCL36RLiq bounty of about 25 bars ***)
L7254
Variable y : set
L7255
Hypothesis H0 : x ∈ real
L7256
Hypothesis H1 : x < ordsucc Empty
L7257
Hypothesis H2 : ¬ (∃z : set, z ∈ real ∧ x * z = ordsucc Empty)
L7258
Hypothesis H3 : SNo x
L7259
Hypothesis H4 : SNo (SNoCut (Sep (SNoS_ ω) (λz : set ⇒ x * z < ordsucc Empty)) (Sep (SNoS_ ω) (λz : set ⇒ ordsucc Empty < x * z)))
L7260
Hypothesis H5 : (∀z : set, z ∈ Sep (SNoS_ ω) (λw : set ⇒ x * w < ordsucc Empty) → z < SNoCut (Sep (SNoS_ ω) (λw : set ⇒ x * w < ordsucc Empty)) (Sep (SNoS_ ω) (λw : set ⇒ ordsucc Empty < x * w)))
L7261
Hypothesis H6 : (∀z : set, z ∈ Sep (SNoS_ ω) (λw : set ⇒ ordsucc Empty < x * w) → SNoCut (Sep (SNoS_ ω) (λw : set ⇒ x * w < ordsucc Empty)) (Sep (SNoS_ ω) (λw : set ⇒ ordsucc Empty < x * w)) < z)
L7262
Hypothesis H8 : (∀z : set, z ∈ ω → abs_SNo (y + - (SNoCut (Sep (SNoS_ ω) (λw : set ⇒ x * w < ordsucc Empty)) (Sep (SNoS_ ω) (λw : set ⇒ ordsucc Empty < x * w)))) < eps_ z)
L7263
Hypothesis H9 : SNo y
L7264
Hypothesis H10 : y ∈ real
L7265
Theorem. (Conj_pos_small_real_recip_ex__11__7)
x * y ∈ real → y = SNoCut (Sep (SNoS_ ω) (λz : set ⇒ x * z < ordsucc Empty)) (Sep (SNoS_ ω) (λz : set ⇒ ordsucc Empty < x * z))
In Proofgold the corresponding term root is aa16b5... and proposition id is 83eb37...
Proof:
Proof not loaded.
End of Section Conj_pos_small_real_recip_ex__11__7
Beginning of Section Conj_pos_small_real_recip_ex__12__0
L7271
Variable x : set
(*** Conj_pos_small_real_recip_ex__12__0 TMXejCpXMA14xRhTB1xfqy7GqeQ7ZCfa8e2 bounty of about 25 bars ***)
L7272
Variable y : set
L7273
Hypothesis H1 : x < ordsucc Empty
L7274
Hypothesis H2 : ¬ (∃z : set, z ∈ real ∧ x * z = ordsucc Empty)
L7275
Hypothesis H3 : SNo x
L7276
Hypothesis H4 : SNo (SNoCut (Sep (SNoS_ ω) (λz : set ⇒ x * z < ordsucc Empty)) (Sep (SNoS_ ω) (λz : set ⇒ ordsucc Empty < x * z)))
L7277
Hypothesis H5 : (∀z : set, z ∈ Sep (SNoS_ ω) (λw : set ⇒ x * w < ordsucc Empty) → z < SNoCut (Sep (SNoS_ ω) (λw : set ⇒ x * w < ordsucc Empty)) (Sep (SNoS_ ω) (λw : set ⇒ ordsucc Empty < x * w)))
L7278
Hypothesis H6 : (∀z : set, z ∈ Sep (SNoS_ ω) (λw : set ⇒ ordsucc Empty < x * w) → SNoCut (Sep (SNoS_ ω) (λw : set ⇒ x * w < ordsucc Empty)) (Sep (SNoS_ ω) (λw : set ⇒ ordsucc Empty < x * w)) < z)
L7279
Hypothesis H7 : y ∈ SNoS_ ω
L7280
Hypothesis H8 : (∀z : set, z ∈ ω → abs_SNo (y + - (SNoCut (Sep (SNoS_ ω) (λw : set ⇒ x * w < ordsucc Empty)) (Sep (SNoS_ ω) (λw : set ⇒ ordsucc Empty < x * w)))) < eps_ z)
L7281
Hypothesis H9 : SNo y
L7282
Theorem. (Conj_pos_small_real_recip_ex__12__0)
y ∈ real → y = SNoCut (Sep (SNoS_ ω) (λz : set ⇒ x * z < ordsucc Empty)) (Sep (SNoS_ ω) (λz : set ⇒ ordsucc Empty < x * z))
In Proofgold the corresponding term root is c9fe64... and proposition id is fa3ae3...
Proof:
Proof not loaded.
End of Section Conj_pos_small_real_recip_ex__12__0
Beginning of Section Conj_pos_small_real_recip_ex__12__9
L7288
Variable x : set
(*** Conj_pos_small_real_recip_ex__12__9 TMXF87PV4nDSoWX5w445ySm7xNHY8nkjPmQ bounty of about 25 bars ***)
L7289
Variable y : set
L7290
Hypothesis H0 : x ∈ real
L7291
Hypothesis H1 : x < ordsucc Empty
L7292
Hypothesis H2 : ¬ (∃z : set, z ∈ real ∧ x * z = ordsucc Empty)
L7293
Hypothesis H3 : SNo x
L7294
Hypothesis H4 : SNo (SNoCut (Sep (SNoS_ ω) (λz : set ⇒ x * z < ordsucc Empty)) (Sep (SNoS_ ω) (λz : set ⇒ ordsucc Empty < x * z)))
L7295
Hypothesis H5 : (∀z : set, z ∈ Sep (SNoS_ ω) (λw : set ⇒ x * w < ordsucc Empty) → z < SNoCut (Sep (SNoS_ ω) (λw : set ⇒ x * w < ordsucc Empty)) (Sep (SNoS_ ω) (λw : set ⇒ ordsucc Empty < x * w)))
L7296
Hypothesis H6 : (∀z : set, z ∈ Sep (SNoS_ ω) (λw : set ⇒ ordsucc Empty < x * w) → SNoCut (Sep (SNoS_ ω) (λw : set ⇒ x * w < ordsucc Empty)) (Sep (SNoS_ ω) (λw : set ⇒ ordsucc Empty < x * w)) < z)
L7297
Hypothesis H7 : y ∈ SNoS_ ω
L7298
Hypothesis H8 : (∀z : set, z ∈ ω → abs_SNo (y + - (SNoCut (Sep (SNoS_ ω) (λw : set ⇒ x * w < ordsucc Empty)) (Sep (SNoS_ ω) (λw : set ⇒ ordsucc Empty < x * w)))) < eps_ z)
L7299
Theorem. (Conj_pos_small_real_recip_ex__12__9)
y ∈ real → y = SNoCut (Sep (SNoS_ ω) (λz : set ⇒ x * z < ordsucc Empty)) (Sep (SNoS_ ω) (λz : set ⇒ ordsucc Empty < x * z))
In Proofgold the corresponding term root is f3ed16... and proposition id is f64317...
Proof:
Proof not loaded.
End of Section Conj_pos_small_real_recip_ex__12__9
Beginning of Section Conj_pos_small_real_recip_ex__13__3
L7305
Variable x : set
(*** Conj_pos_small_real_recip_ex__13__3 TMSQsSq5kKeYas4A4JPvzs9EYjsG9MwRL84 bounty of about 25 bars ***)
L7306
Variable y : set
L7307
Hypothesis H0 : Empty < x
L7308
Hypothesis H1 : SNo x
L7309
Hypothesis H2 : SNo (SNoCut (Sep (SNoS_ ω) (λz : set ⇒ x * z < ordsucc Empty)) (Sep (SNoS_ ω) (λz : set ⇒ ordsucc Empty < x * z)))
L7310
Hypothesis H4 : y ∈ ω
L7311
Hypothesis H5 : eps_ y ≤ x
L7312
Hypothesis H6 : nat_p (exp_SNo_nat (ordsucc (ordsucc Empty)) y)
L7313
Hypothesis H7 : SNo (exp_SNo_nat (ordsucc (ordsucc Empty)) y)
L7314
Theorem. (Conj_pos_small_real_recip_ex__13__3)
exp_SNo_nat (ordsucc (ordsucc Empty)) y + ordsucc Empty ∈ ω → SNoCut (Sep (SNoS_ ω) (λz : set ⇒ x * z < ordsucc Empty)) (Sep (SNoS_ ω) (λz : set ⇒ ordsucc Empty < x * z)) < ω
In Proofgold the corresponding term root is 15471f... and proposition id is 96f39c...
Proof:
Proof not loaded.
End of Section Conj_pos_small_real_recip_ex__13__3
Beginning of Section Conj_pos_small_real_recip_ex__17__1
L7320
Variable x : set
(*** Conj_pos_small_real_recip_ex__17__1 TMMPDKY5xGhBSK9UB7QuJueypeCtTpDLhtB bounty of about 25 bars ***)
L7321
Variable y : set
L7322
Variable z : set
L7323
Hypothesis H0 : Empty < x
L7324
Hypothesis H2 : SNo x
L7325
Hypothesis H3 : (∀w : set, w ∈ Sep (SNoS_ ω) (λu : set ⇒ x * u < ordsucc Empty) → w < SNoCut (Sep (SNoS_ ω) (λu : set ⇒ x * u < ordsucc Empty)) (Sep (SNoS_ ω) (λu : set ⇒ ordsucc Empty < x * u)))
L7326
Hypothesis H4 : (∀w : set, w ∈ Sep (SNoS_ ω) (λu : set ⇒ ordsucc Empty < x * u) → SNoCut (Sep (SNoS_ ω) (λu : set ⇒ x * u < ordsucc Empty)) (Sep (SNoS_ ω) (λu : set ⇒ ordsucc Empty < x * u)) < w)
L7327
Hypothesis H5 : SNo (SNoCut (Sep (SNoS_ ω) (λw : set ⇒ x * w < ordsucc Empty)) (Sep (SNoS_ ω) (λw : set ⇒ ordsucc Empty < x * w)))
L7328
Hypothesis H6 : SNo (x * SNoCut (Sep (SNoS_ ω) (λw : set ⇒ x * w < ordsucc Empty)) (Sep (SNoS_ ω) (λw : set ⇒ ordsucc Empty < x * w)))
L7329
Hypothesis H7 : SNo (x * SNoCut (Sep (SNoS_ ω) (λw : set ⇒ x * w < ordsucc Empty)) (Sep (SNoS_ ω) (λw : set ⇒ ordsucc Empty < x * w)))
L7330
Hypothesis H8 : y ∈ ω
L7331
Hypothesis H9 : z ∈ SNoS_ ω
L7332
Hypothesis H10 : z < SNoCut (Sep (SNoS_ ω) (λw : set ⇒ x * w < ordsucc Empty)) (Sep (SNoS_ ω) (λw : set ⇒ ordsucc Empty < x * w))
L7333
Hypothesis H11 : SNoCut (Sep (SNoS_ ω) (λw : set ⇒ x * w < ordsucc Empty)) (Sep (SNoS_ ω) (λw : set ⇒ ordsucc Empty < x * w)) < z + eps_ (ordsucc y)
L7334
Hypothesis H12 : SNo z
L7335
Hypothesis H13 : SNo (x * z)
L7336
Hypothesis H14 : SNo (eps_ (ordsucc y))
L7337
Hypothesis H15 : SNo (x * eps_ (ordsucc y))
L7338
Hypothesis H16 : SNo (z + eps_ (ordsucc y))
L7339
Hypothesis H17 : SNo (ordsucc Empty + - (x * z))
L7340
Hypothesis H18 : SNo (x * z + - (x * SNoCut (Sep (SNoS_ ω) (λw : set ⇒ x * w < ordsucc Empty)) (Sep (SNoS_ ω) (λw : set ⇒ ordsucc Empty < x * w))))
L7341
Theorem. (Conj_pos_small_real_recip_ex__17__1)
x * (z + eps_ (ordsucc y)) < x * z + eps_ (ordsucc y) → abs_SNo (ordsucc Empty + - (x * SNoCut (Sep (SNoS_ ω) (λw : set ⇒ x * w < ordsucc Empty)) (Sep (SNoS_ ω) (λw : set ⇒ ordsucc Empty < x * w)))) < eps_ y
In Proofgold the corresponding term root is 12ef30... and proposition id is c9fe78...
Proof:
Proof not loaded.
End of Section Conj_pos_small_real_recip_ex__17__1
Beginning of Section Conj_pos_small_real_recip_ex__17__6
L7347
Variable x : set
(*** Conj_pos_small_real_recip_ex__17__6 TMXPsjo4XsgfoQhfLGUwyNDDTMxGWk6pU8Z bounty of about 25 bars ***)
L7348
Variable y : set
L7349
Variable z : set
L7350
Hypothesis H0 : Empty < x
L7351
Hypothesis H1 : x < ordsucc Empty
L7352
Hypothesis H2 : SNo x
L7353
Hypothesis H3 : (∀w : set, w ∈ Sep (SNoS_ ω) (λu : set ⇒ x * u < ordsucc Empty) → w < SNoCut (Sep (SNoS_ ω) (λu : set ⇒ x * u < ordsucc Empty)) (Sep (SNoS_ ω) (λu : set ⇒ ordsucc Empty < x * u)))
L7354
Hypothesis H4 : (∀w : set, w ∈ Sep (SNoS_ ω) (λu : set ⇒ ordsucc Empty < x * u) → SNoCut (Sep (SNoS_ ω) (λu : set ⇒ x * u < ordsucc Empty)) (Sep (SNoS_ ω) (λu : set ⇒ ordsucc Empty < x * u)) < w)
L7355
Hypothesis H5 : SNo (SNoCut (Sep (SNoS_ ω) (λw : set ⇒ x * w < ordsucc Empty)) (Sep (SNoS_ ω) (λw : set ⇒ ordsucc Empty < x * w)))
L7356
Hypothesis H7 : SNo (x * SNoCut (Sep (SNoS_ ω) (λw : set ⇒ x * w < ordsucc Empty)) (Sep (SNoS_ ω) (λw : set ⇒ ordsucc Empty < x * w)))
L7357
Hypothesis H8 : y ∈ ω
L7358
Hypothesis H9 : z ∈ SNoS_ ω
L7359
Hypothesis H10 : z < SNoCut (Sep (SNoS_ ω) (λw : set ⇒ x * w < ordsucc Empty)) (Sep (SNoS_ ω) (λw : set ⇒ ordsucc Empty < x * w))
L7360
Hypothesis H11 : SNoCut (Sep (SNoS_ ω) (λw : set ⇒ x * w < ordsucc Empty)) (Sep (SNoS_ ω) (λw : set ⇒ ordsucc Empty < x * w)) < z + eps_ (ordsucc y)
L7361
Hypothesis H12 : SNo z
L7362
Hypothesis H13 : SNo (x * z)
L7363
Hypothesis H14 : SNo (eps_ (ordsucc y))
L7364
Hypothesis H15 : SNo (x * eps_ (ordsucc y))
L7365
Hypothesis H16 : SNo (z + eps_ (ordsucc y))
L7366
Hypothesis H17 : SNo (ordsucc Empty + - (x * z))
L7367
Hypothesis H18 : SNo (x * z + - (x * SNoCut (Sep (SNoS_ ω) (λw : set ⇒ x * w < ordsucc Empty)) (Sep (SNoS_ ω) (λw : set ⇒ ordsucc Empty < x * w))))
L7368
Theorem. (Conj_pos_small_real_recip_ex__17__6)
x * (z + eps_ (ordsucc y)) < x * z + eps_ (ordsucc y) → abs_SNo (ordsucc Empty + - (x * SNoCut (Sep (SNoS_ ω) (λw : set ⇒ x * w < ordsucc Empty)) (Sep (SNoS_ ω) (λw : set ⇒ ordsucc Empty < x * w)))) < eps_ y
In Proofgold the corresponding term root is f4eaaa... and proposition id is b18ee8...
Proof:
Proof not loaded.
End of Section Conj_pos_small_real_recip_ex__17__6
Beginning of Section Conj_pos_small_real_recip_ex__17__7
L7374
Variable x : set
(*** Conj_pos_small_real_recip_ex__17__7 TMXPsjo4XsgfoQhfLGUwyNDDTMxGWk6pU8Z bounty of about 25 bars ***)
L7375
Variable y : set
L7376
Variable z : set
L7377
Hypothesis H0 : Empty < x
L7378
Hypothesis H1 : x < ordsucc Empty
L7379
Hypothesis H2 : SNo x
L7380
Hypothesis H3 : (∀w : set, w ∈ Sep (SNoS_ ω) (λu : set ⇒ x * u < ordsucc Empty) → w < SNoCut (Sep (SNoS_ ω) (λu : set ⇒ x * u < ordsucc Empty)) (Sep (SNoS_ ω) (λu : set ⇒ ordsucc Empty < x * u)))
L7381
Hypothesis H4 : (∀w : set, w ∈ Sep (SNoS_ ω) (λu : set ⇒ ordsucc Empty < x * u) → SNoCut (Sep (SNoS_ ω) (λu : set ⇒ x * u < ordsucc Empty)) (Sep (SNoS_ ω) (λu : set ⇒ ordsucc Empty < x * u)) < w)
L7382
Hypothesis H5 : SNo (SNoCut (Sep (SNoS_ ω) (λw : set ⇒ x * w < ordsucc Empty)) (Sep (SNoS_ ω) (λw : set ⇒ ordsucc Empty < x * w)))
L7383
Hypothesis H6 : SNo (x * SNoCut (Sep (SNoS_ ω) (λw : set ⇒ x * w < ordsucc Empty)) (Sep (SNoS_ ω) (λw : set ⇒ ordsucc Empty < x * w)))
L7384
Hypothesis H8 : y ∈ ω
L7385
Hypothesis H9 : z ∈ SNoS_ ω
L7386
Hypothesis H10 : z < SNoCut (Sep (SNoS_ ω) (λw : set ⇒ x * w < ordsucc Empty)) (Sep (SNoS_ ω) (λw : set ⇒ ordsucc Empty < x * w))
L7387
Hypothesis H11 : SNoCut (Sep (SNoS_ ω) (λw : set ⇒ x * w < ordsucc Empty)) (Sep (SNoS_ ω) (λw : set ⇒ ordsucc Empty < x * w)) < z + eps_ (ordsucc y)
L7388
Hypothesis H12 : SNo z
L7389
Hypothesis H13 : SNo (x * z)
L7390
Hypothesis H14 : SNo (eps_ (ordsucc y))
L7391
Hypothesis H15 : SNo (x * eps_ (ordsucc y))
L7392
Hypothesis H16 : SNo (z + eps_ (ordsucc y))
L7393
Hypothesis H17 : SNo (ordsucc Empty + - (x * z))
L7394
Hypothesis H18 : SNo (x * z + - (x * SNoCut (Sep (SNoS_ ω) (λw : set ⇒ x * w < ordsucc Empty)) (Sep (SNoS_ ω) (λw : set ⇒ ordsucc Empty < x * w))))
L7395
Theorem. (Conj_pos_small_real_recip_ex__17__7)
x * (z + eps_ (ordsucc y)) < x * z + eps_ (ordsucc y) → abs_SNo (ordsucc Empty + - (x * SNoCut (Sep (SNoS_ ω) (λw : set ⇒ x * w < ordsucc Empty)) (Sep (SNoS_ ω) (λw : set ⇒ ordsucc Empty < x * w)))) < eps_ y
In Proofgold the corresponding term root is f4eaaa... and proposition id is b18ee8...
Proof:
Proof not loaded.
End of Section Conj_pos_small_real_recip_ex__17__7
Beginning of Section Conj_pos_small_real_recip_ex__18__8
L7401
Variable x : set
(*** Conj_pos_small_real_recip_ex__18__8 TMNhPY5S2ATU9ibFKSrWbMMCjJU9SLEJ3DT bounty of about 25 bars ***)
L7402
Variable y : set
L7403
Variable z : set
L7404
Hypothesis H0 : Empty < x
L7405
Hypothesis H1 : x < ordsucc Empty
L7406
Hypothesis H2 : SNo x
L7407
Hypothesis H3 : (∀w : set, w ∈ Sep (SNoS_ ω) (λu : set ⇒ x * u < ordsucc Empty) → w < SNoCut (Sep (SNoS_ ω) (λu : set ⇒ x * u < ordsucc Empty)) (Sep (SNoS_ ω) (λu : set ⇒ ordsucc Empty < x * u)))
L7408
Hypothesis H4 : (∀w : set, w ∈ Sep (SNoS_ ω) (λu : set ⇒ ordsucc Empty < x * u) → SNoCut (Sep (SNoS_ ω) (λu : set ⇒ x * u < ordsucc Empty)) (Sep (SNoS_ ω) (λu : set ⇒ ordsucc Empty < x * u)) < w)
L7409
Hypothesis H5 : SNo (SNoCut (Sep (SNoS_ ω) (λw : set ⇒ x * w < ordsucc Empty)) (Sep (SNoS_ ω) (λw : set ⇒ ordsucc Empty < x * w)))
L7410
Hypothesis H6 : SNo (x * SNoCut (Sep (SNoS_ ω) (λw : set ⇒ x * w < ordsucc Empty)) (Sep (SNoS_ ω) (λw : set ⇒ ordsucc Empty < x * w)))
L7411
Hypothesis H7 : SNo (x * SNoCut (Sep (SNoS_ ω) (λw : set ⇒ x * w < ordsucc Empty)) (Sep (SNoS_ ω) (λw : set ⇒ ordsucc Empty < x * w)))
L7412
Hypothesis H9 : z ∈ SNoS_ ω
L7413
Hypothesis H10 : z < SNoCut (Sep (SNoS_ ω) (λw : set ⇒ x * w < ordsucc Empty)) (Sep (SNoS_ ω) (λw : set ⇒ ordsucc Empty < x * w))
L7414
Hypothesis H11 : SNoCut (Sep (SNoS_ ω) (λw : set ⇒ x * w < ordsucc Empty)) (Sep (SNoS_ ω) (λw : set ⇒ ordsucc Empty < x * w)) < z + eps_ (ordsucc y)
L7415
Hypothesis H12 : SNo z
L7416
Hypothesis H13 : SNo (x * z)
L7417
Hypothesis H14 : SNo (eps_ (ordsucc y))
L7418
Hypothesis H15 : SNo (x * eps_ (ordsucc y))
L7419
Hypothesis H16 : SNo (z + eps_ (ordsucc y))
L7420
Hypothesis H17 : SNo (ordsucc Empty + - (x * z))
L7421
Theorem. (Conj_pos_small_real_recip_ex__18__8)
SNo (x * z + - (x * SNoCut (Sep (SNoS_ ω) (λw : set ⇒ x * w < ordsucc Empty)) (Sep (SNoS_ ω) (λw : set ⇒ ordsucc Empty < x * w)))) → abs_SNo (ordsucc Empty + - (x * SNoCut (Sep (SNoS_ ω) (λw : set ⇒ x * w < ordsucc Empty)) (Sep (SNoS_ ω) (λw : set ⇒ ordsucc Empty < x * w)))) < eps_ y
In Proofgold the corresponding term root is 62427b... and proposition id is 864c57...
Proof:
Proof not loaded.
End of Section Conj_pos_small_real_recip_ex__18__8
Beginning of Section Conj_pos_small_real_recip_ex__19__2
L7427
Variable x : set
(*** Conj_pos_small_real_recip_ex__19__2 TMU9digWpqJEjvHUH9hcZETHWnhbveFMXYv bounty of about 25 bars ***)
L7428
Variable y : set
L7429
Variable z : set
L7430
Hypothesis H0 : Empty < x
L7431
Hypothesis H1 : x < ordsucc Empty
L7432
Hypothesis H3 : (∀w : set, w ∈ Sep (SNoS_ ω) (λu : set ⇒ x * u < ordsucc Empty) → w < SNoCut (Sep (SNoS_ ω) (λu : set ⇒ x * u < ordsucc Empty)) (Sep (SNoS_ ω) (λu : set ⇒ ordsucc Empty < x * u)))
L7433
Hypothesis H4 : (∀w : set, w ∈ Sep (SNoS_ ω) (λu : set ⇒ ordsucc Empty < x * u) → SNoCut (Sep (SNoS_ ω) (λu : set ⇒ x * u < ordsucc Empty)) (Sep (SNoS_ ω) (λu : set ⇒ ordsucc Empty < x * u)) < w)
L7434
Hypothesis H5 : SNo (SNoCut (Sep (SNoS_ ω) (λw : set ⇒ x * w < ordsucc Empty)) (Sep (SNoS_ ω) (λw : set ⇒ ordsucc Empty < x * w)))
L7435
Hypothesis H6 : SNo (x * SNoCut (Sep (SNoS_ ω) (λw : set ⇒ x * w < ordsucc Empty)) (Sep (SNoS_ ω) (λw : set ⇒ ordsucc Empty < x * w)))
L7436
Hypothesis H7 : SNo (x * SNoCut (Sep (SNoS_ ω) (λw : set ⇒ x * w < ordsucc Empty)) (Sep (SNoS_ ω) (λw : set ⇒ ordsucc Empty < x * w)))
L7437
Hypothesis H8 : y ∈ ω
L7438
Hypothesis H9 : z ∈ SNoS_ ω
L7439
Hypothesis H10 : z < SNoCut (Sep (SNoS_ ω) (λw : set ⇒ x * w < ordsucc Empty)) (Sep (SNoS_ ω) (λw : set ⇒ ordsucc Empty < x * w))
L7440
Hypothesis H11 : SNoCut (Sep (SNoS_ ω) (λw : set ⇒ x * w < ordsucc Empty)) (Sep (SNoS_ ω) (λw : set ⇒ ordsucc Empty < x * w)) < z + eps_ (ordsucc y)
L7441
Hypothesis H12 : SNo z
L7442
Hypothesis H13 : SNo (x * z)
L7443
Hypothesis H14 : SNo (eps_ (ordsucc y))
L7444
Hypothesis H15 : SNo (x * eps_ (ordsucc y))
L7445
Hypothesis H16 : SNo (z + eps_ (ordsucc y))
L7446
Theorem. (Conj_pos_small_real_recip_ex__19__2)
SNo (ordsucc Empty + - (x * z)) → abs_SNo (ordsucc Empty + - (x * SNoCut (Sep (SNoS_ ω) (λw : set ⇒ x * w < ordsucc Empty)) (Sep (SNoS_ ω) (λw : set ⇒ ordsucc Empty < x * w)))) < eps_ y
In Proofgold the corresponding term root is 729ea8... and proposition id is 35e126...
Proof:
Proof not loaded.
End of Section Conj_pos_small_real_recip_ex__19__2
Beginning of Section Conj_pos_small_real_recip_ex__23__0
L7452
Variable x : set
(*** Conj_pos_small_real_recip_ex__23__0 TMX7GHFNnh4sXd8mZC6EALQPWCbn5Sq8xHg bounty of about 25 bars ***)
L7453
Variable y : set
L7454
Variable z : set
L7455
Hypothesis H1 : x < ordsucc Empty
L7456
Hypothesis H2 : SNo x
L7457
Hypothesis H3 : (∀w : set, w ∈ Sep (SNoS_ ω) (λu : set ⇒ x * u < ordsucc Empty) → w < SNoCut (Sep (SNoS_ ω) (λu : set ⇒ x * u < ordsucc Empty)) (Sep (SNoS_ ω) (λu : set ⇒ ordsucc Empty < x * u)))
L7458
Hypothesis H4 : (∀w : set, w ∈ Sep (SNoS_ ω) (λu : set ⇒ ordsucc Empty < x * u) → SNoCut (Sep (SNoS_ ω) (λu : set ⇒ x * u < ordsucc Empty)) (Sep (SNoS_ ω) (λu : set ⇒ ordsucc Empty < x * u)) < w)
L7459
Hypothesis H5 : SNo (SNoCut (Sep (SNoS_ ω) (λw : set ⇒ x * w < ordsucc Empty)) (Sep (SNoS_ ω) (λw : set ⇒ ordsucc Empty < x * w)))
L7460
Hypothesis H6 : SNo (x * SNoCut (Sep (SNoS_ ω) (λw : set ⇒ x * w < ordsucc Empty)) (Sep (SNoS_ ω) (λw : set ⇒ ordsucc Empty < x * w)))
L7461
Hypothesis H7 : SNo (x * SNoCut (Sep (SNoS_ ω) (λw : set ⇒ x * w < ordsucc Empty)) (Sep (SNoS_ ω) (λw : set ⇒ ordsucc Empty < x * w)))
L7462
Hypothesis H8 : y ∈ ω
L7463
Hypothesis H9 : z ∈ SNoS_ ω
L7464
Hypothesis H10 : z < SNoCut (Sep (SNoS_ ω) (λw : set ⇒ x * w < ordsucc Empty)) (Sep (SNoS_ ω) (λw : set ⇒ ordsucc Empty < x * w))
L7465
Hypothesis H11 : SNoCut (Sep (SNoS_ ω) (λw : set ⇒ x * w < ordsucc Empty)) (Sep (SNoS_ ω) (λw : set ⇒ ordsucc Empty < x * w)) < z + eps_ (ordsucc y)
L7466
Hypothesis H12 : SNo z
L7467
Theorem. (Conj_pos_small_real_recip_ex__23__0)
SNo (x * z) → abs_SNo (ordsucc Empty + - (x * SNoCut (Sep (SNoS_ ω) (λw : set ⇒ x * w < ordsucc Empty)) (Sep (SNoS_ ω) (λw : set ⇒ ordsucc Empty < x * w)))) < eps_ y
In Proofgold the corresponding term root is 194138... and proposition id is b680d5...
Proof:
Proof not loaded.
End of Section Conj_pos_small_real_recip_ex__23__0
Beginning of Section Conj_pos_small_real_recip_ex__24__3
L7473
Variable x : set
(*** Conj_pos_small_real_recip_ex__24__3 TMZHcw7XtrAH23Q4QiyN9KZuUR2MDo2wHtp bounty of about 25 bars ***)
L7474
Hypothesis H0 : Empty < x
L7475
Hypothesis H1 : x < ordsucc Empty
L7476
Hypothesis H2 : ¬ (∃y : set, y ∈ real ∧ x * y = ordsucc Empty)
L7477
Hypothesis H4 : (∀y : set, y ∈ Sep (SNoS_ ω) (λz : set ⇒ x * z < ordsucc Empty) → y < SNoCut (Sep (SNoS_ ω) (λz : set ⇒ x * z < ordsucc Empty)) (Sep (SNoS_ ω) (λz : set ⇒ ordsucc Empty < x * z)))
L7478
Hypothesis H5 : (∀y : set, y ∈ Sep (SNoS_ ω) (λz : set ⇒ ordsucc Empty < x * z) → SNoCut (Sep (SNoS_ ω) (λz : set ⇒ x * z < ordsucc Empty)) (Sep (SNoS_ ω) (λz : set ⇒ ordsucc Empty < x * z)) < y)
L7479
Hypothesis H6 : SNoCut (Sep (SNoS_ ω) (λy : set ⇒ x * y < ordsucc Empty)) (Sep (SNoS_ ω) (λy : set ⇒ ordsucc Empty < x * y)) ∈ real
L7480
Hypothesis H7 : SNo (SNoCut (Sep (SNoS_ ω) (λy : set ⇒ x * y < ordsucc Empty)) (Sep (SNoS_ ω) (λy : set ⇒ ordsucc Empty < x * y)))
L7481
Hypothesis H8 : (∀y : set, y ∈ ω → (∃z : set, z ∈ SNoS_ ω ∧ (z < SNoCut (Sep (SNoS_ ω) (λw : set ⇒ x * w < ordsucc Empty)) (Sep (SNoS_ ω) (λw : set ⇒ ordsucc Empty < x * w)) ∧ SNoCut (Sep (SNoS_ ω) (λw : set ⇒ x * w < ordsucc Empty)) (Sep (SNoS_ ω) (λw : set ⇒ ordsucc Empty < x * w)) < z + eps_ y)))
L7482
Hypothesis H9 : x * SNoCut (Sep (SNoS_ ω) (λy : set ⇒ x * y < ordsucc Empty)) (Sep (SNoS_ ω) (λy : set ⇒ ordsucc Empty < x * y)) ∈ real
L7483
Theorem. (Conj_pos_small_real_recip_ex__24__3)
¬ SNo (x * SNoCut (Sep (SNoS_ ω) (λy : set ⇒ x * y < ordsucc Empty)) (Sep (SNoS_ ω) (λy : set ⇒ ordsucc Empty < x * y)))
In Proofgold the corresponding term root is a9c87b... and proposition id is 5a94b2...
Proof:
Proof not loaded.
End of Section Conj_pos_small_real_recip_ex__24__3
Beginning of Section Conj_pos_small_real_recip_ex__24__8
L7489
Variable x : set
(*** Conj_pos_small_real_recip_ex__24__8 TMPWakYUV4PUbKqaTyBdMCjwNLuAF2WEcQh bounty of about 25 bars ***)
L7490
Hypothesis H0 : Empty < x
L7491
Hypothesis H1 : x < ordsucc Empty
L7492
Hypothesis H2 : ¬ (∃y : set, y ∈ real ∧ x * y = ordsucc Empty)
L7493
Hypothesis H3 : SNo x
L7494
Hypothesis H4 : (∀y : set, y ∈ Sep (SNoS_ ω) (λz : set ⇒ x * z < ordsucc Empty) → y < SNoCut (Sep (SNoS_ ω) (λz : set ⇒ x * z < ordsucc Empty)) (Sep (SNoS_ ω) (λz : set ⇒ ordsucc Empty < x * z)))
L7495
Hypothesis H5 : (∀y : set, y ∈ Sep (SNoS_ ω) (λz : set ⇒ ordsucc Empty < x * z) → SNoCut (Sep (SNoS_ ω) (λz : set ⇒ x * z < ordsucc Empty)) (Sep (SNoS_ ω) (λz : set ⇒ ordsucc Empty < x * z)) < y)
L7496
Hypothesis H6 : SNoCut (Sep (SNoS_ ω) (λy : set ⇒ x * y < ordsucc Empty)) (Sep (SNoS_ ω) (λy : set ⇒ ordsucc Empty < x * y)) ∈ real
L7497
Hypothesis H7 : SNo (SNoCut (Sep (SNoS_ ω) (λy : set ⇒ x * y < ordsucc Empty)) (Sep (SNoS_ ω) (λy : set ⇒ ordsucc Empty < x * y)))
L7498
Hypothesis H9 : x * SNoCut (Sep (SNoS_ ω) (λy : set ⇒ x * y < ordsucc Empty)) (Sep (SNoS_ ω) (λy : set ⇒ ordsucc Empty < x * y)) ∈ real
L7499
Theorem. (Conj_pos_small_real_recip_ex__24__8)
¬ SNo (x * SNoCut (Sep (SNoS_ ω) (λy : set ⇒ x * y < ordsucc Empty)) (Sep (SNoS_ ω) (λy : set ⇒ ordsucc Empty < x * y)))
In Proofgold the corresponding term root is 83c3cf... and proposition id is d37e25...
Proof:
Proof not loaded.
End of Section Conj_pos_small_real_recip_ex__24__8
Beginning of Section Conj_pos_small_real_recip_ex__24__9
L7505
Variable x : set
(*** Conj_pos_small_real_recip_ex__24__9 TMX5TjYbGMH25Z4fZpjthbpdd6dwTmDfzFT bounty of about 25 bars ***)
L7506
Hypothesis H0 : Empty < x
L7507
Hypothesis H1 : x < ordsucc Empty
L7508
Hypothesis H2 : ¬ (∃y : set, y ∈ real ∧ x * y = ordsucc Empty)
L7509
Hypothesis H3 : SNo x
L7510
Hypothesis H4 : (∀y : set, y ∈ Sep (SNoS_ ω) (λz : set ⇒ x * z < ordsucc Empty) → y < SNoCut (Sep (SNoS_ ω) (λz : set ⇒ x * z < ordsucc Empty)) (Sep (SNoS_ ω) (λz : set ⇒ ordsucc Empty < x * z)))
L7511
Hypothesis H5 : (∀y : set, y ∈ Sep (SNoS_ ω) (λz : set ⇒ ordsucc Empty < x * z) → SNoCut (Sep (SNoS_ ω) (λz : set ⇒ x * z < ordsucc Empty)) (Sep (SNoS_ ω) (λz : set ⇒ ordsucc Empty < x * z)) < y)
L7512
Hypothesis H6 : SNoCut (Sep (SNoS_ ω) (λy : set ⇒ x * y < ordsucc Empty)) (Sep (SNoS_ ω) (λy : set ⇒ ordsucc Empty < x * y)) ∈ real
L7513
Hypothesis H7 : SNo (SNoCut (Sep (SNoS_ ω) (λy : set ⇒ x * y < ordsucc Empty)) (Sep (SNoS_ ω) (λy : set ⇒ ordsucc Empty < x * y)))
L7514
Hypothesis H8 : (∀y : set, y ∈ ω → (∃z : set, z ∈ SNoS_ ω ∧ (z < SNoCut (Sep (SNoS_ ω) (λw : set ⇒ x * w < ordsucc Empty)) (Sep (SNoS_ ω) (λw : set ⇒ ordsucc Empty < x * w)) ∧ SNoCut (Sep (SNoS_ ω) (λw : set ⇒ x * w < ordsucc Empty)) (Sep (SNoS_ ω) (λw : set ⇒ ordsucc Empty < x * w)) < z + eps_ y)))
L7515
Theorem. (Conj_pos_small_real_recip_ex__24__9)
¬ SNo (x * SNoCut (Sep (SNoS_ ω) (λy : set ⇒ x * y < ordsucc Empty)) (Sep (SNoS_ ω) (λy : set ⇒ ordsucc Empty < x * y)))
In Proofgold the corresponding term root is 50ed02... and proposition id is 1e8f9e...
Proof:
Proof not loaded.
End of Section Conj_pos_small_real_recip_ex__24__9
Beginning of Section Conj_pos_small_real_recip_ex__26__3
L7521
Variable x : set
(*** Conj_pos_small_real_recip_ex__26__3 TMZAVZB6SwcbSPJRUG28LRHBFF6LQ1SwZz1 bounty of about 25 bars ***)
L7522
Hypothesis H0 : x ∈ real
L7523
Hypothesis H1 : Empty < x
L7524
Hypothesis H2 : x < ordsucc Empty
L7525
Hypothesis H4 : SNo x
L7526
Hypothesis H5 : (∀y : set, y ∈ SNoS_ ω → (∀z : set, z ∈ ω → abs_SNo (y + - x) < eps_ z) → y = x)
L7527
Hypothesis H6 : SNo (SNoCut (Sep (SNoS_ ω) (λy : set ⇒ x * y < ordsucc Empty)) (Sep (SNoS_ ω) (λy : set ⇒ ordsucc Empty < x * y)))
L7528
Hypothesis H7 : (∀y : set, y ∈ Sep (SNoS_ ω) (λz : set ⇒ x * z < ordsucc Empty) → y < SNoCut (Sep (SNoS_ ω) (λz : set ⇒ x * z < ordsucc Empty)) (Sep (SNoS_ ω) (λz : set ⇒ ordsucc Empty < x * z)))
L7529
Hypothesis H8 : (∀y : set, y ∈ Sep (SNoS_ ω) (λz : set ⇒ ordsucc Empty < x * z) → SNoCut (Sep (SNoS_ ω) (λz : set ⇒ x * z < ordsucc Empty)) (Sep (SNoS_ ω) (λz : set ⇒ ordsucc Empty < x * z)) < y)
L7530
Hypothesis H9 : SNoCut (Sep (SNoS_ ω) (λy : set ⇒ x * y < ordsucc Empty)) (Sep (SNoS_ ω) (λy : set ⇒ ordsucc Empty < x * y)) ∈ SNoS_ (ordsucc ω)
L7531
Theorem. (Conj_pos_small_real_recip_ex__26__3)
¬ SNoCut (Sep (SNoS_ ω) (λy : set ⇒ x * y < ordsucc Empty)) (Sep (SNoS_ ω) (λy : set ⇒ ordsucc Empty < x * y)) ∈ real
In Proofgold the corresponding term root is 4f73d4... and proposition id is 15b947...
Proof:
Proof not loaded.
End of Section Conj_pos_small_real_recip_ex__26__3
Beginning of Section Conj_pos_small_real_recip_ex__28__6
L7537
Variable x : set
(*** Conj_pos_small_real_recip_ex__28__6 TMWkBcL9rtKJG7Skoz4xgqiWb2fQLfW49ez bounty of about 25 bars ***)
L7538
Hypothesis H0 : x ∈ real
L7539
Hypothesis H1 : Empty < x
L7540
Hypothesis H2 : x < ordsucc Empty
L7541
Hypothesis H3 : ¬ (∃y : set, y ∈ real ∧ x * y = ordsucc Empty)
L7542
Hypothesis H4 : SNo x
L7543
Hypothesis H5 : (∀y : set, y ∈ SNoS_ ω → (∀z : set, z ∈ ω → abs_SNo (y + - x) < eps_ z) → y = x)
L7544
Hypothesis H7 : (∀y : set, y ∈ Sep (SNoS_ ω) (λz : set ⇒ ordsucc Empty < x * z) → (∀P : prop, (SNo y → SNoLev y ∈ ω → ordsucc Empty < x * y → P) → P))
L7545
Theorem. (Conj_pos_small_real_recip_ex__28__6)
¬ SNoCutP (Sep (SNoS_ ω) (λy : set ⇒ x * y < ordsucc Empty)) (Sep (SNoS_ ω) (λy : set ⇒ ordsucc Empty < x * y))
In Proofgold the corresponding term root is a629d1... and proposition id is 522a26...
Proof:
Proof not loaded.
End of Section Conj_pos_small_real_recip_ex__28__6
Beginning of Section Conj_pos_real_recip_ex__1__3
L7551
Variable x : set
(*** Conj_pos_real_recip_ex__1__3 TMYxfzWP3TyvFGP5mgke3usKuW1W2dYceXn bounty of about 25 bars ***)
L7552
Variable y : set
L7553
Variable z : set
L7554
Hypothesis H0 : SNo x
L7555
Hypothesis H1 : y ∈ ω
L7556
Hypothesis H2 : z ∈ real
L7557
Theorem. (Conj_pos_real_recip_ex__1__3)
SNo (eps_ y) → (∃w : set, w ∈ real ∧ x * w = ordsucc Empty)
In Proofgold the corresponding term root is a42ff1... and proposition id is 3c9d21...
Proof:
Proof not loaded.
End of Section Conj_pos_real_recip_ex__1__3
Beginning of Section Conj_pos_real_recip_ex__2__4
L7563
Variable x : set
(*** Conj_pos_real_recip_ex__2__4 TMHduHoFc1HwuDUrnuvnEh1t2XvbuPBKnfg bounty of about 25 bars ***)
L7564
Variable y : set
L7565
Hypothesis H0 : x ∈ real
L7566
Hypothesis H1 : Empty < x
L7567
Hypothesis H2 : SNo x
L7568
Hypothesis H3 : y ∈ ω
L7569
Hypothesis H5 : eps_ y ∈ real
L7570
Theorem. (Conj_pos_real_recip_ex__2__4)
Empty < eps_ y * x → (∃z : set, z ∈ real ∧ x * z = ordsucc Empty)
In Proofgold the corresponding term root is fdf778... and proposition id is f4ed2c...
Proof:
Proof not loaded.
End of Section Conj_pos_real_recip_ex__2__4
Beginning of Section Conj_real_Archimedean__2__3
L7576
Variable x : set
(*** Conj_real_Archimedean__2__3 TMZ22KSEEWvE1jsZxsfmtuBURBgnGUx8TPh bounty of about 25 bars ***)
L7577
Variable y : set
L7578
Variable z : set
L7579
Hypothesis H0 : SNo x
L7580
Hypothesis H1 : SNo (exp_SNo_nat (ordsucc (ordsucc Empty)) y)
L7581
Hypothesis H2 : ordsucc Empty ≤ exp_SNo_nat (ordsucc (ordsucc Empty)) y * x
L7582
Hypothesis H4 : SNo z
L7583
Theorem. (Conj_real_Archimedean__2__3)
Empty ≤ z → z * ordsucc Empty ≤ z * exp_SNo_nat (ordsucc (ordsucc Empty)) y * x
In Proofgold the corresponding term root is c68794... and proposition id is 93b87e...
Proof:
Proof not loaded.
End of Section Conj_real_Archimedean__2__3
Beginning of Section Conj_real_Archimedean__8__2
L7589
Variable x : set
(*** Conj_real_Archimedean__8__2 TMHtre3BvKpSrkGACbgJDx89321ys2Q2ymN bounty of about 25 bars ***)
L7590
Variable y : set
L7591
Variable z : set
L7592
Hypothesis H0 : y ∈ real
L7593
Hypothesis H1 : SNo x
L7594
Hypothesis H3 : eps_ z ≤ x
L7595
Theorem. (Conj_real_Archimedean__8__2)
nat_p (exp_SNo_nat (ordsucc (ordsucc Empty)) z) → (∃w : set, w ∈ ω ∧ y ≤ w * x)
In Proofgold the corresponding term root is 60691e... and proposition id is 4b8b6b...
Proof:
Proof not loaded.
End of Section Conj_real_Archimedean__8__2
Beginning of Section Conj_real_complete1__3__1
L7601
Variable x : set
(*** Conj_real_complete1__3__1 TMHYkQPEwgBGgMwub78UEBPqwSGWyuy8qK6 bounty of about 25 bars ***)
L7602
Variable y : set
L7603
Variable z : set
L7604
Variable w : set
L7605
Hypothesis H0 : (∀u : set, u ∈ ω → ap x u ≤ ap y u ∧ ap x u ≤ ap x (ordsucc u) ∧ ap y (ordsucc u) ≤ ap y u)
L7606
Hypothesis H2 : (∀u : set, u ∈ z → ap x u ≤ ap x z)
L7607
Hypothesis H3 : w ∈ ordsucc z
L7608
Hypothesis H4 : z ∈ ω
L7609
Theorem. (Conj_real_complete1__3__1)
ordsucc z ∈ ω → ap x w ≤ ap x (ordsucc z)
In Proofgold the corresponding term root is 360d3d... and proposition id is 721433...
Proof:
Proof not loaded.
End of Section Conj_real_complete1__3__1
Beginning of Section Conj_real_complete1__3__3
L7615
Variable x : set
(*** Conj_real_complete1__3__3 TMTP39b7t9hBLGCVUgKEhEwYFGg9Ugy4vfF bounty of about 25 bars ***)
L7616
Variable y : set
L7617
Variable z : set
L7618
Variable w : set
L7619
Hypothesis H0 : (∀u : set, u ∈ ω → ap x u ≤ ap y u ∧ ap x u ≤ ap x (ordsucc u) ∧ ap y (ordsucc u) ≤ ap y u)
L7620
Hypothesis H1 : (∀u : set, u ∈ ω → SNo (ap x u))
L7621
Hypothesis H2 : (∀u : set, u ∈ z → ap x u ≤ ap x z)
L7622
Hypothesis H4 : z ∈ ω
L7623
Theorem. (Conj_real_complete1__3__3)
ordsucc z ∈ ω → ap x w ≤ ap x (ordsucc z)
In Proofgold the corresponding term root is 21b361... and proposition id is ed23ad...
Proof:
Proof not loaded.
End of Section Conj_real_complete1__3__3
Beginning of Section Conj_real_complete1__4__3
L7629
Variable x : set
(*** Conj_real_complete1__4__3 TMTdUussKNkA6rwhnezaxmLyw9kt9yom4vH bounty of about 25 bars ***)
L7630
Variable y : set
L7631
Variable z : set
L7632
Variable w : set
L7633
Hypothesis H0 : (∀u : set, u ∈ ω → ap x u ≤ ap y u ∧ ap x u ≤ ap x (ordsucc u) ∧ ap y (ordsucc u) ≤ ap y u)
L7634
Hypothesis H1 : (∀u : set, u ∈ ω → SNo (ap x u))
L7635
Hypothesis H2 : nat_p z
L7636
Hypothesis H4 : w ∈ ordsucc z
L7637
Theorem. (Conj_real_complete1__4__3)
z ∈ ω → ap x w ≤ ap x (ordsucc z)
In Proofgold the corresponding term root is 7d3135... and proposition id is 1f897a...
Proof:
Proof not loaded.
End of Section Conj_real_complete1__4__3
Beginning of Section Conj_real_complete1__9__1
L7643
Variable x : set
(*** Conj_real_complete1__9__1 TMLsAQBkCCJTxwoGjtCzWefeKmuV12qcAYs bounty of about 25 bars ***)
L7644
Variable y : set
L7645
Variable z : set
L7646
Variable w : set
L7647
Hypothesis H0 : (∀u : set, u ∈ ω → ap x u ≤ ap y u ∧ ap x u ≤ ap x (ordsucc u) ∧ ap y (ordsucc u) ≤ ap y u)
L7648
Hypothesis H2 : (∀u : set, u ∈ ω → SNo (ap y u))
L7649
Hypothesis H3 : (∀u : set, nat_p u → (∀v : set, v ∈ u → ap x v ≤ ap x u))
L7650
Hypothesis H4 : (∀u : set, nat_p u → (∀v : set, v ∈ u → ap y u ≤ ap y v))
L7651
Hypothesis H5 : z ∈ ω
L7652
Hypothesis H6 : w ∈ ω
L7653
Hypothesis H7 : nat_p z
L7654
Hypothesis H8 : nat_p w
L7655
Theorem. (Conj_real_complete1__9__1)
ap x z ≤ ap y z → ap x z ≤ ap y w
In Proofgold the corresponding term root is 998d24... and proposition id is a20b17...
Proof:
Proof not loaded.
End of Section Conj_real_complete1__9__1
Beginning of Section Conj_real_complete1__12__0
L7661
Variable x : set
(*** Conj_real_complete1__12__0 TMMSiP4yJQCTTuknYDALik3YjGSFMxFzFeR bounty of about 25 bars ***)
L7662
Variable y : set
L7663
Variable z : set
L7664
Hypothesis H1 : (∀w : set, w ∈ ω → SNo (ap y w))
L7665
Hypothesis H2 : SNo (SNoCut (Sep (SNoS_ ω) (λw : set ⇒ ∃u : set, u ∈ ω ∧ w < ap x u)) (Sep (SNoS_ ω) (λw : set ⇒ ∃u : set, u ∈ ω ∧ ap y u < w)))
L7666
Hypothesis H3 : (∀w : set, SNo w → (∀u : set, u ∈ Sep (SNoS_ ω) (λv : set ⇒ ∃x2 : set, x2 ∈ ω ∧ v < ap x x2) → u < w) → (∀u : set, u ∈ Sep (SNoS_ ω) (λv : set ⇒ ∃x2 : set, x2 ∈ ω ∧ ap y x2 < v) → w < u) → Subq (SNoLev (SNoCut (Sep (SNoS_ ω) (λu : set ⇒ ∃v : set, v ∈ ω ∧ u < ap x v)) (Sep (SNoS_ ω) (λu : set ⇒ ∃v : set, v ∈ ω ∧ ap y v < u)))) (SNoLev w) ∧ SNoEq_ (SNoLev (SNoCut (Sep (SNoS_ ω) (λu : set ⇒ ∃v : set, v ∈ ω ∧ u < ap x v)) (Sep (SNoS_ ω) (λu : set ⇒ ∃v : set, v ∈ ω ∧ ap y v < u)))) (SNoCut (Sep (SNoS_ ω) (λu : set ⇒ ∃v : set, v ∈ ω ∧ u < ap x v)) (Sep (SNoS_ ω) (λu : set ⇒ ∃v : set, v ∈ ω ∧ ap y v < u))) w)
L7667
Hypothesis H4 : (∀w : set, w ∈ ω → ap x w ≤ SNoCut (Sep (SNoS_ ω) (λu : set ⇒ ∃v : set, v ∈ ω ∧ u < ap x v)) (Sep (SNoS_ ω) (λu : set ⇒ ∃v : set, v ∈ ω ∧ ap y v < u)) ∧ SNoCut (Sep (SNoS_ ω) (λu : set ⇒ ∃v : set, v ∈ ω ∧ u < ap x v)) (Sep (SNoS_ ω) (λu : set ⇒ ∃v : set, v ∈ ω ∧ ap y v < u)) ≤ ap y w)
L7668
Hypothesis H5 : z ∈ SNoS_ ω
L7669
Hypothesis H6 : ¬ (z ∈ Sep (SNoS_ ω) (λw : set ⇒ ∃u : set, u ∈ ω ∧ w < ap x u) ∨ z ∈ Sep (SNoS_ ω) (λw : set ⇒ ∃u : set, u ∈ ω ∧ ap y u < w))
L7670
Hypothesis H7 : SNoLev z ∈ ω
L7671
Hypothesis H8 : ordinal (SNoLev z)
L7672
Hypothesis H9 : SNo z
L7673
Hypothesis H10 : (∀w : set, w ∈ Sep (SNoS_ ω) (λu : set ⇒ ∃v : set, v ∈ ω ∧ u < ap x v) → w < z)
L7674
Theorem. (Conj_real_complete1__12__0)
¬ (∀w : set, w ∈ Sep (SNoS_ ω) (λu : set ⇒ ∃v : set, v ∈ ω ∧ ap y v < u) → z < w)
In Proofgold the corresponding term root is 84da14... and proposition id is 93e190...
Proof:
Proof not loaded.
End of Section Conj_real_complete1__12__0
Beginning of Section Conj_real_complete1__12__2
L7680
Variable x : set
(*** Conj_real_complete1__12__2 TMQDih3V9NmyYNj6zh1PL2niw62w8n1C2Cs bounty of about 25 bars ***)
L7681
Variable y : set
L7682
Variable z : set
L7683
Hypothesis H0 : ¬ (∃w : set, w ∈ real ∧ (∀u : set, u ∈ ω → ap x u ≤ w ∧ w ≤ ap y u))
L7684
Hypothesis H1 : (∀w : set, w ∈ ω → SNo (ap y w))
L7685
Hypothesis H3 : (∀w : set, SNo w → (∀u : set, u ∈ Sep (SNoS_ ω) (λv : set ⇒ ∃x2 : set, x2 ∈ ω ∧ v < ap x x2) → u < w) → (∀u : set, u ∈ Sep (SNoS_ ω) (λv : set ⇒ ∃x2 : set, x2 ∈ ω ∧ ap y x2 < v) → w < u) → Subq (SNoLev (SNoCut (Sep (SNoS_ ω) (λu : set ⇒ ∃v : set, v ∈ ω ∧ u < ap x v)) (Sep (SNoS_ ω) (λu : set ⇒ ∃v : set, v ∈ ω ∧ ap y v < u)))) (SNoLev w) ∧ SNoEq_ (SNoLev (SNoCut (Sep (SNoS_ ω) (λu : set ⇒ ∃v : set, v ∈ ω ∧ u < ap x v)) (Sep (SNoS_ ω) (λu : set ⇒ ∃v : set, v ∈ ω ∧ ap y v < u)))) (SNoCut (Sep (SNoS_ ω) (λu : set ⇒ ∃v : set, v ∈ ω ∧ u < ap x v)) (Sep (SNoS_ ω) (λu : set ⇒ ∃v : set, v ∈ ω ∧ ap y v < u))) w)
L7686
Hypothesis H4 : (∀w : set, w ∈ ω → ap x w ≤ SNoCut (Sep (SNoS_ ω) (λu : set ⇒ ∃v : set, v ∈ ω ∧ u < ap x v)) (Sep (SNoS_ ω) (λu : set ⇒ ∃v : set, v ∈ ω ∧ ap y v < u)) ∧ SNoCut (Sep (SNoS_ ω) (λu : set ⇒ ∃v : set, v ∈ ω ∧ u < ap x v)) (Sep (SNoS_ ω) (λu : set ⇒ ∃v : set, v ∈ ω ∧ ap y v < u)) ≤ ap y w)
L7687
Hypothesis H5 : z ∈ SNoS_ ω
L7688
Hypothesis H6 : ¬ (z ∈ Sep (SNoS_ ω) (λw : set ⇒ ∃u : set, u ∈ ω ∧ w < ap x u) ∨ z ∈ Sep (SNoS_ ω) (λw : set ⇒ ∃u : set, u ∈ ω ∧ ap y u < w))
L7689
Hypothesis H7 : SNoLev z ∈ ω
L7690
Hypothesis H8 : ordinal (SNoLev z)
L7691
Hypothesis H9 : SNo z
L7692
Hypothesis H10 : (∀w : set, w ∈ Sep (SNoS_ ω) (λu : set ⇒ ∃v : set, v ∈ ω ∧ u < ap x v) → w < z)
L7693
Theorem. (Conj_real_complete1__12__2)
¬ (∀w : set, w ∈ Sep (SNoS_ ω) (λu : set ⇒ ∃v : set, v ∈ ω ∧ ap y v < u) → z < w)
In Proofgold the corresponding term root is d14294... and proposition id is 9c678a...
Proof:
Proof not loaded.
End of Section Conj_real_complete1__12__2
Beginning of Section Conj_real_complete1__13__7
L7699
Variable x : set
(*** Conj_real_complete1__13__7 TMSzrrfisUb6ZZ9opJymXNtpxQ7bC7h1RmT bounty of about 25 bars ***)
L7700
Variable y : set
L7701
Variable z : set
L7702
Hypothesis H0 : ¬ (∃w : set, w ∈ real ∧ (∀u : set, u ∈ ω → ap x u ≤ w ∧ w ≤ ap y u))
L7703
Hypothesis H1 : (∀w : set, w ∈ ω → SNo (ap x w))
L7704
Hypothesis H2 : (∀w : set, w ∈ ω → SNo (ap y w))
L7705
Hypothesis H3 : SNo (SNoCut (Sep (SNoS_ ω) (λw : set ⇒ ∃u : set, u ∈ ω ∧ w < ap x u)) (Sep (SNoS_ ω) (λw : set ⇒ ∃u : set, u ∈ ω ∧ ap y u < w)))
L7706
Hypothesis H4 : (∀w : set, SNo w → (∀u : set, u ∈ Sep (SNoS_ ω) (λv : set ⇒ ∃x2 : set, x2 ∈ ω ∧ v < ap x x2) → u < w) → (∀u : set, u ∈ Sep (SNoS_ ω) (λv : set ⇒ ∃x2 : set, x2 ∈ ω ∧ ap y x2 < v) → w < u) → Subq (SNoLev (SNoCut (Sep (SNoS_ ω) (λu : set ⇒ ∃v : set, v ∈ ω ∧ u < ap x v)) (Sep (SNoS_ ω) (λu : set ⇒ ∃v : set, v ∈ ω ∧ ap y v < u)))) (SNoLev w) ∧ SNoEq_ (SNoLev (SNoCut (Sep (SNoS_ ω) (λu : set ⇒ ∃v : set, v ∈ ω ∧ u < ap x v)) (Sep (SNoS_ ω) (λu : set ⇒ ∃v : set, v ∈ ω ∧ ap y v < u)))) (SNoCut (Sep (SNoS_ ω) (λu : set ⇒ ∃v : set, v ∈ ω ∧ u < ap x v)) (Sep (SNoS_ ω) (λu : set ⇒ ∃v : set, v ∈ ω ∧ ap y v < u))) w)
L7707
Hypothesis H5 : (∀w : set, w ∈ ω → ap x w ≤ SNoCut (Sep (SNoS_ ω) (λu : set ⇒ ∃v : set, v ∈ ω ∧ u < ap x v)) (Sep (SNoS_ ω) (λu : set ⇒ ∃v : set, v ∈ ω ∧ ap y v < u)) ∧ SNoCut (Sep (SNoS_ ω) (λu : set ⇒ ∃v : set, v ∈ ω ∧ u < ap x v)) (Sep (SNoS_ ω) (λu : set ⇒ ∃v : set, v ∈ ω ∧ ap y v < u)) ≤ ap y w)
L7708
Hypothesis H6 : z ∈ SNoS_ ω
L7709
Hypothesis H8 : SNoLev z ∈ ω
L7710
Hypothesis H9 : ordinal (SNoLev z)
L7711
Hypothesis H10 : SNo z
L7712
Theorem. (Conj_real_complete1__13__7)
¬ (∀w : set, w ∈ Sep (SNoS_ ω) (λu : set ⇒ ∃v : set, v ∈ ω ∧ u < ap x v) → w < z)
In Proofgold the corresponding term root is baf0c1... and proposition id is 84e21a...
Proof:
Proof not loaded.
End of Section Conj_real_complete1__13__7
Beginning of Section Conj_real_complete1__14__5
L7718
Variable x : set
(*** Conj_real_complete1__14__5 TMVsfdvd4ET23WBhN2akt6MDBS2hvAKukPP bounty of about 25 bars ***)
L7719
Variable y : set
L7720
Variable z : set
L7721
Variable w : set
L7722
Variable u : set
L7723
Hypothesis H0 : (∀v : set, v ∈ ω → SNo (ap y v))
L7724
Hypothesis H1 : SNo (SNoCut (Sep (SNoS_ ω) (λv : set ⇒ ∃x2 : set, x2 ∈ ω ∧ v < ap x x2)) (Sep (SNoS_ ω) (λv : set ⇒ ∃x2 : set, x2 ∈ ω ∧ ap y x2 < v)))
L7725
Hypothesis H2 : (∀v : set, v ∈ ω → abs_SNo (z + - (SNoCut (Sep (SNoS_ ω) (λx2 : set ⇒ ∃y2 : set, y2 ∈ ω ∧ x2 < ap x y2)) (Sep (SNoS_ ω) (λx2 : set ⇒ ∃y2 : set, y2 ∈ ω ∧ ap y y2 < x2)))) < eps_ v)
L7726
Hypothesis H3 : SNo z
L7727
Hypothesis H4 : w ∈ ω
L7728
Hypothesis H6 : u ∈ ω
L7729
Hypothesis H7 : Empty < z + - (ap y w)
L7730
Hypothesis H8 : SNoCut (Sep (SNoS_ ω) (λv : set ⇒ ∃x2 : set, x2 ∈ ω ∧ v < ap x x2)) (Sep (SNoS_ ω) (λv : set ⇒ ∃x2 : set, x2 ∈ ω ∧ ap y x2 < v)) ≤ ap y w
L7731
Theorem. (Conj_real_complete1__14__5)
Empty < z + - (SNoCut (Sep (SNoS_ ω) (λv : set ⇒ ∃x2 : set, x2 ∈ ω ∧ v < ap x x2)) (Sep (SNoS_ ω) (λv : set ⇒ ∃x2 : set, x2 ∈ ω ∧ ap y x2 < v))) → abs_SNo (z + - (ap y w)) < eps_ u
In Proofgold the corresponding term root is 2a314f... and proposition id is 396c21...
Proof:
Proof not loaded.
End of Section Conj_real_complete1__14__5
Beginning of Section Conj_real_complete1__15__6
L7737
Variable x : set
(*** Conj_real_complete1__15__6 TMdW3M8me4jck3p1HzyEF7sB1VQYNsooomR bounty of about 25 bars ***)
L7738
Variable y : set
L7739
Variable z : set
L7740
Variable w : set
L7741
Variable u : set
L7742
Hypothesis H0 : (∀v : set, v ∈ ω → SNo (ap y v))
L7743
Hypothesis H1 : SNo (SNoCut (Sep (SNoS_ ω) (λv : set ⇒ ∃x2 : set, x2 ∈ ω ∧ v < ap x x2)) (Sep (SNoS_ ω) (λv : set ⇒ ∃x2 : set, x2 ∈ ω ∧ ap y x2 < v)))
L7744
Hypothesis H2 : (∀v : set, v ∈ ω → SNoCut (Sep (SNoS_ ω) (λx2 : set ⇒ ∃y2 : set, y2 ∈ ω ∧ x2 < ap x y2)) (Sep (SNoS_ ω) (λx2 : set ⇒ ∃y2 : set, y2 ∈ ω ∧ ap y y2 < x2)) ≤ ap y v)
L7745
Hypothesis H3 : (∀v : set, v ∈ ω → abs_SNo (z + - (SNoCut (Sep (SNoS_ ω) (λx2 : set ⇒ ∃y2 : set, y2 ∈ ω ∧ x2 < ap x y2)) (Sep (SNoS_ ω) (λx2 : set ⇒ ∃y2 : set, y2 ∈ ω ∧ ap y y2 < x2)))) < eps_ v)
L7746
Hypothesis H4 : SNo z
L7747
Hypothesis H5 : w ∈ ω
L7748
Hypothesis H7 : u ∈ ω
L7749
Hypothesis H8 : Empty < z + - (ap y w)
L7750
Theorem. (Conj_real_complete1__15__6)
SNoCut (Sep (SNoS_ ω) (λv : set ⇒ ∃x2 : set, x2 ∈ ω ∧ v < ap x x2)) (Sep (SNoS_ ω) (λv : set ⇒ ∃x2 : set, x2 ∈ ω ∧ ap y x2 < v)) ≤ ap y w → abs_SNo (z + - (ap y w)) < eps_ u
In Proofgold the corresponding term root is c9bdd8... and proposition id is a01216...
Proof:
Proof not loaded.
End of Section Conj_real_complete1__15__6
Beginning of Section Conj_real_complete1__20__7
L7756
Variable x : set
(*** Conj_real_complete1__20__7 TMTous61Px84n5jVyrkxN13CSWJsz1ioskB bounty of about 25 bars ***)
L7757
Variable y : set
L7758
Variable z : set
L7759
Variable w : set
L7760
Variable u : set
L7761
Hypothesis H0 : (∀v : set, v ∈ ω → SNo (ap x v))
L7762
Hypothesis H1 : SNo (SNoCut (Sep (SNoS_ ω) (λv : set ⇒ ∃x2 : set, x2 ∈ ω ∧ v < ap x x2)) (Sep (SNoS_ ω) (λv : set ⇒ ∃x2 : set, x2 ∈ ω ∧ ap y x2 < v)))
L7763
Hypothesis H2 : (∀v : set, v ∈ ω → ap x v ≤ SNoCut (Sep (SNoS_ ω) (λx2 : set ⇒ ∃y2 : set, y2 ∈ ω ∧ x2 < ap x y2)) (Sep (SNoS_ ω) (λx2 : set ⇒ ∃y2 : set, y2 ∈ ω ∧ ap y y2 < x2)))
L7764
Hypothesis H3 : (∀v : set, v ∈ ω → abs_SNo (z + - (SNoCut (Sep (SNoS_ ω) (λx2 : set ⇒ ∃y2 : set, y2 ∈ ω ∧ x2 < ap x y2)) (Sep (SNoS_ ω) (λx2 : set ⇒ ∃y2 : set, y2 ∈ ω ∧ ap y y2 < x2)))) < eps_ v)
L7765
Hypothesis H4 : SNo z
L7766
Hypothesis H5 : w ∈ ω
L7767
Hypothesis H6 : z < ap x w
L7768
Theorem. (Conj_real_complete1__20__7)
Empty < ap x w + - z → abs_SNo (z + - (ap x w)) < eps_ u
In Proofgold the corresponding term root is 950ace... and proposition id is 28e2e9...
Proof:
Proof not loaded.
End of Section Conj_real_complete1__20__7
Beginning of Section Conj_real_complete1__25__7
L7774
Variable x : set
(*** Conj_real_complete1__25__7 TMPVaLP5LsFCgAEccyv1DMAQTtsK2YnG9nS bounty of about 25 bars ***)
L7775
Variable y : set
L7776
Hypothesis H0 : x ∈ setexp real ω
L7777
Hypothesis H1 : y ∈ setexp real ω
L7778
Hypothesis H2 : ¬ (∃z : set, z ∈ real ∧ (∀w : set, w ∈ ω → ap x w ≤ z ∧ z ≤ ap y w))
L7779
Hypothesis H3 : (∀z : set, z ∈ ω → SNo (ap x z))
L7780
Hypothesis H4 : (∀z : set, z ∈ ω → SNo (ap y z))
L7781
Hypothesis H5 : Subq (Sep (SNoS_ ω) (λz : set ⇒ ∃w : set, w ∈ ω ∧ z < ap x w)) (SNoS_ ω)
L7782
Hypothesis H6 : Subq (Sep (SNoS_ ω) (λz : set ⇒ ∃w : set, w ∈ ω ∧ ap y w < z)) (SNoS_ ω)
L7783
Hypothesis H8 : SNo (SNoCut (Sep (SNoS_ ω) (λz : set ⇒ ∃w : set, w ∈ ω ∧ z < ap x w)) (Sep (SNoS_ ω) (λz : set ⇒ ∃w : set, w ∈ ω ∧ ap y w < z)))
L7784
Hypothesis H9 : (∀z : set, SNo z → (∀w : set, w ∈ Sep (SNoS_ ω) (λu : set ⇒ ∃v : set, v ∈ ω ∧ u < ap x v) → w < z) → (∀w : set, w ∈ Sep (SNoS_ ω) (λu : set ⇒ ∃v : set, v ∈ ω ∧ ap y v < u) → z < w) → Subq (SNoLev (SNoCut (Sep (SNoS_ ω) (λw : set ⇒ ∃u : set, u ∈ ω ∧ w < ap x u)) (Sep (SNoS_ ω) (λw : set ⇒ ∃u : set, u ∈ ω ∧ ap y u < w)))) (SNoLev z) ∧ SNoEq_ (SNoLev (SNoCut (Sep (SNoS_ ω) (λw : set ⇒ ∃u : set, u ∈ ω ∧ w < ap x u)) (Sep (SNoS_ ω) (λw : set ⇒ ∃u : set, u ∈ ω ∧ ap y u < w)))) (SNoCut (Sep (SNoS_ ω) (λw : set ⇒ ∃u : set, u ∈ ω ∧ w < ap x u)) (Sep (SNoS_ ω) (λw : set ⇒ ∃u : set, u ∈ ω ∧ ap y u < w))) z)
L7785
Hypothesis H10 : (∀z : set, z ∈ ω → ap x z ≤ SNoCut (Sep (SNoS_ ω) (λw : set ⇒ ∃u : set, u ∈ ω ∧ w < ap x u)) (Sep (SNoS_ ω) (λw : set ⇒ ∃u : set, u ∈ ω ∧ ap y u < w)))
L7786
Hypothesis H11 : (∀z : set, z ∈ ω → SNoCut (Sep (SNoS_ ω) (λw : set ⇒ ∃u : set, u ∈ ω ∧ w < ap x u)) (Sep (SNoS_ ω) (λw : set ⇒ ∃u : set, u ∈ ω ∧ ap y u < w)) ≤ ap y z)
L7787
Theorem. (Conj_real_complete1__25__7)
¬ (∀z : set, z ∈ ω → ap x z ≤ SNoCut (Sep (SNoS_ ω) (λw : set ⇒ ∃u : set, u ∈ ω ∧ w < ap x u)) (Sep (SNoS_ ω) (λw : set ⇒ ∃u : set, u ∈ ω ∧ ap y u < w)) ∧ SNoCut (Sep (SNoS_ ω) (λw : set ⇒ ∃u : set, u ∈ ω ∧ w < ap x u)) (Sep (SNoS_ ω) (λw : set ⇒ ∃u : set, u ∈ ω ∧ ap y u < w)) ≤ ap y z)
In Proofgold the corresponding term root is 318d0e... and proposition id is 037698...
Proof:
Proof not loaded.
End of Section Conj_real_complete1__25__7
Beginning of Section Conj_real_complete1__26__11
L7793
Variable x : set
(*** Conj_real_complete1__26__11 TMKB3eGR57KrnConpMuX3dqK973krYn4LVU bounty of about 25 bars ***)
L7794
Variable y : set
L7795
Hypothesis H0 : x ∈ setexp real ω
L7796
Hypothesis H1 : y ∈ setexp real ω
L7797
Hypothesis H2 : ¬ (∃z : set, z ∈ real ∧ (∀w : set, w ∈ ω → ap x w ≤ z ∧ z ≤ ap y w))
L7798
Hypothesis H3 : (∀z : set, z ∈ ω → SNo (ap x z))
L7799
Hypothesis H4 : (∀z : set, z ∈ ω → SNo (ap y z))
L7800
Hypothesis H5 : (∀z : set, z ∈ ω → (∀w : set, w ∈ ω → ap x z ≤ ap y w))
L7801
Hypothesis H6 : Subq (Sep (SNoS_ ω) (λz : set ⇒ ∃w : set, w ∈ ω ∧ z < ap x w)) (SNoS_ ω)
L7802
Hypothesis H7 : Subq (Sep (SNoS_ ω) (λz : set ⇒ ∃w : set, w ∈ ω ∧ ap y w < z)) (SNoS_ ω)
L7803
Hypothesis H8 : SNoCutP (Sep (SNoS_ ω) (λz : set ⇒ ∃w : set, w ∈ ω ∧ z < ap x w)) (Sep (SNoS_ ω) (λz : set ⇒ ∃w : set, w ∈ ω ∧ ap y w < z))
L7804
Hypothesis H9 : SNo (SNoCut (Sep (SNoS_ ω) (λz : set ⇒ ∃w : set, w ∈ ω ∧ z < ap x w)) (Sep (SNoS_ ω) (λz : set ⇒ ∃w : set, w ∈ ω ∧ ap y w < z)))
L7805
Hypothesis H10 : (∀z : set, z ∈ Sep (SNoS_ ω) (λw : set ⇒ ∃u : set, u ∈ ω ∧ ap y u < w) → SNoCut (Sep (SNoS_ ω) (λw : set ⇒ ∃u : set, u ∈ ω ∧ w < ap x u)) (Sep (SNoS_ ω) (λw : set ⇒ ∃u : set, u ∈ ω ∧ ap y u < w)) < z)
L7806
Hypothesis H12 : (∀z : set, z ∈ ω → ap x z ≤ SNoCut (Sep (SNoS_ ω) (λw : set ⇒ ∃u : set, u ∈ ω ∧ w < ap x u)) (Sep (SNoS_ ω) (λw : set ⇒ ∃u : set, u ∈ ω ∧ ap y u < w)))
L7807
Theorem. (Conj_real_complete1__26__11)
¬ (∀z : set, z ∈ ω → SNoCut (Sep (SNoS_ ω) (λw : set ⇒ ∃u : set, u ∈ ω ∧ w < ap x u)) (Sep (SNoS_ ω) (λw : set ⇒ ∃u : set, u ∈ ω ∧ ap y u < w)) ≤ ap y z)
In Proofgold the corresponding term root is 17788f... and proposition id is ca4c7a...
Proof:
Proof not loaded.
End of Section Conj_real_complete1__26__11
Beginning of Section Conj_real_complete1__27__11
L7813
Variable x : set
(*** Conj_real_complete1__27__11 TMdxSfFcq3FdqmSNeoLt7E3REakvYwcepxA bounty of about 25 bars ***)
L7814
Variable y : set
L7815
Hypothesis H0 : x ∈ setexp real ω
L7816
Hypothesis H1 : y ∈ setexp real ω
L7817
Hypothesis H2 : ¬ (∃z : set, z ∈ real ∧ (∀w : set, w ∈ ω → ap x w ≤ z ∧ z ≤ ap y w))
L7818
Hypothesis H3 : (∀z : set, z ∈ ω → SNo (ap x z))
L7819
Hypothesis H4 : (∀z : set, z ∈ ω → SNo (ap y z))
L7820
Hypothesis H5 : (∀z : set, z ∈ ω → (∀w : set, w ∈ ω → ap x z ≤ ap y w))
L7821
Hypothesis H6 : Subq (Sep (SNoS_ ω) (λz : set ⇒ ∃w : set, w ∈ ω ∧ z < ap x w)) (SNoS_ ω)
L7822
Hypothesis H7 : Subq (Sep (SNoS_ ω) (λz : set ⇒ ∃w : set, w ∈ ω ∧ ap y w < z)) (SNoS_ ω)
L7823
Hypothesis H8 : SNoCutP (Sep (SNoS_ ω) (λz : set ⇒ ∃w : set, w ∈ ω ∧ z < ap x w)) (Sep (SNoS_ ω) (λz : set ⇒ ∃w : set, w ∈ ω ∧ ap y w < z))
L7824
Hypothesis H9 : SNo (SNoCut (Sep (SNoS_ ω) (λz : set ⇒ ∃w : set, w ∈ ω ∧ z < ap x w)) (Sep (SNoS_ ω) (λz : set ⇒ ∃w : set, w ∈ ω ∧ ap y w < z)))
L7825
Hypothesis H10 : (∀z : set, z ∈ Sep (SNoS_ ω) (λw : set ⇒ ∃u : set, u ∈ ω ∧ w < ap x u) → z < SNoCut (Sep (SNoS_ ω) (λw : set ⇒ ∃u : set, u ∈ ω ∧ w < ap x u)) (Sep (SNoS_ ω) (λw : set ⇒ ∃u : set, u ∈ ω ∧ ap y u < w)))
L7826
Hypothesis H12 : (∀z : set, SNo z → (∀w : set, w ∈ Sep (SNoS_ ω) (λu : set ⇒ ∃v : set, v ∈ ω ∧ u < ap x v) → w < z) → (∀w : set, w ∈ Sep (SNoS_ ω) (λu : set ⇒ ∃v : set, v ∈ ω ∧ ap y v < u) → z < w) → Subq (SNoLev (SNoCut (Sep (SNoS_ ω) (λw : set ⇒ ∃u : set, u ∈ ω ∧ w < ap x u)) (Sep (SNoS_ ω) (λw : set ⇒ ∃u : set, u ∈ ω ∧ ap y u < w)))) (SNoLev z) ∧ SNoEq_ (SNoLev (SNoCut (Sep (SNoS_ ω) (λw : set ⇒ ∃u : set, u ∈ ω ∧ w < ap x u)) (Sep (SNoS_ ω) (λw : set ⇒ ∃u : set, u ∈ ω ∧ ap y u < w)))) (SNoCut (Sep (SNoS_ ω) (λw : set ⇒ ∃u : set, u ∈ ω ∧ w < ap x u)) (Sep (SNoS_ ω) (λw : set ⇒ ∃u : set, u ∈ ω ∧ ap y u < w))) z)
L7827
Theorem. (Conj_real_complete1__27__11)
¬ (∀z : set, z ∈ ω → ap x z ≤ SNoCut (Sep (SNoS_ ω) (λw : set ⇒ ∃u : set, u ∈ ω ∧ w < ap x u)) (Sep (SNoS_ ω) (λw : set ⇒ ∃u : set, u ∈ ω ∧ ap y u < w)))
In Proofgold the corresponding term root is 9d1e39... and proposition id is fbf7cc...
Proof:
Proof not loaded.
End of Section Conj_real_complete1__27__11
Beginning of Section Conj_real_complete1__31__1
L7833
Variable x : set
(*** Conj_real_complete1__31__1 TMbqvnFNHXeaZjHK6c7WjL8VdnvXcYkau4M bounty of about 25 bars ***)
L7834
Variable y : set
L7835
Hypothesis H0 : x ∈ setexp real ω
L7836
Hypothesis H2 : (∀z : set, z ∈ ω → ap x z ≤ ap y z ∧ ap x z ≤ ap x (ordsucc z) ∧ ap y (ordsucc z) ≤ ap y z)
L7837
Hypothesis H3 : ¬ (∃z : set, z ∈ real ∧ (∀w : set, w ∈ ω → ap x w ≤ z ∧ z ≤ ap y w))
L7838
Hypothesis H4 : (∀z : set, z ∈ ω → SNo (ap x z))
L7839
Hypothesis H5 : (∀z : set, z ∈ ω → SNo (ap y z))
L7840
Hypothesis H6 : (∀z : set, nat_p z → (∀w : set, w ∈ z → ap x w ≤ ap x z))
L7841
Hypothesis H7 : (∀z : set, nat_p z → (∀w : set, w ∈ z → ap y z ≤ ap y w))
L7842
Theorem. (Conj_real_complete1__31__1)
¬ (∀z : set, z ∈ ω → (∀w : set, w ∈ ω → ap x z ≤ ap y w))
In Proofgold the corresponding term root is 939a47... and proposition id is 8c99f0...
Proof:
Proof not loaded.
End of Section Conj_real_complete1__31__1
Beginning of Section Conj_ctagged_notin_SNo__3__0
L7848
Variable x : set
(*** Conj_ctagged_notin_SNo__3__0 TMWyfVXUfbVzc7ZDWZFZYv2ibcna3c7cihA bounty of about 25 bars ***)
L7849
Variable y : set
L7850
Hypothesis H1 : ordinal (SNoLev x)
L7851
Hypothesis H2 : Subq x (SNoElts_ (SNoLev x))
L7852
Theorem. (Conj_ctagged_notin_SNo__3__0)
¬ SetAdjoin y (Sing (ordsucc (ordsucc Empty))) ∈ binunion (SNoLev x) (Repl (SNoLev x) (λz : set ⇒ SetAdjoin z (Sing (ordsucc Empty))))
In Proofgold the corresponding term root is 3db93a... and proposition id is 0154ba...
Proof:
Proof not loaded.
End of Section Conj_ctagged_notin_SNo__3__0
Beginning of Section Conj_ctagged_eqE_Subq__1__1
L7858
Variable x : set
(*** Conj_ctagged_eqE_Subq__1__1 TMTfP8MMHDXqkkgbohA8fMfmHKxpg55HEbL bounty of about 25 bars ***)
L7859
Variable y : set
L7860
Hypothesis H0 : Sing (ordsucc (ordsucc Empty)) ∈ x
L7861
Hypothesis H2 : ordinal y
L7862
Theorem. (Conj_ctagged_eqE_Subq__1__1)
¬ Sing (ordsucc (ordsucc Empty)) ∈ SetAdjoin y (Sing (ordsucc Empty))
In Proofgold the corresponding term root is cfbd0a... and proposition id is 34c672...
Proof:
Proof not loaded.
End of Section Conj_ctagged_eqE_Subq__1__1
Beginning of Section Conj_ctagged_eqE_Subq__2__3
L7868
Variable x : set
(*** Conj_ctagged_eqE_Subq__2__3 TMHXXjw8cG92xF9SogmGhUZzP7F4VrnCjj9 bounty of about 25 bars ***)
L7869
Variable y : set
L7870
Variable z : set
L7871
Hypothesis H0 : Sing (ordsucc (ordsucc Empty)) ∈ y
L7872
Hypothesis H1 : ordinal (SNoLev x)
L7873
Hypothesis H2 : z ∈ SNoLev x
L7874
Theorem. (Conj_ctagged_eqE_Subq__2__3)
¬ ordinal z
In Proofgold the corresponding term root is 018440... and proposition id is a53944...
Proof:
Proof not loaded.
End of Section Conj_ctagged_eqE_Subq__2__3
Beginning of Section Conj_ctagged_eqE_Subq__5__2
L7880
Variable x : set
(*** Conj_ctagged_eqE_Subq__5__2 TMPjtiRn3YTJppVfNqxCxUQRtvMHvSPjTuJ bounty of about 25 bars ***)
L7881
Variable y : set
L7882
Variable z : set
L7883
Hypothesis H0 : SNo x
L7884
Hypothesis H1 : y ∈ x
L7885
Hypothesis H3 : z ∈ Sing (Sing (ordsucc (ordsucc Empty)))
L7886
Theorem. (Conj_ctagged_eqE_Subq__5__2)
z ≠ Sing (ordsucc (ordsucc Empty))
In Proofgold the corresponding term root is 3bbee7... and proposition id is f1351b...
Proof:
Proof not loaded.
End of Section Conj_ctagged_eqE_Subq__5__2
Beginning of Section Conj_SNo_pair_prop_1_Subq__1__0
L7892
Variable x : set
(*** Conj_SNo_pair_prop_1_Subq__1__0 TMFWVR9bMV7faSWpjCcRWUHVwFqA8xNAxvK bounty of about 25 bars ***)
L7893
Variable y : set
L7894
Variable z : set
L7895
Variable w : set
L7896
Variable u : set
L7897
Hypothesis H1 : SNo_pair x y = SNo_pair z w
L7898
Hypothesis H2 : u ∈ x
L7899
Theorem. (Conj_SNo_pair_prop_1_Subq__1__0)
u ∈ SNo_pair z w → u ∈ z
In Proofgold the corresponding term root is 853287... and proposition id is 79d12f...
Proof:
Proof not loaded.
End of Section Conj_SNo_pair_prop_1_Subq__1__0
Beginning of Section Conj_add_CSNo_minus_CSNo_rinv__1__1
L7905
Variable x : set
(*** Conj_add_CSNo_minus_CSNo_rinv__1__1 TMGvMPu6FNJFNzzxkUsrD8j7o5dvBWLqayu bounty of about 25 bars ***)
L7906
Hypothesis H0 : CSNo x
L7907
Theorem. (Conj_add_CSNo_minus_CSNo_rinv__1__1)
SNo (- (CSNo_Im x)) → SNo_pair (CSNo_Re x + CSNo_Re (SNo_pair (- (CSNo_Re x)) (- (CSNo_Im x)))) (CSNo_Im x + CSNo_Im (SNo_pair (- (CSNo_Re x)) (- (CSNo_Im x)))) = Empty
In Proofgold the corresponding term root is b97ff8... and proposition id is 56123f...
Proof:
Proof not loaded.
End of Section Conj_add_CSNo_minus_CSNo_rinv__1__1
Beginning of Section Conj_add_CSNo_com__1__2
L7913
Variable x : set
(*** Conj_add_CSNo_com__1__2 TMJR3mdFNNJgxoneGne75ELM1mva7cdPb4H bounty of about 25 bars ***)
L7914
Variable y : set
L7915
Hypothesis H0 : CSNo x
L7916
Hypothesis H1 : CSNo y
L7917
Theorem. (Conj_add_CSNo_com__1__2)
CSNo (add_CSNo y x) → add_CSNo x y = add_CSNo y x
In Proofgold the corresponding term root is 5df675... and proposition id is a44cb8...
Proof:
Proof not loaded.
End of Section Conj_add_CSNo_com__1__2
Beginning of Section Conj_add_CSNo_assoc__2__4
L7923
Variable x : set
(*** Conj_add_CSNo_assoc__2__4 TMMeS8ujRWaUQhLZd9Z2BYTSkkL7jZnKZbt bounty of about 25 bars ***)
L7924
Variable y : set
L7925
Variable z : set
L7926
Hypothesis H0 : CSNo x
L7927
Hypothesis H1 : CSNo y
L7928
Hypothesis H2 : CSNo z
L7929
Hypothesis H3 : CSNo (add_CSNo y z)
L7930
Theorem. (Conj_add_CSNo_assoc__2__4)
CSNo (add_CSNo x (add_CSNo y z)) → add_CSNo x (add_CSNo y z) = add_CSNo (add_CSNo x y) z
In Proofgold the corresponding term root is 8e5f0e... and proposition id is 31641f...
Proof:
Proof not loaded.
End of Section Conj_add_CSNo_assoc__2__4
Beginning of Section Conj_mul_CSNo_assoc__2__8
L7936
Variable x : set
(*** Conj_mul_CSNo_assoc__2__8 TMKfXsHxyzeYigzgFi6jQKquTV161zM4ThW bounty of about 25 bars ***)
L7937
Variable y : set
L7938
Variable z : set
L7939
Hypothesis H0 : CSNo x
L7940
Hypothesis H1 : CSNo y
L7941
Hypothesis H2 : CSNo z
L7942
Hypothesis H3 : CSNo (mul_CSNo y z)
L7943
Hypothesis H4 : CSNo (mul_CSNo x y)
L7944
Hypothesis H5 : CSNo (mul_CSNo x (mul_CSNo y z))
L7945
Hypothesis H6 : CSNo (mul_CSNo (mul_CSNo x y) z)
L7946
Hypothesis H7 : SNo (CSNo_Re x)
L7947
Hypothesis H9 : SNo (CSNo_Re z)
L7948
Hypothesis H10 : SNo (CSNo_Im x)
L7949
Theorem. (Conj_mul_CSNo_assoc__2__8)
SNo (CSNo_Im y) → mul_CSNo x (mul_CSNo y z) = mul_CSNo (mul_CSNo x y) z
In Proofgold the corresponding term root is 571933... and proposition id is a861a6...
Proof:
Proof not loaded.
End of Section Conj_mul_CSNo_assoc__2__8
Beginning of Section Conj_mul_CSNo_assoc__3__1
L7955
Variable x : set
(*** Conj_mul_CSNo_assoc__3__1 TMXhHMj7FhJXndiqL2TkBP8x79MN8HpkMwc bounty of about 25 bars ***)
L7956
Variable y : set
L7957
Variable z : set
L7958
Hypothesis H0 : CSNo x
L7959
Hypothesis H2 : CSNo z
L7960
Hypothesis H3 : CSNo (mul_CSNo y z)
L7961
Hypothesis H4 : CSNo (mul_CSNo x y)
L7962
Hypothesis H5 : CSNo (mul_CSNo x (mul_CSNo y z))
L7963
Hypothesis H6 : CSNo (mul_CSNo (mul_CSNo x y) z)
L7964
Hypothesis H7 : SNo (CSNo_Re x)
L7965
Hypothesis H8 : SNo (CSNo_Re y)
L7966
Hypothesis H9 : SNo (CSNo_Re z)
L7967
Theorem. (Conj_mul_CSNo_assoc__3__1)
SNo (CSNo_Im x) → mul_CSNo x (mul_CSNo y z) = mul_CSNo (mul_CSNo x y) z
In Proofgold the corresponding term root is 7a9c04... and proposition id is dfc1e1...
Proof:
Proof not loaded.
End of Section Conj_mul_CSNo_assoc__3__1
Beginning of Section Conj_mul_CSNo_assoc__4__6
L7973
Variable x : set
(*** Conj_mul_CSNo_assoc__4__6 TMNpWNxQPQr62KPRcmVDoUdw8J8m8YjXpqh bounty of about 25 bars ***)
L7974
Variable y : set
L7975
Variable z : set
L7976
Hypothesis H0 : CSNo x
L7977
Hypothesis H1 : CSNo y
L7978
Hypothesis H2 : CSNo z
L7979
Hypothesis H3 : CSNo (mul_CSNo y z)
L7980
Hypothesis H4 : CSNo (mul_CSNo x y)
L7981
Hypothesis H5 : CSNo (mul_CSNo x (mul_CSNo y z))
L7982
Hypothesis H7 : SNo (CSNo_Re x)
L7983
Hypothesis H8 : SNo (CSNo_Re y)
L7984
Theorem. (Conj_mul_CSNo_assoc__4__6)
SNo (CSNo_Re z) → mul_CSNo x (mul_CSNo y z) = mul_CSNo (mul_CSNo x y) z
In Proofgold the corresponding term root is 1bb7dc... and proposition id is c814dc...
Proof:
Proof not loaded.
End of Section Conj_mul_CSNo_assoc__4__6
Beginning of Section Conj_mul_CSNo_distrL__2__0
L7990
Variable x : set
(*** Conj_mul_CSNo_distrL__2__0 TMb4GJkZzkgg5ZLTZQf1qd98F95hEErVQvQ bounty of about 25 bars ***)
L7991
Variable y : set
L7992
Variable z : set
L7993
Hypothesis H1 : CSNo y
L7994
Hypothesis H2 : CSNo z
L7995
Hypothesis H3 : CSNo (add_CSNo y z)
L7996
Hypothesis H4 : CSNo (mul_CSNo x y)
L7997
Hypothesis H5 : CSNo (mul_CSNo x z)
L7998
Hypothesis H6 : CSNo (mul_CSNo x (add_CSNo y z))
L7999
Hypothesis H7 : CSNo (add_CSNo (mul_CSNo x y) (mul_CSNo x z))
L8000
Hypothesis H8 : SNo (CSNo_Re x)
L8001
Hypothesis H9 : SNo (CSNo_Re y)
L8002
Hypothesis H10 : SNo (CSNo_Re z)
L8003
Hypothesis H11 : SNo (CSNo_Im x)
L8004
Theorem. (Conj_mul_CSNo_distrL__2__0)
SNo (CSNo_Im y) → mul_CSNo x (add_CSNo y z) = add_CSNo (mul_CSNo x y) (mul_CSNo x z)
In Proofgold the corresponding term root is 8ee5c4... and proposition id is d14351...
Proof:
Proof not loaded.
End of Section Conj_mul_CSNo_distrL__2__0
Beginning of Section Conj_CSNo_relative_recip__5__2
L8010
Variable x : set
(*** Conj_CSNo_relative_recip__5__2 TMX8WVqJ1byoEumpMvwXnXEXSESYsRiwKad bounty of about 25 bars ***)
L8011
Variable y : set
L8012
Hypothesis H0 : CSNo x
L8013
Hypothesis H1 : SNo y
L8014
Hypothesis H3 : SNo (CSNo_Re x)
L8015
Hypothesis H4 : SNo (CSNo_Im x)
L8016
Hypothesis H5 : SNo (y * CSNo_Re x)
L8017
Hypothesis H6 : CSNo (y * CSNo_Re x)
L8018
Hypothesis H7 : SNo (y * CSNo_Im x)
L8019
Hypothesis H8 : CSNo (y * CSNo_Im x)
L8020
Theorem. (Conj_CSNo_relative_recip__5__2)
CSNo (mul_CSNo Complex_i (y * CSNo_Im x)) → mul_CSNo x (add_CSNo (y * CSNo_Re x) (minus_CSNo (mul_CSNo Complex_i (y * CSNo_Im x)))) = ordsucc Empty
In Proofgold the corresponding term root is 4bf189... and proposition id is 0058e8...
Proof:
Proof not loaded.
End of Section Conj_CSNo_relative_recip__5__2
Beginning of Section Conj_nonzero_complex_recip_ex__2__4
L8026
Variable x : set
(*** Conj_nonzero_complex_recip_ex__2__4 TMHDEdxPXwPMBssSQ7ddwei6EedB6BucqEs bounty of about 25 bars ***)
L8027
Hypothesis H0 : x ∈ complex
L8028
Hypothesis H1 : x ≠ Empty
L8029
Hypothesis H2 : CSNo_Re x ∈ real
L8030
Hypothesis H3 : CSNo_Im x ∈ real
L8031
Hypothesis H5 : SNo (CSNo_Im x * CSNo_Im x)
L8032
Hypothesis H6 : CSNo_Re x * CSNo_Re x + CSNo_Im x * CSNo_Im x ∈ real
L8033
Theorem. (Conj_nonzero_complex_recip_ex__2__4)
SNo (CSNo_Re x * CSNo_Re x + CSNo_Im x * CSNo_Im x) → (∃y : set, y ∈ complex ∧ mul_CSNo x y = ordsucc Empty)
In Proofgold the corresponding term root is f40d1b... and proposition id is 6a366f...
Proof:
Proof not loaded.
End of Section Conj_nonzero_complex_recip_ex__2__4