Beginning of Section Conj_mul_SNo_assoc_lem2__104__20
Variable g : (set → (set → set))
Variable x : set
Variable y : set
Variable z : set
Variable w : set
Variable u : set
Hypothesis H0 : (∀v : set, ∀x2 : set, SNo v → SNo x2 → SNo (g v x2))
Hypothesis H1 : (∀v : set, ∀x2 : set, ∀y2 : set, SNo v → SNo x2 → SNo y2 → g v (x2 + y2) = g v x2 + g v y2)
Hypothesis H2 : (∀v : set, ∀x2 : set, ∀y2 : set, SNo v → SNo x2 → SNo y2 → g (v + x2) y2 = g v y2 + g x2 y2)
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)))
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)))
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)
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)
Hypothesis H7 : SNo x
Hypothesis H8 : SNo y
Hypothesis H9 : SNo z
Hypothesis H10 : (∀v : set, v ∈ SNoS_ (SNoLev x) → g v (g y z) = g (g v y) z)
Hypothesis H11 : (∀v : set, v ∈ SNoS_ (SNoLev y) → g x (g v z) = g (g x v) z)
Hypothesis H12 : (∀v : set, v ∈ SNoS_ (SNoLev z) → g x (g y v) = g (g x y) v)
Hypothesis H13 : (∀v : set, v ∈ SNoS_ (SNoLev x) → (∀x2 : set, x2 ∈ SNoS_ (SNoLev y) → g v (g x2 z) = g (g v x2) z))
Hypothesis H14 : (∀v : set, v ∈ SNoS_ (SNoLev x) → (∀x2 : set, x2 ∈ SNoS_ (SNoLev z) → g v (g y x2) = g (g v y) x2))
Hypothesis H15 : (∀v : set, v ∈ SNoS_ (SNoLev y) → (∀x2 : set, x2 ∈ SNoS_ (SNoLev z) → g x (g v x2) = g (g x v) x2))
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)))
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))
Hypothesis H18 : u ∈ w
Hypothesis H19 : SNo (g x y)
Hypothesis H21 : SNo (g (g x y) z)
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
Variable g : (set → (set → set))
Variable x : set
Variable y : set
Variable z : set
Variable w : set
Variable u : set
Hypothesis H0 : (∀v : set, ∀x2 : set, SNo v → SNo x2 → SNo (g v x2))
Hypothesis H1 : (∀v : set, ∀x2 : set, ∀y2 : set, SNo v → SNo x2 → SNo y2 → g v (x2 + y2) = g v x2 + g v y2)
Hypothesis H2 : (∀v : set, ∀x2 : set, ∀y2 : set, SNo v → SNo x2 → SNo y2 → g (v + x2) y2 = g v y2 + g x2 y2)
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)))
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)
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)
Hypothesis H7 : SNo x
Hypothesis H8 : SNo y
Hypothesis H9 : SNo z
Hypothesis H10 : (∀v : set, v ∈ SNoS_ (SNoLev x) → g v (g y z) = g (g v y) z)
Hypothesis H11 : (∀v : set, v ∈ SNoS_ (SNoLev y) → g x (g v z) = g (g x v) z)
Hypothesis H12 : (∀v : set, v ∈ SNoS_ (SNoLev z) → g x (g y v) = g (g x y) v)
Hypothesis H13 : (∀v : set, v ∈ SNoS_ (SNoLev x) → (∀x2 : set, x2 ∈ SNoS_ (SNoLev y) → g v (g x2 z) = g (g v x2) z))
Hypothesis H14 : (∀v : set, v ∈ SNoS_ (SNoLev x) → (∀x2 : set, x2 ∈ SNoS_ (SNoLev z) → g v (g y x2) = g (g v y) x2))
Hypothesis H15 : (∀v : set, v ∈ SNoS_ (SNoLev y) → (∀x2 : set, x2 ∈ SNoS_ (SNoLev z) → g x (g v x2) = g (g x v) x2))
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)))
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))
Hypothesis H18 : u ∈ w
Hypothesis H19 : SNo (g x y)
Hypothesis H20 : SNo (g y z)
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
Variable g : (set → (set → set))
Variable x : set
Variable y : set
Variable z : set
Variable w : set
Variable u : set
Hypothesis H0 : (∀v : set, ∀x2 : set, SNo v → SNo x2 → SNo (g v x2))
Hypothesis H1 : (∀v : set, ∀x2 : set, ∀y2 : set, SNo v → SNo x2 → SNo y2 → g v (x2 + y2) = g v x2 + g v y2)
Hypothesis H2 : (∀v : set, ∀x2 : set, ∀y2 : set, SNo v → SNo x2 → SNo y2 → g (v + x2) y2 = g v y2 + g x2 y2)
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)))
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)))
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)
Hypothesis H7 : SNo x
Hypothesis H8 : SNo y
Hypothesis H9 : SNo z
Hypothesis H10 : (∀v : set, v ∈ SNoS_ (SNoLev x) → g v (g y z) = g (g v y) z)
Hypothesis H11 : (∀v : set, v ∈ SNoS_ (SNoLev y) → g x (g v z) = g (g x v) z)
Hypothesis H12 : (∀v : set, v ∈ SNoS_ (SNoLev z) → g x (g y v) = g (g x y) v)
Hypothesis H13 : (∀v : set, v ∈ SNoS_ (SNoLev x) → (∀x2 : set, x2 ∈ SNoS_ (SNoLev y) → g v (g x2 z) = g (g v x2) z))
Hypothesis H14 : (∀v : set, v ∈ SNoS_ (SNoLev x) → (∀x2 : set, x2 ∈ SNoS_ (SNoLev z) → g v (g y x2) = g (g v y) x2))
Hypothesis H15 : (∀v : set, v ∈ SNoS_ (SNoLev y) → (∀x2 : set, x2 ∈ SNoS_ (SNoLev z) → g x (g v x2) = g (g x v) x2))
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)))
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))
Hypothesis H18 : u ∈ w
Hypothesis H19 : SNo (g x y)
Hypothesis H20 : SNo (g y z)
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
Variable g : (set → (set → set))
Variable x : set
Variable y : set
Variable z : set
Variable w : set
Variable u : set
Hypothesis H0 : (∀v : set, ∀x2 : set, SNo v → SNo x2 → SNo (g v x2))
Hypothesis H1 : (∀v : set, ∀x2 : set, ∀y2 : set, SNo v → SNo x2 → SNo y2 → g v (x2 + y2) = g v x2 + g v y2)
Hypothesis H2 : (∀v : set, ∀x2 : set, ∀y2 : set, SNo v → SNo x2 → SNo y2 → g (v + x2) y2 = g v y2 + g x2 y2)
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)))
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)))
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)
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)
Hypothesis H7 : SNo x
Hypothesis H9 : SNo z
Hypothesis H10 : (∀v : set, v ∈ SNoS_ (SNoLev x) → g v (g y z) = g (g v y) z)
Hypothesis H11 : (∀v : set, v ∈ SNoS_ (SNoLev y) → g x (g v z) = g (g x v) z)
Hypothesis H12 : (∀v : set, v ∈ SNoS_ (SNoLev z) → g x (g y v) = g (g x y) v)
Hypothesis H13 : (∀v : set, v ∈ SNoS_ (SNoLev x) → (∀x2 : set, x2 ∈ SNoS_ (SNoLev y) → g v (g x2 z) = g (g v x2) z))
Hypothesis H14 : (∀v : set, v ∈ SNoS_ (SNoLev x) → (∀x2 : set, x2 ∈ SNoS_ (SNoLev z) → g v (g y x2) = g (g v y) x2))
Hypothesis H15 : (∀v : set, v ∈ SNoS_ (SNoLev y) → (∀x2 : set, x2 ∈ SNoS_ (SNoLev z) → g x (g v x2) = g (g x v) x2))
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)))
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))
Hypothesis H18 : u ∈ w
Hypothesis H19 : SNo (g x y)
Hypothesis H20 : SNo (g y z)
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
Variable g : (set → (set → set))
Variable x : set
Variable y : set
Variable z : set
Variable w : set
Variable u : set
Hypothesis H0 : (∀v : set, ∀x2 : set, SNo v → SNo x2 → SNo (g v x2))
Hypothesis H1 : (∀v : set, ∀x2 : set, ∀y2 : set, SNo v → SNo x2 → SNo y2 → g v (x2 + y2) = g v x2 + g v y2)
Hypothesis H2 : (∀v : set, ∀x2 : set, ∀y2 : set, SNo v → SNo x2 → SNo y2 → g (v + x2) y2 = g v y2 + g x2 y2)
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)))
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)))
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)
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)
Hypothesis H7 : SNo x
Hypothesis H9 : SNo z
Hypothesis H10 : (∀v : set, v ∈ SNoS_ (SNoLev x) → g v (g y z) = g (g v y) z)
Hypothesis H11 : (∀v : set, v ∈ SNoS_ (SNoLev y) → g x (g v z) = g (g x v) z)
Hypothesis H12 : (∀v : set, v ∈ SNoS_ (SNoLev z) → g x (g y v) = g (g x y) v)
Hypothesis H13 : (∀v : set, v ∈ SNoS_ (SNoLev x) → (∀x2 : set, x2 ∈ SNoS_ (SNoLev y) → g v (g x2 z) = g (g v x2) z))
Hypothesis H14 : (∀v : set, v ∈ SNoS_ (SNoLev x) → (∀x2 : set, x2 ∈ SNoS_ (SNoLev z) → g v (g y x2) = g (g v y) x2))
Hypothesis H15 : (∀v : set, v ∈ SNoS_ (SNoLev y) → (∀x2 : set, x2 ∈ SNoS_ (SNoLev z) → g x (g v x2) = g (g x v) x2))
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)))
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))
Hypothesis H18 : u ∈ w
Hypothesis H19 : SNo (g x y)
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
Variable g : (set → (set → set))
Variable x : set
Variable y : set
Variable z : set
Variable w : set
Variable u : set
Hypothesis H0 : (∀v : set, ∀x2 : set, SNo v → SNo x2 → SNo (g v x2))
Hypothesis H1 : (∀v : set, ∀x2 : set, ∀y2 : set, SNo v → SNo x2 → SNo y2 → g v (x2 + y2) = g v x2 + g v y2)
Hypothesis H2 : (∀v : set, ∀x2 : set, ∀y2 : set, SNo v → SNo x2 → SNo y2 → g (v + x2) y2 = g v y2 + g x2 y2)
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)))
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)
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)
Hypothesis H7 : SNo x
Hypothesis H8 : SNo y
Hypothesis H9 : SNo z
Hypothesis H10 : (∀v : set, v ∈ SNoS_ (SNoLev x) → g v (g y z) = g (g v y) z)
Hypothesis H11 : (∀v : set, v ∈ SNoS_ (SNoLev y) → g x (g v z) = g (g x v) z)
Hypothesis H12 : (∀v : set, v ∈ SNoS_ (SNoLev z) → g x (g y v) = g (g x y) v)
Hypothesis H13 : (∀v : set, v ∈ SNoS_ (SNoLev x) → (∀x2 : set, x2 ∈ SNoS_ (SNoLev y) → g v (g x2 z) = g (g v x2) z))
Hypothesis H14 : (∀v : set, v ∈ SNoS_ (SNoLev x) → (∀x2 : set, x2 ∈ SNoS_ (SNoLev z) → g v (g y x2) = g (g v y) x2))
Hypothesis H15 : (∀v : set, v ∈ SNoS_ (SNoLev y) → (∀x2 : set, x2 ∈ SNoS_ (SNoLev z) → g x (g v x2) = g (g x v) x2))
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)))
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))
Hypothesis H18 : u ∈ w
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
Variable x : set
Variable y : set
Variable z : set
Variable w : set
Variable u : set
Variable v : set
Variable x2 : set
Hypothesis H0 : SNo x
Hypothesis H1 : SNo y
Hypothesis H2 : SNo z
Hypothesis H3 : (∀y2 : set, y2 ∈ SNoS_ (SNoLev x) → y2 * y * z = (y2 * y) * z)
Hypothesis H4 : (∀y2 : set, y2 ∈ SNoS_ (SNoLev y) → x * y2 * z = (x * y2) * z)
Hypothesis H5 : (∀y2 : set, y2 ∈ SNoS_ (SNoLev z) → x * y * y2 = (x * y) * y2)
Hypothesis H6 : (∀y2 : set, y2 ∈ SNoS_ (SNoLev x) → (∀z2 : set, z2 ∈ SNoS_ (SNoLev y) → y2 * z2 * z = (y2 * z2) * z))
Hypothesis H7 : (∀y2 : set, y2 ∈ SNoS_ (SNoLev x) → (∀z2 : set, z2 ∈ SNoS_ (SNoLev z) → y2 * y * z2 = (y2 * y) * z2))
Hypothesis H8 : (∀y2 : set, y2 ∈ SNoS_ (SNoLev y) → (∀z2 : set, z2 ∈ SNoS_ (SNoLev z) → x * y2 * z2 = (x * y2) * z2))
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)))
Hypothesis H10 : SNo (x * y)
Hypothesis H11 : SNoCutP w u
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))
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))
Hypothesis H14 : x * y * z = SNoCut w u
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))
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))
Hypothesis H18 : (x * y) * z = SNoCut v x2
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)))
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)))
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)
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)
Hypothesis H23 : (∀y2 : set, y2 ∈ SNoS_ (SNoLev x) → (y2 * y) * z = y2 * y * z)
Hypothesis H24 : (∀y2 : set, y2 ∈ SNoS_ (SNoLev y) → (x * y2) * z = x * y2 * z)
Hypothesis H25 : (∀y2 : set, y2 ∈ SNoS_ (SNoLev z) → (x * y) * y2 = x * y * y2)
Hypothesis H26 : (∀y2 : set, y2 ∈ SNoS_ (SNoLev y) → (∀z2 : set, z2 ∈ SNoS_ (SNoLev x) → (z2 * y2) * z = z2 * y2 * z))
Hypothesis H27 : (∀y2 : set, y2 ∈ SNoS_ (SNoLev z) → (∀z2 : set, z2 ∈ SNoS_ (SNoLev x) → (z2 * y) * y2 = z2 * y * y2))
Hypothesis H28 : (∀y2 : set, y2 ∈ SNoS_ (SNoLev z) → (∀z2 : set, z2 ∈ SNoS_ (SNoLev y) → (x * z2) * y2 = x * z2 * y2))
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
Variable x : set
Variable y : set
Variable z : set
Variable w : set
Variable u : set
Variable v : set
Variable x2 : set
Hypothesis H0 : SNo x
Hypothesis H1 : SNo y
Hypothesis H2 : SNo z
Hypothesis H3 : (∀y2 : set, y2 ∈ SNoS_ (SNoLev x) → y2 * y * z = (y2 * y) * z)
Hypothesis H4 : (∀y2 : set, y2 ∈ SNoS_ (SNoLev y) → x * y2 * z = (x * y2) * z)
Hypothesis H5 : (∀y2 : set, y2 ∈ SNoS_ (SNoLev z) → x * y * y2 = (x * y) * y2)
Hypothesis H6 : (∀y2 : set, y2 ∈ SNoS_ (SNoLev x) → (∀z2 : set, z2 ∈ SNoS_ (SNoLev y) → y2 * z2 * z = (y2 * z2) * z))
Hypothesis H7 : (∀y2 : set, y2 ∈ SNoS_ (SNoLev x) → (∀z2 : set, z2 ∈ SNoS_ (SNoLev z) → y2 * y * z2 = (y2 * y) * z2))
Hypothesis H8 : (∀y2 : set, y2 ∈ SNoS_ (SNoLev y) → (∀z2 : set, z2 ∈ SNoS_ (SNoLev z) → x * y2 * z2 = (x * y2) * z2))
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)))
Hypothesis H10 : SNo (x * y)
Hypothesis H11 : SNoCutP w u
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))
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))
Hypothesis H14 : x * y * z = SNoCut w u
Hypothesis H15 : SNoCutP v x2
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))
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))
Hypothesis H18 : (x * y) * z = SNoCut v x2
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)))
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)))
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)
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)
Hypothesis H23 : (∀y2 : set, y2 ∈ SNoS_ (SNoLev x) → (y2 * y) * z = y2 * y * z)
Hypothesis H24 : (∀y2 : set, y2 ∈ SNoS_ (SNoLev y) → (x * y2) * z = x * y2 * z)
Hypothesis H26 : (∀y2 : set, y2 ∈ SNoS_ (SNoLev y) → (∀z2 : set, z2 ∈ SNoS_ (SNoLev x) → (z2 * y2) * z = z2 * y2 * z))
Hypothesis H27 : (∀y2 : set, y2 ∈ SNoS_ (SNoLev z) → (∀z2 : set, z2 ∈ SNoS_ (SNoLev x) → (z2 * y) * y2 = z2 * y * y2))
Hypothesis H28 : (∀y2 : set, y2 ∈ SNoS_ (SNoLev z) → (∀z2 : set, z2 ∈ SNoS_ (SNoLev y) → (x * z2) * y2 = x * z2 * y2))
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
Variable x : set
Variable y : set
Variable z : set
Variable w : set
Variable u : set
Variable v : set
Variable x2 : set
Hypothesis H0 : SNo x
Hypothesis H1 : SNo y
Hypothesis H2 : SNo z
Hypothesis H3 : (∀y2 : set, y2 ∈ SNoS_ (SNoLev x) → y2 * y * z = (y2 * y) * z)
Hypothesis H4 : (∀y2 : set, y2 ∈ SNoS_ (SNoLev y) → x * y2 * z = (x * y2) * z)
Hypothesis H6 : (∀y2 : set, y2 ∈ SNoS_ (SNoLev x) → (∀z2 : set, z2 ∈ SNoS_ (SNoLev y) → y2 * z2 * z = (y2 * z2) * z))
Hypothesis H7 : (∀y2 : set, y2 ∈ SNoS_ (SNoLev x) → (∀z2 : set, z2 ∈ SNoS_ (SNoLev z) → y2 * y * z2 = (y2 * y) * z2))
Hypothesis H8 : (∀y2 : set, y2 ∈ SNoS_ (SNoLev y) → (∀z2 : set, z2 ∈ SNoS_ (SNoLev z) → x * y2 * z2 = (x * y2) * z2))
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)))
Hypothesis H10 : SNo (x * y)
Hypothesis H11 : SNoCutP w u
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))
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))
Hypothesis H14 : x * y * z = SNoCut w u
Hypothesis H15 : SNoCutP v x2
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))
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))
Hypothesis H18 : (x * y) * z = SNoCut v x2
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)))
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)))
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)
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)
Hypothesis H23 : (∀y2 : set, y2 ∈ SNoS_ (SNoLev x) → (y2 * y) * z = y2 * y * z)
Hypothesis H24 : (∀y2 : set, y2 ∈ SNoS_ (SNoLev y) → (x * y2) * z = x * y2 * z)
Hypothesis H25 : (∀y2 : set, y2 ∈ SNoS_ (SNoLev z) → (x * y) * y2 = x * y * y2)
Hypothesis H26 : (∀y2 : set, y2 ∈ SNoS_ (SNoLev y) → (∀z2 : set, z2 ∈ SNoS_ (SNoLev x) → (z2 * y2) * z = z2 * y2 * z))
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
Variable x : set
Variable y : set
Variable z : set
Variable w : set
Variable u : set
Variable v : set
Variable x2 : set
Hypothesis H0 : SNo x
Hypothesis H1 : SNo y
Hypothesis H2 : SNo z
Hypothesis H3 : (∀y2 : set, y2 ∈ SNoS_ (SNoLev x) → y2 * y * z = (y2 * y) * z)
Hypothesis H4 : (∀y2 : set, y2 ∈ SNoS_ (SNoLev y) → x * y2 * z = (x * y2) * z)
Hypothesis H5 : (∀y2 : set, y2 ∈ SNoS_ (SNoLev z) → x * y * y2 = (x * y) * y2)
Hypothesis H6 : (∀y2 : set, y2 ∈ SNoS_ (SNoLev x) → (∀z2 : set, z2 ∈ SNoS_ (SNoLev y) → y2 * z2 * z = (y2 * z2) * z))
Hypothesis H7 : (∀y2 : set, y2 ∈ SNoS_ (SNoLev x) → (∀z2 : set, z2 ∈ SNoS_ (SNoLev z) → y2 * y * z2 = (y2 * y) * z2))
Hypothesis H8 : (∀y2 : set, y2 ∈ SNoS_ (SNoLev y) → (∀z2 : set, z2 ∈ SNoS_ (SNoLev z) → x * y2 * z2 = (x * y2) * z2))
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)))
Hypothesis H10 : SNo (x * y)
Hypothesis H11 : SNoCutP w u
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))
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))
Hypothesis H14 : x * y * z = SNoCut w u
Hypothesis H15 : SNoCutP v x2
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))
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))
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)))
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)))
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)
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)
Hypothesis H23 : (∀y2 : set, y2 ∈ SNoS_ (SNoLev x) → (y2 * y) * z = y2 * y * z)
Hypothesis H24 : (∀y2 : set, y2 ∈ SNoS_ (SNoLev y) → (x * y2) * z = x * y2 * z)
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
Variable x : set
Variable y : set
Variable z : set
Variable w : set
Variable u : set
Variable v : set
Variable x2 : set
Hypothesis H0 : SNo x
Hypothesis H1 : SNo y
Hypothesis H2 : SNo z
Hypothesis H3 : (∀y2 : set, y2 ∈ SNoS_ (SNoLev x) → y2 * y * z = (y2 * y) * z)
Hypothesis H4 : (∀y2 : set, y2 ∈ SNoS_ (SNoLev y) → x * y2 * z = (x * y2) * z)
Hypothesis H5 : (∀y2 : set, y2 ∈ SNoS_ (SNoLev z) → x * y * y2 = (x * y) * y2)
Hypothesis H6 : (∀y2 : set, y2 ∈ SNoS_ (SNoLev x) → (∀z2 : set, z2 ∈ SNoS_ (SNoLev y) → y2 * z2 * z = (y2 * z2) * z))
Hypothesis H7 : (∀y2 : set, y2 ∈ SNoS_ (SNoLev x) → (∀z2 : set, z2 ∈ SNoS_ (SNoLev z) → y2 * y * z2 = (y2 * y) * z2))
Hypothesis H8 : (∀y2 : set, y2 ∈ SNoS_ (SNoLev y) → (∀z2 : set, z2 ∈ SNoS_ (SNoLev z) → x * y2 * z2 = (x * y2) * z2))
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)))
Hypothesis H10 : SNo (x * y)
Hypothesis H11 : SNoCutP w u
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))
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))
Hypothesis H14 : x * y * z = SNoCut w u
Hypothesis H15 : SNoCutP v x2
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))
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))
Hypothesis H18 : (x * y) * z = SNoCut v x2
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)))
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)
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)
Hypothesis H23 : (∀y2 : set, y2 ∈ SNoS_ (SNoLev x) → (y2 * y) * z = y2 * y * z)
Hypothesis H24 : (∀y2 : set, y2 ∈ SNoS_ (SNoLev y) → (x * y2) * z = x * y2 * z)
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
Variable x : set
Variable y : set
Variable z : set
Variable w : set
Variable u : set
Variable v : set
Variable x2 : set
Hypothesis H0 : SNo x
Hypothesis H1 : SNo y
Hypothesis H2 : SNo z
Hypothesis H3 : (∀y2 : set, y2 ∈ SNoS_ (SNoLev x) → y2 * y * z = (y2 * y) * z)
Hypothesis H4 : (∀y2 : set, y2 ∈ SNoS_ (SNoLev y) → x * y2 * z = (x * y2) * z)
Hypothesis H5 : (∀y2 : set, y2 ∈ SNoS_ (SNoLev z) → x * y * y2 = (x * y) * y2)
Hypothesis H6 : (∀y2 : set, y2 ∈ SNoS_ (SNoLev x) → (∀z2 : set, z2 ∈ SNoS_ (SNoLev y) → y2 * z2 * z = (y2 * z2) * z))
Hypothesis H7 : (∀y2 : set, y2 ∈ SNoS_ (SNoLev x) → (∀z2 : set, z2 ∈ SNoS_ (SNoLev z) → y2 * y * z2 = (y2 * y) * z2))
Hypothesis H8 : (∀y2 : set, y2 ∈ SNoS_ (SNoLev y) → (∀z2 : set, z2 ∈ SNoS_ (SNoLev z) → x * y2 * z2 = (x * y2) * z2))
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)))
Hypothesis H10 : SNo (x * y)
Hypothesis H11 : SNoCutP w u
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))
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))
Hypothesis H14 : x * y * z = SNoCut w u
Hypothesis H15 : SNoCutP v x2
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))
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))
Hypothesis H18 : (x * y) * z = SNoCut v x2
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)))
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)))
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)
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)
Hypothesis H24 : (∀y2 : set, y2 ∈ SNoS_ (SNoLev y) → (x * y2) * z = x * y2 * z)
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
Variable x : set
Variable y : set
Variable z : set
Variable w : set
Variable u : set
Variable v : set
Variable x2 : set
Hypothesis H0 : SNo x
Hypothesis H1 : SNo y
Hypothesis H2 : SNo z
Hypothesis H3 : (∀y2 : set, y2 ∈ SNoS_ (SNoLev x) → y2 * y * z = (y2 * y) * z)
Hypothesis H4 : (∀y2 : set, y2 ∈ SNoS_ (SNoLev y) → x * y2 * z = (x * y2) * z)
Hypothesis H5 : (∀y2 : set, y2 ∈ SNoS_ (SNoLev z) → x * y * y2 = (x * y) * y2)
Hypothesis H6 : (∀y2 : set, y2 ∈ SNoS_ (SNoLev x) → (∀z2 : set, z2 ∈ SNoS_ (SNoLev y) → y2 * z2 * z = (y2 * z2) * z))
Hypothesis H7 : (∀y2 : set, y2 ∈ SNoS_ (SNoLev x) → (∀z2 : set, z2 ∈ SNoS_ (SNoLev z) → y2 * y * z2 = (y2 * y) * z2))
Hypothesis H8 : (∀y2 : set, y2 ∈ SNoS_ (SNoLev y) → (∀z2 : set, z2 ∈ SNoS_ (SNoLev z) → x * y2 * z2 = (x * y2) * z2))
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)))
Hypothesis H10 : SNo (x * y)
Hypothesis H11 : SNoCutP w u
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))
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))
Hypothesis H14 : x * y * z = SNoCut w u
Hypothesis H15 : SNoCutP v x2
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))
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))
Hypothesis H18 : (x * y) * z = SNoCut v x2
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)))
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)
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)
Hypothesis H23 : (∀y2 : set, y2 ∈ SNoS_ (SNoLev x) → (y2 * y) * z = y2 * y * z)
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
Variable x : set
Variable y : set
Variable z : set
Variable w : set
Variable u : set
Variable v : set
Variable x2 : set
Hypothesis H0 : SNo x
Hypothesis H1 : SNo y
Hypothesis H2 : SNo z
Hypothesis H3 : (∀y2 : set, y2 ∈ SNoS_ (SNoLev x) → y2 * y * z = (y2 * y) * z)
Hypothesis H4 : (∀y2 : set, y2 ∈ SNoS_ (SNoLev y) → x * y2 * z = (x * y2) * z)
Hypothesis H5 : (∀y2 : set, y2 ∈ SNoS_ (SNoLev z) → x * y * y2 = (x * y) * y2)
Hypothesis H7 : (∀y2 : set, y2 ∈ SNoS_ (SNoLev x) → (∀z2 : set, z2 ∈ SNoS_ (SNoLev z) → y2 * y * z2 = (y2 * y) * z2))
Hypothesis H8 : (∀y2 : set, y2 ∈ SNoS_ (SNoLev y) → (∀z2 : set, z2 ∈ SNoS_ (SNoLev z) → x * y2 * z2 = (x * y2) * z2))
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)))
Hypothesis H10 : SNo (x * y)
Hypothesis H11 : SNoCutP w u
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))
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))
Hypothesis H14 : x * y * z = SNoCut w u
Hypothesis H15 : SNoCutP v x2
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))
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))
Hypothesis H18 : (x * y) * z = SNoCut v x2
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)))
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)))
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)
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)
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
Variable x : set
Variable y : set
Variable z : set
Variable w : set
Variable u : set
Variable v : set
Variable x2 : set
Hypothesis H1 : SNo y
Hypothesis H2 : SNo z
Hypothesis H3 : (∀y2 : set, y2 ∈ SNoS_ (SNoLev x) → y2 * y * z = (y2 * y) * z)
Hypothesis H4 : (∀y2 : set, y2 ∈ SNoS_ (SNoLev y) → x * y2 * z = (x * y2) * z)
Hypothesis H5 : (∀y2 : set, y2 ∈ SNoS_ (SNoLev z) → x * y * y2 = (x * y) * y2)
Hypothesis H6 : (∀y2 : set, y2 ∈ SNoS_ (SNoLev x) → (∀z2 : set, z2 ∈ SNoS_ (SNoLev y) → y2 * z2 * z = (y2 * z2) * z))
Hypothesis H7 : (∀y2 : set, y2 ∈ SNoS_ (SNoLev x) → (∀z2 : set, z2 ∈ SNoS_ (SNoLev z) → y2 * y * z2 = (y2 * y) * z2))
Hypothesis H8 : (∀y2 : set, y2 ∈ SNoS_ (SNoLev y) → (∀z2 : set, z2 ∈ SNoS_ (SNoLev z) → x * y2 * z2 = (x * y2) * z2))
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)))
Hypothesis H10 : SNo (x * y)
Hypothesis H11 : SNoCutP w u
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))
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))
Hypothesis H14 : x * y * z = SNoCut w u
Hypothesis H15 : SNoCutP v x2
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))
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))
Hypothesis H18 : (x * y) * z = SNoCut v x2
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)))
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)))
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
Variable x : set
Variable y : set
Variable z : set
Variable w : set
Variable u : set
Variable v : set
Variable x2 : set
Hypothesis H0 : SNo x
Hypothesis H1 : SNo y
Hypothesis H2 : SNo z
Hypothesis H3 : (∀y2 : set, y2 ∈ SNoS_ (SNoLev x) → y2 * y * z = (y2 * y) * z)
Hypothesis H4 : (∀y2 : set, y2 ∈ SNoS_ (SNoLev y) → x * y2 * z = (x * y2) * z)
Hypothesis H5 : (∀y2 : set, y2 ∈ SNoS_ (SNoLev z) → x * y * y2 = (x * y) * y2)
Hypothesis H6 : (∀y2 : set, y2 ∈ SNoS_ (SNoLev x) → (∀z2 : set, z2 ∈ SNoS_ (SNoLev y) → y2 * z2 * z = (y2 * z2) * z))
Hypothesis H7 : (∀y2 : set, y2 ∈ SNoS_ (SNoLev x) → (∀z2 : set, z2 ∈ SNoS_ (SNoLev z) → y2 * y * z2 = (y2 * y) * z2))
Hypothesis H8 : (∀y2 : set, y2 ∈ SNoS_ (SNoLev y) → (∀z2 : set, z2 ∈ SNoS_ (SNoLev z) → x * y2 * z2 = (x * y2) * z2))
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)))
Hypothesis H10 : SNo (x * y)
Hypothesis H11 : SNoCutP w u
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))
Hypothesis H14 : x * y * z = SNoCut w u
Hypothesis H15 : SNoCutP v x2
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))
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))
Hypothesis H18 : (x * y) * z = SNoCut v x2
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)))
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
Variable x : set
Variable y : set
Variable z : set
Variable w : set
Variable u : set
Variable v : set
Variable x2 : set
Hypothesis H0 : SNo x
Hypothesis H1 : SNo y
Hypothesis H2 : SNo z
Hypothesis H3 : (∀y2 : set, y2 ∈ SNoS_ (SNoLev x) → y2 * y * z = (y2 * y) * z)
Hypothesis H4 : (∀y2 : set, y2 ∈ SNoS_ (SNoLev y) → x * y2 * z = (x * y2) * z)
Hypothesis H5 : (∀y2 : set, y2 ∈ SNoS_ (SNoLev z) → x * y * y2 = (x * y) * y2)
Hypothesis H6 : (∀y2 : set, y2 ∈ SNoS_ (SNoLev x) → (∀z2 : set, z2 ∈ SNoS_ (SNoLev y) → y2 * z2 * z = (y2 * z2) * z))
Hypothesis H7 : (∀y2 : set, y2 ∈ SNoS_ (SNoLev x) → (∀z2 : set, z2 ∈ SNoS_ (SNoLev z) → y2 * y * z2 = (y2 * y) * z2))
Hypothesis H8 : (∀y2 : set, y2 ∈ SNoS_ (SNoLev y) → (∀z2 : set, z2 ∈ SNoS_ (SNoLev z) → x * y2 * z2 = (x * y2) * z2))
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)))
Hypothesis H10 : SNo (x * y)
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))
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))
Hypothesis H14 : x * y * z = SNoCut w u
Hypothesis H15 : SNoCutP v x2
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))
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))
Hypothesis H18 : (x * y) * z = SNoCut v x2
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
Variable x : set
Variable y : set
Variable z : set
Hypothesis H0 : SNo x
Hypothesis H1 : Empty ≤ x
Hypothesis H3 : SNo z
Hypothesis H4 : y ≤ z
Hypothesis H5 : Empty * z + x * y = x * y
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
Variable x : set
Variable y : set
Variable z : set
Hypothesis H0 : SNo x
Hypothesis H1 : x < Empty
Hypothesis H2 : SNo y
Hypothesis H4 : z < y
Hypothesis H5 : x * y + Empty * z = x * y
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
Variable x : set
Variable y : set
Variable z : set
Hypothesis H1 : x < Empty
Hypothesis H2 : SNo y
Hypothesis H3 : SNo z
Hypothesis H4 : z < y
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
Variable x : set
Variable y : set
Variable z : set
Variable w : set
Hypothesis H0 : SNo x
Hypothesis H2 : SNo z
Hypothesis H3 : SNo w
Hypothesis H4 : SNo (- y)
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
Variable x : set
Variable y : set
Variable z : set
Variable w : set
Hypothesis H1 : SNo y
Hypothesis H2 : SNo z
Hypothesis H3 : SNo w
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
Variable x : set
Variable y : set
Hypothesis H1 : x ∈ ω
Hypothesis H2 : SNo (eps_ (ordsucc x))
Hypothesis H3 : SNo y
Hypothesis H4 : SNoLev y ∈ ordsucc (ordsucc x)
Hypothesis H5 : y < eps_ (ordsucc x)
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
Variable x : set
Variable y : set
Hypothesis H0 : nat_p x
Hypothesis H2 : SNo (eps_ (ordsucc x))
Hypothesis H3 : SNo y
Hypothesis H4 : SNoLev y ∈ ordsucc (ordsucc x)
Hypothesis H5 : y < eps_ (ordsucc x)
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
Variable x : set
Hypothesis H0 : nat_p x
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
Variable x : set
Variable y : set
Variable z : set
Hypothesis H0 : SNo x
Hypothesis H2 : SNo z
Hypothesis H3 : x + x = y + z
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
Variable x : set
Variable y : set
Hypothesis H0 : SNo x
Hypothesis H2 : nat_p y
Hypothesis H3 : ordsucc Empty ≤ exp_SNo_nat x y
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
Variable x : set
Variable y : set
Hypothesis H1 : y ∈ ω
Hypothesis H2 : x + y ∈ ω
Hypothesis H3 : nat_p (x + y)
Hypothesis H4 : SNo (eps_ x)
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
Variable x : set
Variable y : set
Hypothesis H1 : y ∈ ω
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
Variable x : set
Variable f : (set → set)
Variable f2 : (set → set)
Variable y : set
Variable z : set
Hypothesis H1 : (∀w : set, x ∈ w → f2 w = f (binintersect w (SNoElts_ x)))
Hypothesis H2 : f z = y
Hypothesis H3 : SNoLev z = x
Hypothesis H4 : SNo z
Hypothesis H5 : SNoLev (SNo_extend1 z) = ordsucc x
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
Variable x : set
Variable f : (set → set)
Variable f2 : (set → set)
Variable y : set
Variable z : set
Hypothesis H0 : nat_p x
Hypothesis H2 : f z = y
Hypothesis H3 : SNoLev z = x
Hypothesis H4 : SNo z
Hypothesis H5 : SNoLev (SNo_extend1 z) = ordsucc x
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
Variable x : set
Variable f : (set → set)
Variable f2 : (set → set)
Variable y : set
Hypothesis H0 : nat_p x
Hypothesis H1 : nat_p (exp_SNo_nat (ordsucc (ordsucc Empty)) x)
Hypothesis H2 : SNo (exp_SNo_nat (ordsucc (ordsucc Empty)) x)
Hypothesis H3 : nat_p (exp_SNo_nat (ordsucc (ordsucc Empty)) x + exp_SNo_nat (ordsucc (ordsucc Empty)) x)
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))
Hypothesis H5 : (∀z : set, x ∈ z → f2 z = f (binintersect z (SNoElts_ x)))
Hypothesis H6 : (∀z : set, nIn x z → f2 z = exp_SNo_nat (ordsucc (ordsucc Empty)) x + f (binintersect z (SNoElts_ x)))
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
Variable x : set
Variable f : (set → set)
Variable f2 : (set → set)
Variable y : set
Variable z : set
Hypothesis H0 : SNo (exp_SNo_nat (ordsucc (ordsucc Empty)) x)
Hypothesis H2 : (∀w : set, x ∈ w → f2 w = f (binintersect w (SNoElts_ x)))
Hypothesis H3 : (∀w : set, nIn x w → f2 w = exp_SNo_nat (ordsucc (ordsucc Empty)) x + f (binintersect w (SNoElts_ x)))
Hypothesis H4 : SNo y
Hypothesis H5 : SNoLev y = ordsucc x
Hypothesis H6 : binintersect y (SNoElts_ x) ∈ Sep (SNoS_ ω) (λw : set ⇒ SNoLev w = x)
Hypothesis H7 : nat_p (f (binintersect y (SNoElts_ x)))
Hypothesis H8 : SNo (f (binintersect y (SNoElts_ x)))
Hypothesis H9 : f (binintersect y (SNoElts_ x)) < exp_SNo_nat (ordsucc (ordsucc Empty)) x
Hypothesis H10 : SNo z
Hypothesis H11 : SNoLev z = ordsucc x
Hypothesis H12 : binintersect z (SNoElts_ x) ∈ Sep (SNoS_ ω) (λw : set ⇒ SNoLev w = x)
Hypothesis H13 : nat_p (f (binintersect z (SNoElts_ x)))
Hypothesis H14 : SNo (f (binintersect z (SNoElts_ x)))
Hypothesis H15 : f (binintersect z (SNoElts_ x)) < exp_SNo_nat (ordsucc (ordsucc Empty)) x
Hypothesis H16 : x ∈ SNoLev y
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
Variable x : set
Variable f : (set → set)
Variable f2 : (set → set)
Variable y : set
Variable z : set
Hypothesis H0 : SNo (exp_SNo_nat (ordsucc (ordsucc Empty)) x)
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))
Hypothesis H2 : (∀w : set, x ∈ w → f2 w = f (binintersect w (SNoElts_ x)))
Hypothesis H3 : (∀w : set, nIn x w → f2 w = exp_SNo_nat (ordsucc (ordsucc Empty)) x + f (binintersect w (SNoElts_ x)))
Hypothesis H5 : SNoLev y = ordsucc x
Hypothesis H6 : binintersect y (SNoElts_ x) ∈ Sep (SNoS_ ω) (λw : set ⇒ SNoLev w = x)
Hypothesis H7 : nat_p (f (binintersect y (SNoElts_ x)))
Hypothesis H8 : SNo (f (binintersect y (SNoElts_ x)))
Hypothesis H9 : f (binintersect y (SNoElts_ x)) < exp_SNo_nat (ordsucc (ordsucc Empty)) x
Hypothesis H10 : SNo z
Hypothesis H11 : SNoLev z = ordsucc x
Hypothesis H12 : binintersect z (SNoElts_ x) ∈ Sep (SNoS_ ω) (λw : set ⇒ SNoLev w = x)
Hypothesis H13 : nat_p (f (binintersect z (SNoElts_ x)))
Hypothesis H14 : SNo (f (binintersect z (SNoElts_ x)))
Hypothesis H15 : f (binintersect z (SNoElts_ x)) < exp_SNo_nat (ordsucc (ordsucc Empty)) x
Hypothesis H16 : x ∈ SNoLev y
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
Variable x : set
Variable f : (set → set)
Variable f2 : (set → set)
Variable y : set
Variable z : set
Hypothesis H0 : SNo (exp_SNo_nat (ordsucc (ordsucc Empty)) x)
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))
Hypothesis H2 : (∀w : set, x ∈ w → f2 w = f (binintersect w (SNoElts_ x)))
Hypothesis H3 : (∀w : set, nIn x w → f2 w = exp_SNo_nat (ordsucc (ordsucc Empty)) x + f (binintersect w (SNoElts_ x)))
Hypothesis H4 : SNo y
Hypothesis H5 : SNoLev y = ordsucc x
Hypothesis H7 : nat_p (f (binintersect y (SNoElts_ x)))
Hypothesis H8 : SNo (f (binintersect y (SNoElts_ x)))
Hypothesis H9 : f (binintersect y (SNoElts_ x)) < exp_SNo_nat (ordsucc (ordsucc Empty)) x
Hypothesis H10 : SNo z
Hypothesis H11 : SNoLev z = ordsucc x
Hypothesis H12 : binintersect z (SNoElts_ x) ∈ Sep (SNoS_ ω) (λw : set ⇒ SNoLev w = x)
Hypothesis H13 : nat_p (f (binintersect z (SNoElts_ x)))
Hypothesis H14 : SNo (f (binintersect z (SNoElts_ x)))
Hypothesis H15 : f (binintersect z (SNoElts_ x)) < exp_SNo_nat (ordsucc (ordsucc Empty)) x
Hypothesis H16 : x ∈ SNoLev y
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
Variable x : set
Variable f : (set → set)
Variable f2 : (set → set)
Variable y : set
Variable z : set
Hypothesis H0 : SNo (exp_SNo_nat (ordsucc (ordsucc Empty)) x)
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))
Hypothesis H2 : (∀w : set, x ∈ w → f2 w = f (binintersect w (SNoElts_ x)))
Hypothesis H3 : (∀w : set, nIn x w → f2 w = exp_SNo_nat (ordsucc (ordsucc Empty)) x + f (binintersect w (SNoElts_ x)))
Hypothesis H4 : SNo y
Hypothesis H5 : SNoLev y = ordsucc x
Hypothesis H6 : binintersect y (SNoElts_ x) ∈ Sep (SNoS_ ω) (λw : set ⇒ SNoLev w = x)
Hypothesis H7 : nat_p (f (binintersect y (SNoElts_ x)))
Hypothesis H8 : SNo (f (binintersect y (SNoElts_ x)))
Hypothesis H9 : f (binintersect y (SNoElts_ x)) < exp_SNo_nat (ordsucc (ordsucc Empty)) x
Hypothesis H10 : SNo z
Hypothesis H11 : SNoLev z = ordsucc x
Hypothesis H12 : binintersect z (SNoElts_ x) ∈ Sep (SNoS_ ω) (λw : set ⇒ SNoLev w = x)
Hypothesis H13 : nat_p (f (binintersect z (SNoElts_ x)))
Hypothesis H15 : f (binintersect z (SNoElts_ x)) < exp_SNo_nat (ordsucc (ordsucc Empty)) x
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
Variable x : set
Variable f : (set → set)
Hypothesis H1 : nat_p (exp_SNo_nat (ordsucc (ordsucc Empty)) x)
Hypothesis H2 : ordinal (exp_SNo_nat (ordsucc (ordsucc Empty)) x)
Hypothesis H3 : SNo (exp_SNo_nat (ordsucc (ordsucc Empty)) x)
Hypothesis H4 : nat_p (exp_SNo_nat (ordsucc (ordsucc Empty)) x + exp_SNo_nat (ordsucc (ordsucc Empty)) x)
Hypothesis H5 : ordinal (exp_SNo_nat (ordsucc (ordsucc Empty)) x + exp_SNo_nat (ordsucc (ordsucc Empty)) x)
Hypothesis H6 : SNo (exp_SNo_nat (ordsucc (ordsucc Empty)) x + exp_SNo_nat (ordsucc (ordsucc Empty)) x)
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)
Hypothesis H8 : exp_SNo_nat (ordsucc (ordsucc Empty)) x < exp_SNo_nat (ordsucc (ordsucc Empty)) x + exp_SNo_nat (ordsucc (ordsucc Empty)) x
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))
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))
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))
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))
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
Variable x : set
Variable f : (set → set)
Hypothesis H0 : nat_p x
Hypothesis H1 : nat_p (exp_SNo_nat (ordsucc (ordsucc Empty)) x)
Hypothesis H2 : ordinal (exp_SNo_nat (ordsucc (ordsucc Empty)) x)
Hypothesis H4 : nat_p (exp_SNo_nat (ordsucc (ordsucc Empty)) x + exp_SNo_nat (ordsucc (ordsucc Empty)) x)
Hypothesis H5 : ordinal (exp_SNo_nat (ordsucc (ordsucc Empty)) x + exp_SNo_nat (ordsucc (ordsucc Empty)) x)
Hypothesis H6 : SNo (exp_SNo_nat (ordsucc (ordsucc Empty)) x + exp_SNo_nat (ordsucc (ordsucc Empty)) x)
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)
Hypothesis H8 : exp_SNo_nat (ordsucc (ordsucc Empty)) x < exp_SNo_nat (ordsucc (ordsucc Empty)) x + exp_SNo_nat (ordsucc (ordsucc Empty)) x
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))
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))
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))
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))
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
Variable x : set
Variable f : (set → set)
Hypothesis H0 : nat_p x
Hypothesis H1 : nat_p (exp_SNo_nat (ordsucc (ordsucc Empty)) x)
Hypothesis H2 : ordinal (exp_SNo_nat (ordsucc (ordsucc Empty)) x)
Hypothesis H3 : SNo (exp_SNo_nat (ordsucc (ordsucc Empty)) x)
Hypothesis H4 : nat_p (exp_SNo_nat (ordsucc (ordsucc Empty)) x + exp_SNo_nat (ordsucc (ordsucc Empty)) x)
Hypothesis H5 : ordinal (exp_SNo_nat (ordsucc (ordsucc Empty)) x + exp_SNo_nat (ordsucc (ordsucc Empty)) x)
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)
Hypothesis H8 : exp_SNo_nat (ordsucc (ordsucc Empty)) x < exp_SNo_nat (ordsucc (ordsucc Empty)) x + exp_SNo_nat (ordsucc (ordsucc Empty)) x
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))
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))
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))
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))
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
Variable x : set
Hypothesis H0 : nat_p x
Hypothesis H2 : nat_p (exp_SNo_nat (ordsucc (ordsucc Empty)) x)
Hypothesis H3 : ordinal (exp_SNo_nat (ordsucc (ordsucc Empty)) x)
Hypothesis H4 : SNo (exp_SNo_nat (ordsucc (ordsucc Empty)) x)
Hypothesis H5 : nat_p (exp_SNo_nat (ordsucc (ordsucc Empty)) x + exp_SNo_nat (ordsucc (ordsucc Empty)) x)
Hypothesis H6 : ordinal (exp_SNo_nat (ordsucc (ordsucc Empty)) x + exp_SNo_nat (ordsucc (ordsucc Empty)) x)
Hypothesis H7 : SNo (exp_SNo_nat (ordsucc (ordsucc Empty)) x + exp_SNo_nat (ordsucc (ordsucc Empty)) x)
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
Variable x : set
Hypothesis H0 : nat_p x
Hypothesis H1 : equip (Sep (SNoS_ ω) (λy : set ⇒ SNoLev y = x)) (exp_SNo_nat (ordsucc (ordsucc Empty)) x)
Hypothesis H2 : nat_p (exp_SNo_nat (ordsucc (ordsucc Empty)) x)
Hypothesis H3 : ordinal (exp_SNo_nat (ordsucc (ordsucc Empty)) x)
Hypothesis H5 : nat_p (exp_SNo_nat (ordsucc (ordsucc Empty)) x + exp_SNo_nat (ordsucc (ordsucc Empty)) x)
Hypothesis H6 : ordinal (exp_SNo_nat (ordsucc (ordsucc Empty)) x + exp_SNo_nat (ordsucc (ordsucc Empty)) x)
Hypothesis H7 : SNo (exp_SNo_nat (ordsucc (ordsucc Empty)) x + exp_SNo_nat (ordsucc (ordsucc Empty)) x)
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
Variable x : set
Hypothesis H0 : nat_p x
Hypothesis H1 : equip (Sep (SNoS_ ω) (λy : set ⇒ SNoLev y = x)) (exp_SNo_nat (ordsucc (ordsucc Empty)) x)
Hypothesis H2 : nat_p (exp_SNo_nat (ordsucc (ordsucc Empty)) x)
Hypothesis H3 : ordinal (exp_SNo_nat (ordsucc (ordsucc Empty)) x)
Hypothesis H4 : SNo (exp_SNo_nat (ordsucc (ordsucc Empty)) x)
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
Variable x : set
Variable y : set
Hypothesis H0 : x ∈ ω
Hypothesis H2 : SNo x
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
Variable x : set
Variable y : set
Hypothesis H0 : x ∈ ω
Hypothesis H1 : SNo x
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
Variable x : set
Variable y : set
Hypothesis H0 : x ∈ ω
Hypothesis H1 : SNo x
Hypothesis H3 : ordinal y
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
Variable x : set
Variable y : set
Variable z : set
Hypothesis H1 : SNo y
Hypothesis H2 : SNo z
Hypothesis H3 : SNo (- y)
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
Variable x : set
Variable y : set
Variable z : set
Hypothesis H0 : SNo x
Hypothesis H1 : SNo y
Hypothesis H2 : (∀w : set, w ∈ SNoL x → SNo w → w ≤ y)
Hypothesis H3 : SNoLev y ∈ SNoLev x
Hypothesis H4 : SNo z
Hypothesis H5 : SNoLev z ∈ SNoLev y
Hypothesis H7 : z < x
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
Variable x : set
Variable y : set
Variable z : set
Variable w : set
Hypothesis H0 : SNo x
Hypothesis H1 : SNo z
Hypothesis H3 : x < z
Hypothesis H4 : w ∈ SNoR z
Hypothesis H5 : (y + w) < x + x
Hypothesis H6 : SNo w
Hypothesis H7 : SNoLev w ∈ SNoLev z
Hypothesis H8 : z < w
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
Variable x : set
Variable y : set
Variable z : set
Hypothesis H0 : SNo x
Hypothesis H2 : SNo y
Hypothesis H3 : SNo z
Hypothesis H4 : z < x
Hypothesis H5 : (x + x) < y + z
Hypothesis H6 : SNo (- x)
Hypothesis H7 : SNo (- y)
Hypothesis H8 : SNo (- z)
Hypothesis H9 : SNo (x + x)
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
Variable x : set
Variable y : set
Variable z : set
Hypothesis H0 : SNo x
Hypothesis H1 : SNo_min_of (SNoR x) y
Hypothesis H2 : SNo y
Hypothesis H3 : SNo z
Hypothesis H4 : z < x
Hypothesis H6 : SNo (- x)
Hypothesis H7 : SNo (- y)
Hypothesis H8 : SNo (- z)
Hypothesis H9 : SNo (x + x)
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
Variable x : set
Variable y : set
Variable z : set
Hypothesis H0 : SNo x
Hypothesis H1 : SNo_min_of (SNoR x) y
Hypothesis H2 : SNo y
Hypothesis H3 : SNo z
Hypothesis H4 : z < x
Hypothesis H5 : (x + x) < y + z
Hypothesis H7 : SNo (- y)
Hypothesis H8 : SNo (- z)
Hypothesis H9 : SNo (x + x)
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
Variable x : set
Variable y : set
Variable z : set
Hypothesis H0 : SNo x
Hypothesis H1 : SNo_min_of (SNoR x) y
Hypothesis H2 : SNo y
Hypothesis H3 : SNo z
Hypothesis H4 : z < x
Hypothesis H5 : (x + x) < y + z
Hypothesis H6 : SNo (- x)
Hypothesis H7 : SNo (- y)
Hypothesis H8 : SNo (- z)
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
Variable x : set
Variable y : set
Variable z : set
Hypothesis H0 : SNo x
Hypothesis H2 : SNo y
Hypothesis H3 : SNo z
Hypothesis H4 : z < x
Hypothesis H5 : (x + x) < y + z
Hypothesis H6 : SNo (- x)
Hypothesis H7 : SNo (- y)
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
Variable x : set
Variable y : set
Variable f : (set → set)
Hypothesis H0 : (∀z : set, (∀w : set, w ∈ z → SNo w) → equip z (ordsucc x) → (∃w : set, SNo_max_of z w))
Hypothesis H2 : (∀z : set, z ∈ ordsucc (ordsucc x) → f z ∈ y)
Hypothesis H3 : (∀z : set, z ∈ ordsucc (ordsucc x) → (∀w : set, w ∈ ordsucc (ordsucc x) → f z = f w → z = w))
Hypothesis H4 : (∀z : set, z ∈ y → (∃w : set, w ∈ ordsucc (ordsucc x) ∧ f w = z))
Hypothesis H5 : Subq (Repl (ordsucc x) f) y
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
Variable x : set
Hypothesis H1 : SNoL x ≠ Empty
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
Variable x : set
Hypothesis H1 : SNoR x ≠ Empty
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
Variable x : set
Variable y : set
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
Variable x : set
Variable y : set
Variable z : set
Variable w : set
Variable u : set
Variable v : set
Hypothesis H0 : z ∈ ω
Hypothesis H1 : SNo (eps_ z)
Hypothesis H2 : w ∈ int
Hypothesis H3 : x = eps_ z * w
Hypothesis H4 : SNo w
Hypothesis H5 : u ∈ ω
Hypothesis H6 : SNo (eps_ u)
Hypothesis H8 : SNo v
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
Variable x : set
Variable y : set
Variable z : set
Variable w : set
Variable u : set
Hypothesis H0 : z ∈ ω
Hypothesis H1 : SNo (eps_ z)
Hypothesis H3 : x = eps_ z * w
Hypothesis H4 : SNo w
Hypothesis H5 : u ∈ ω
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
Variable x : set
Variable y : set
Variable z : set
Variable w : set
Variable u : set
Variable v : set
Hypothesis H0 : z ∈ ω
Hypothesis H1 : SNo (eps_ z)
Hypothesis H2 : w ∈ int
Hypothesis H3 : x = eps_ z * w
Hypothesis H4 : u ∈ ω
Hypothesis H5 : SNo (eps_ u)
Hypothesis H6 : v ∈ int
Hypothesis H7 : y = eps_ u * v
Hypothesis H9 : exp_SNo_nat (ordsucc (ordsucc Empty)) u ∈ int
Hypothesis H10 : exp_SNo_nat (ordsucc (ordsucc Empty)) u * w ∈ int
Hypothesis H11 : exp_SNo_nat (ordsucc (ordsucc Empty)) z ∈ int
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
Variable x : set
Variable y : set
Variable z : set
Variable w : set
Variable u : set
Variable v : set
Hypothesis H0 : z ∈ ω
Hypothesis H1 : SNo (eps_ z)
Hypothesis H2 : w ∈ int
Hypothesis H3 : x = eps_ z * w
Hypothesis H4 : u ∈ ω
Hypothesis H5 : SNo (eps_ u)
Hypothesis H6 : v ∈ int
Hypothesis H7 : y = eps_ u * v
Hypothesis H8 : SNo (eps_ u * v)
Hypothesis H9 : exp_SNo_nat (ordsucc (ordsucc Empty)) u ∈ int
Hypothesis H10 : exp_SNo_nat (ordsucc (ordsucc Empty)) u * w ∈ int
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
Variable x : set
Variable y : set
Variable z : set
Variable w : set
Variable u : set
Variable v : set
Hypothesis H1 : SNo (eps_ z)
Hypothesis H2 : w ∈ int
Hypothesis H3 : x = eps_ z * w
Hypothesis H4 : u ∈ ω
Hypothesis H5 : SNo (eps_ u)
Hypothesis H6 : v ∈ int
Hypothesis H7 : y = eps_ u * v
Hypothesis H8 : SNo (eps_ u * v)
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
Variable x : set
Variable y : set
Variable z : set
Variable w : set
Variable u : set
Variable v : set
Hypothesis H0 : z ∈ ω
Hypothesis H1 : SNo (eps_ z)
Hypothesis H2 : w ∈ int
Hypothesis H3 : x = eps_ z * w
Hypothesis H4 : u ∈ ω
Hypothesis H5 : SNo (eps_ u)
Hypothesis H6 : v ∈ int
Hypothesis H8 : SNo (eps_ u * v)
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
Variable x : set
Variable y : set
Variable z : set
Variable w : set
Variable u : set
Variable v : set
Hypothesis H0 : z ∈ ω
Hypothesis H1 : SNo (eps_ z)
Hypothesis H2 : w ∈ int
Hypothesis H3 : x = eps_ z * w
Hypothesis H4 : u ∈ ω
Hypothesis H5 : SNo (eps_ u)
Hypothesis H7 : y = eps_ u * v
Hypothesis H8 : SNo v
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
Variable x : set
Variable y : set
Variable z : set
Variable w : set
Variable u : set
Variable v : set
Hypothesis H0 : z ∈ ω
Hypothesis H1 : SNo (eps_ z)
Hypothesis H2 : w ∈ int
Hypothesis H3 : x = eps_ z * w
Hypothesis H4 : u ∈ ω
Hypothesis H5 : SNo (eps_ u)
Hypothesis H6 : v ∈ int
Hypothesis H8 : SNo v
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
Variable x : set
Variable y : set
Variable z : set
Variable w : set
Variable u : set
Hypothesis H0 : z ∈ ω
Hypothesis H2 : w ∈ int
Hypothesis H3 : x = eps_ z * w
Hypothesis H4 : u ∈ ω
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
Variable x : set
Variable y : set
Hypothesis H0 : nat_p x
Hypothesis H2 : ¬ diadic_rational_p y
Hypothesis H3 : ordinal y
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
Variable x : set
Variable y : set
Variable z : set
Variable w : set
Variable u : set
Hypothesis H0 : nat_p x
Hypothesis H1 : (∀v : set, v ∈ x → (∀x2 : set, SNo x2 → SNoLev x2 = v → diadic_rational_p x2))
Hypothesis H3 : SNoLev y = x
Hypothesis H4 : ¬ diadic_rational_p y
Hypothesis H5 : SNoLev z ∈ SNoLev y
Hypothesis H6 : SNo w
Hypothesis H7 : diadic_rational_p w
Hypothesis H8 : w + u = y + y
Hypothesis H9 : SNo u
Hypothesis H10 : SNoLev u ∈ SNoLev z
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
Variable x : set
Variable y : set
Variable z : set
Variable w : set
Hypothesis H0 : nat_p x
Hypothesis H1 : (∀u : set, u ∈ x → (∀v : set, SNo v → SNoLev v = u → diadic_rational_p v))
Hypothesis H2 : SNo y
Hypothesis H3 : SNoLev y = x
Hypothesis H4 : ¬ diadic_rational_p y
Hypothesis H5 : SNo_max_of (SNoL y) z
Hypothesis H6 : SNo z
Hypothesis H7 : SNoLev z ∈ SNoLev y
Hypothesis H8 : z < y
Hypothesis H9 : SNo_min_of (SNoR y) w
Hypothesis H11 : SNoLev w ∈ SNoLev y
Hypothesis H12 : y < w
Hypothesis H13 : SNo (y + y)
Hypothesis H14 : SNo (z + w)
Hypothesis H15 : diadic_rational_p z
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
Variable x : set
Variable y : set
Variable z : set
Variable w : set
Hypothesis H0 : nat_p x
Hypothesis H1 : (∀u : set, u ∈ x → (∀v : set, SNo v → SNoLev v = u → diadic_rational_p v))
Hypothesis H2 : SNo y
Hypothesis H3 : SNoLev y = x
Hypothesis H4 : ¬ diadic_rational_p y
Hypothesis H5 : SNo_max_of (SNoL y) z
Hypothesis H6 : SNo z
Hypothesis H7 : SNoLev z ∈ SNoLev y
Hypothesis H8 : z < y
Hypothesis H9 : SNo_min_of (SNoR y) w
Hypothesis H10 : SNo w
Hypothesis H11 : SNoLev w ∈ SNoLev y
Hypothesis H13 : SNo (y + y)
Hypothesis H14 : SNo (z + w)
Hypothesis H15 : diadic_rational_p z
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
Variable x : set
Variable y : set
Variable z : set
Variable w : set
Hypothesis H0 : nat_p x
Hypothesis H1 : (∀u : set, u ∈ x → (∀v : set, SNo v → SNoLev v = u → diadic_rational_p v))
Hypothesis H2 : SNo y
Hypothesis H4 : ¬ diadic_rational_p y
Hypothesis H5 : SNo_max_of (SNoL y) z
Hypothesis H6 : SNo z
Hypothesis H7 : SNoLev z ∈ SNoLev y
Hypothesis H8 : z < y
Hypothesis H9 : SNo_min_of (SNoR y) w
Hypothesis H10 : SNo w
Hypothesis H11 : SNoLev w ∈ SNoLev y
Hypothesis H12 : y < w
Hypothesis H13 : SNo (y + y)
Hypothesis H14 : SNo (z + w)
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
Variable x : set
Variable y : set
Variable z : set
Variable w : set
Hypothesis H0 : nat_p x
Hypothesis H1 : (∀u : set, u ∈ x → (∀v : set, SNo v → SNoLev v = u → diadic_rational_p v))
Hypothesis H2 : SNo y
Hypothesis H3 : SNoLev y = x
Hypothesis H4 : ¬ diadic_rational_p y
Hypothesis H6 : SNo z
Hypothesis H7 : SNoLev z ∈ SNoLev y
Hypothesis H8 : z < y
Hypothesis H9 : SNo_min_of (SNoR y) w
Hypothesis H10 : SNo w
Hypothesis H11 : SNoLev w ∈ SNoLev y
Hypothesis H12 : y < w
Hypothesis H13 : SNo (y + y)
Hypothesis H14 : SNo (z + w)
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
Variable x : set
Variable y : set
Hypothesis H1 : SNo y
Hypothesis H2 : x < y
Hypothesis H3 : SNoLev y ∈ ω
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
Variable x : set
Variable y : set
Hypothesis H1 : nIn x (SNoS_ ω)
Hypothesis H2 : nat_p y
Hypothesis H3 : - y < x → x < y → (∃z : set, z ∈ SNoS_ ω ∧ (z < x ∧ x < z + ordsucc Empty))
Hypothesis H4 : - (ordsucc y) < x
Hypothesis H5 : x < ordsucc y
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
Variable x : set
Hypothesis H0 : x ∈ SNoS_ (ordsucc ω)
Hypothesis H1 : - ω < x
Hypothesis H3 : SNo x
Hypothesis H4 : ¬ (∀y : set, y ∈ ω → (∃z : set, z ∈ SNoS_ ω ∧ (z < x ∧ x < z + eps_ y)))
Hypothesis H5 : nIn x (SNoS_ ω)
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
Variable x : set
Variable P : prop
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
Hypothesis H1 : x ∈ SNoS_ (ordsucc ω)
Hypothesis H2 : (∀y : set, y ∈ SNoS_ ω → (∀z : set, z ∈ ω → abs_SNo (y + - x) < eps_ z) → y = x)
Hypothesis H3 : SNoLev x ∈ ordsucc ω
Hypothesis H4 : SNo x
Hypothesis H5 : x < ω
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
Variable x : set
Variable P : prop
Hypothesis H1 : x ∈ SNoS_ (ordsucc ω)
Hypothesis H2 : x ≠ - ω
Hypothesis H3 : (∀y : set, y ∈ SNoS_ ω → (∀z : set, z ∈ ω → abs_SNo (y + - x) < eps_ z) → y = x)
Hypothesis H4 : SNoLev x ∈ ordsucc ω
Hypothesis H5 : SNo x
Hypothesis H6 : x < ω
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
Variable x : set
Variable P : prop
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
Hypothesis H1 : x ∈ SNoS_ (ordsucc ω)
Hypothesis H2 : x ≠ - ω
Hypothesis H3 : (∀y : set, y ∈ SNoS_ ω → (∀z : set, z ∈ ω → abs_SNo (y + - x) < eps_ z) → y = x)
Hypothesis H4 : SNoLev x ∈ ordsucc ω
Hypothesis H5 : SNo x
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
Variable x : set
Variable y : set
Hypothesis H1 : y ∈ SNoS_ ω
Hypothesis H2 : (∀z : set, z ∈ ω → abs_SNo (y + - x) < eps_ z)
Hypothesis H3 : Empty < y + - x
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
Variable x : set
Variable y : set
Hypothesis H0 : x ∈ SNoS_ ω
Hypothesis H2 : y ∈ SNoS_ ω
Hypothesis H3 : (∀z : set, z ∈ ω → abs_SNo (y + - x) < eps_ z)
Hypothesis H4 : SNo y
Hypothesis H5 : Empty < x + - y
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
Variable x : set
Variable y : set
Variable P : prop
Variable z : set
Variable w : set
Hypothesis H0 : SNo x
Hypothesis H1 : y ∈ ω
Hypothesis H2 : (∀u : set, u ∈ SNoS_ ω → Empty < u → u < x → x < u + eps_ y → P)
Hypothesis H3 : x < z + eps_ y
Hypothesis H4 : SNo z
Hypothesis H5 : z ≤ Empty
Hypothesis H7 : w ∈ ω
Hypothesis H8 : eps_ w ≤ x
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
Variable x : set
Variable y : set
Variable z : set
Hypothesis H0 : ordinal x
Hypothesis H1 : (∀w : set, w ∈ x → ordsucc w ∈ x)
Hypothesis H2 : Subq y (SNoS_ x)
Hypothesis H3 : Subq z (SNoS_ x)
Hypothesis H5 : (∀w : set, w ∈ y → SNoLev w ∈ x)
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
Variable x : set
Variable y : set
Variable z : set
Variable w : set
Hypothesis H0 : SNo x
Hypothesis H1 : (∀u : set, u ∈ ω → SNo (ap y u))
Hypothesis H2 : z ∈ SNoS_ ω
Hypothesis H3 : (∀u : set, u ∈ ω → abs_SNo (z + - x) < eps_ u)
Hypothesis H4 : SNo z
Hypothesis H5 : SNo (- z)
Hypothesis H6 : SNo (x + - z)
Hypothesis H7 : w ∈ ω
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))
Hypothesis H9 : SNo (ap y (ordsucc w))
Hypothesis H10 : z < ap y (ordsucc w)
Hypothesis H11 : Empty < ap y (ordsucc w) + - z
Hypothesis H13 : ap y (ordsucc w) < x
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
Variable x : set
Variable y : set
Variable z : set
Variable w : set
Hypothesis H1 : (∀u : set, u ∈ ω → ap y u < x)
Hypothesis H2 : (∀u : set, u ∈ ω → (∀v : set, v ∈ u → ap y v < ap y u))
Hypothesis H3 : (∀u : set, u ∈ ω → SNo (ap y u))
Hypothesis H4 : z ∈ SNoS_ ω
Hypothesis H5 : (∀u : set, u ∈ ω → abs_SNo (z + - x) < eps_ u)
Hypothesis H6 : SNo z
Hypothesis H7 : SNo (- z)
Hypothesis H8 : SNo (x + - z)
Hypothesis H9 : w ∈ ω
Hypothesis H10 : z ≤ ap y w
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))
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
Variable x : set
Variable y : set
Variable z : set
Variable w : set
Hypothesis H0 : SNo x
Hypothesis H1 : (∀u : set, u ∈ ω → ap y u < x)
Hypothesis H2 : (∀u : set, u ∈ ω → (∀v : set, v ∈ u → ap y v < ap y u))
Hypothesis H3 : (∀u : set, u ∈ ω → SNo (ap y u))
Hypothesis H4 : z ∈ SNoS_ ω
Hypothesis H5 : (∀u : set, u ∈ ω → abs_SNo (z + - x) < eps_ u)
Hypothesis H6 : SNo z
Hypothesis H7 : SNo (- z)
Hypothesis H8 : SNo (x + - z)
Hypothesis H10 : z ≤ ap y w
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))
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
Variable x : set
Variable y : set
Variable z : set
Variable w : set
Hypothesis H0 : SNo x
Hypothesis H1 : SNo (- x)
Hypothesis H2 : z ∈ SNoS_ ω
Hypothesis H3 : (∀u : set, u ∈ ω → abs_SNo (z + - x) < eps_ u)
Hypothesis H4 : SNo z
Hypothesis H5 : SNo (z + - x)
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))
Hypothesis H7 : SNo (ap y (ordsucc w))
Hypothesis H8 : ap y (ordsucc w) < z
Hypothesis H9 : Empty < z + - (ap y (ordsucc w))
Hypothesis H11 : x < ap y (ordsucc w)
Hypothesis H12 : abs_SNo (z + - x) = z + - x
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
Variable x : set
Variable y : set
Variable z : set
Variable w : set
Hypothesis H0 : SNo x
Hypothesis H1 : SNo (- x)
Hypothesis H2 : z ∈ SNoS_ ω
Hypothesis H3 : (∀u : set, u ∈ ω → abs_SNo (z + - x) < eps_ u)
Hypothesis H4 : SNo z
Hypothesis H5 : SNo (z + - x)
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))
Hypothesis H8 : ap y (ordsucc w) < z
Hypothesis H9 : Empty < z + - (ap y (ordsucc w))
Hypothesis H10 : SNo (z + - (ap y (ordsucc w)))
Hypothesis H11 : x < ap y (ordsucc w)
Hypothesis H12 : x < z
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
Variable x : set
Variable y : set
Variable z : set
Variable w : set
Hypothesis H0 : SNo x
Hypothesis H1 : SNo (- x)
Hypothesis H2 : z ∈ SNoS_ ω
Hypothesis H3 : (∀u : set, u ∈ ω → abs_SNo (z + - x) < eps_ u)
Hypothesis H4 : SNo z
Hypothesis H5 : SNo (z + - x)
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))
Hypothesis H7 : SNo (ap y (ordsucc w))
Hypothesis H8 : ap y (ordsucc w) < z
Hypothesis H10 : SNo (z + - (ap y (ordsucc w)))
Hypothesis H11 : x < ap y (ordsucc w)
Hypothesis H12 : x < z
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
Variable x : set
Variable y : set
Variable z : set
Variable w : set
Hypothesis H0 : SNo x
Hypothesis H1 : (∀u : set, u ∈ ω → x < ap y u)
Hypothesis H2 : SNo (- x)
Hypothesis H3 : z ∈ SNoS_ ω
Hypothesis H4 : (∀u : set, u ∈ ω → abs_SNo (z + - x) < eps_ u)
Hypothesis H5 : SNo z
Hypothesis H6 : SNo (z + - x)
Hypothesis H7 : w ∈ ω
Hypothesis H9 : SNo (ap y (ordsucc w))
Hypothesis H10 : ap y (ordsucc w) < z
Hypothesis H11 : Empty < z + - (ap y (ordsucc w))
Hypothesis H12 : SNo (z + - (ap y (ordsucc w)))
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
Variable x : set
Variable y : set
Variable z : set
Variable w : set
Hypothesis H0 : SNo x
Hypothesis H1 : (∀u : set, u ∈ ω → x < ap y u)
Hypothesis H2 : SNo (- x)
Hypothesis H3 : z ∈ SNoS_ ω
Hypothesis H4 : (∀u : set, u ∈ ω → abs_SNo (z + - x) < eps_ u)
Hypothesis H5 : SNo z
Hypothesis H6 : SNo (z + - x)
Hypothesis H7 : w ∈ ω
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))
Hypothesis H9 : SNo (ap y (ordsucc w))
Hypothesis H11 : Empty < z + - (ap y (ordsucc w))
Hypothesis H12 : SNo (z + - (ap y (ordsucc w)))
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
Variable x : set
Variable y : set
Variable z : set
Variable w : set
Hypothesis H1 : (∀u : set, u ∈ ω → x < ap y u)
Hypothesis H2 : (∀u : set, u ∈ ω → (∀v : set, v ∈ u → ap y u < ap y v))
Hypothesis H3 : SNo (- x)
Hypothesis H4 : (∀u : set, u ∈ ω → SNo (ap y u))
Hypothesis H5 : z ∈ SNoS_ ω
Hypothesis H6 : (∀u : set, u ∈ ω → abs_SNo (z + - x) < eps_ u)
Hypothesis H7 : SNo z
Hypothesis H8 : SNo (z + - x)
Hypothesis H9 : w ∈ ω
Hypothesis H10 : ap y w ≤ z
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))
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
Variable x : set
Variable y : set
Variable z : set
Variable w : set
Hypothesis H0 : SNo x
Hypothesis H1 : (∀u : set, u ∈ ω → x < ap y u)
Hypothesis H2 : (∀u : set, u ∈ ω → (∀v : set, v ∈ u → ap y u < ap y v))
Hypothesis H3 : SNo (- x)
Hypothesis H4 : (∀u : set, u ∈ ω → SNo (ap y u))
Hypothesis H5 : z ∈ SNoS_ ω
Hypothesis H6 : (∀u : set, u ∈ ω → abs_SNo (z + - x) < eps_ u)
Hypothesis H7 : SNo z
Hypothesis H9 : w ∈ ω
Hypothesis H10 : ap y w ≤ z
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))
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
Variable x : set
Variable y : set
Variable z : set
Variable w : set
Hypothesis H0 : SNo x
Hypothesis H1 : z ∈ setexp (SNoS_ ω) ω
Hypothesis H2 : (∀u : set, u ∈ ω → x < ap z u)
Hypothesis H3 : (∀u : set, u ∈ ω → (∀v : set, v ∈ u → ap z u < ap z v))
Hypothesis H4 : x = SNoCut (Repl ω (ap y)) (Repl ω (ap z))
Hypothesis H5 : SNo (- x)
Hypothesis H6 : (∀u : set, u ∈ ω → SNo (ap z u))
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)
Hypothesis H8 : SNoLev x = ω
Hypothesis H9 : w ∈ SNoS_ ω
Hypothesis H10 : (∀u : set, u ∈ ω → abs_SNo (w + - x) < eps_ u)
Hypothesis H11 : SNoLev w ∈ ω
Hypothesis H12 : SNo w
Hypothesis H14 : (∀u : set, u ∈ Repl ω (ap y) → u < w)
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
Variable x : set
Variable y : set
Variable z : set
Variable w : set
Hypothesis H0 : SNo x
Hypothesis H1 : y ∈ setexp (SNoS_ ω) ω
Hypothesis H2 : z ∈ setexp (SNoS_ ω) ω
Hypothesis H3 : (∀u : set, u ∈ ω → ap y u < x)
Hypothesis H4 : (∀u : set, u ∈ ω → (∀v : set, v ∈ u → ap y v < ap y u))
Hypothesis H5 : (∀u : set, u ∈ ω → x < ap z u)
Hypothesis H6 : (∀u : set, u ∈ ω → (∀v : set, v ∈ u → ap z u < ap z v))
Hypothesis H7 : x = SNoCut (Repl ω (ap y)) (Repl ω (ap z))
Hypothesis H8 : SNo (- x)
Hypothesis H9 : (∀u : set, u ∈ ω → SNo (ap y u))
Hypothesis H10 : (∀u : set, u ∈ ω → SNo (ap z u))
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)
Hypothesis H12 : SNoLev x = ω
Hypothesis H13 : w ∈ SNoS_ ω
Hypothesis H14 : (∀u : set, u ∈ ω → abs_SNo (w + - x) < eps_ u)
Hypothesis H15 : SNoLev w ∈ ω
Hypothesis H17 : SNo (- w)
Hypothesis H18 : SNo (x + - w)
Hypothesis H19 : SNo (w + - x)
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
Variable x : set
Variable y : set
Variable z : set
Variable w : set
Hypothesis H0 : SNo x
Hypothesis H2 : z ∈ setexp (SNoS_ ω) ω
Hypothesis H3 : (∀u : set, u ∈ ω → ap y u < x)
Hypothesis H4 : (∀u : set, u ∈ ω → (∀v : set, v ∈ u → ap y v < ap y u))
Hypothesis H5 : (∀u : set, u ∈ ω → x < ap z u)
Hypothesis H6 : (∀u : set, u ∈ ω → (∀v : set, v ∈ u → ap z u < ap z v))
Hypothesis H7 : x = SNoCut (Repl ω (ap y)) (Repl ω (ap z))
Hypothesis H8 : SNo (- x)
Hypothesis H9 : (∀u : set, u ∈ ω → SNo (ap y u))
Hypothesis H10 : (∀u : set, u ∈ ω → SNo (ap z u))
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)
Hypothesis H12 : SNoLev x = ω
Hypothesis H13 : w ∈ SNoS_ ω
Hypothesis H14 : (∀u : set, u ∈ ω → abs_SNo (w + - x) < eps_ u)
Hypothesis H15 : SNoLev w ∈ ω
Hypothesis H16 : SNo w
Hypothesis H17 : SNo (- w)
Hypothesis H18 : SNo (x + - w)
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
Variable x : set
Variable y : set
Variable z : set
Variable w : set
Hypothesis H0 : SNo x
Hypothesis H1 : y ∈ setexp (SNoS_ ω) ω
Hypothesis H3 : (∀u : set, u ∈ ω → ap y u < x)
Hypothesis H4 : (∀u : set, u ∈ ω → (∀v : set, v ∈ u → ap y v < ap y u))
Hypothesis H5 : (∀u : set, u ∈ ω → x < ap z u)
Hypothesis H6 : (∀u : set, u ∈ ω → (∀v : set, v ∈ u → ap z u < ap z v))
Hypothesis H7 : x = SNoCut (Repl ω (ap y)) (Repl ω (ap z))
Hypothesis H8 : SNo (- x)
Hypothesis H9 : (∀u : set, u ∈ ω → SNo (ap y u))
Hypothesis H10 : (∀u : set, u ∈ ω → SNo (ap z u))
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)
Hypothesis H12 : SNoLev x = ω
Hypothesis H13 : w ∈ SNoS_ ω
Hypothesis H14 : (∀u : set, u ∈ ω → abs_SNo (w + - x) < eps_ u)
Hypothesis H15 : SNoLev w ∈ ω
Hypothesis H16 : SNo w
Hypothesis H17 : SNo (- w)
Hypothesis H18 : SNo (x + - w)
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
Variable x : set
Variable y : set
Variable z : set
Variable w : set
Hypothesis H0 : SNo x
Hypothesis H1 : y ∈ setexp (SNoS_ ω) ω
Hypothesis H2 : z ∈ setexp (SNoS_ ω) ω
Hypothesis H4 : (∀u : set, u ∈ ω → (∀v : set, v ∈ u → ap y v < ap y u))
Hypothesis H5 : (∀u : set, u ∈ ω → x < ap z u)
Hypothesis H6 : (∀u : set, u ∈ ω → (∀v : set, v ∈ u → ap z u < ap z v))
Hypothesis H7 : x = SNoCut (Repl ω (ap y)) (Repl ω (ap z))
Hypothesis H8 : SNo (- x)
Hypothesis H9 : (∀u : set, u ∈ ω → SNo (ap y u))
Hypothesis H10 : (∀u : set, u ∈ ω → SNo (ap z u))
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)
Hypothesis H12 : SNoLev x = ω
Hypothesis H13 : w ∈ SNoS_ ω
Hypothesis H14 : (∀u : set, u ∈ ω → abs_SNo (w + - x) < eps_ u)
Hypothesis H15 : SNoLev w ∈ ω
Hypothesis H16 : SNo w
Hypothesis H17 : SNo (- w)
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
Variable x : set
Variable y : set
Variable z : set
Variable w : set
Hypothesis H0 : SNo x
Hypothesis H1 : y ∈ setexp (SNoS_ ω) ω
Hypothesis H2 : z ∈ setexp (SNoS_ ω) ω
Hypothesis H3 : (∀u : set, u ∈ ω → ap y u < x)
Hypothesis H4 : (∀u : set, u ∈ ω → (∀v : set, v ∈ u → ap y v < ap y u))
Hypothesis H5 : (∀u : set, u ∈ ω → x < ap z u)
Hypothesis H6 : (∀u : set, u ∈ ω → (∀v : set, v ∈ u → ap z u < ap z v))
Hypothesis H7 : x = SNoCut (Repl ω (ap y)) (Repl ω (ap z))
Hypothesis H8 : SNo (- x)
Hypothesis H10 : (∀u : set, u ∈ ω → SNo (ap z u))
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)
Hypothesis H12 : SNoLev x = ω
Hypothesis H13 : w ∈ SNoS_ ω
Hypothesis H14 : (∀u : set, u ∈ ω → abs_SNo (w + - x) < eps_ u)
Hypothesis H15 : SNoLev w ∈ ω
Hypothesis H16 : SNo w
Hypothesis H17 : SNo (- w)
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
Variable x : set
Variable y : set
Variable z : set
Variable w : set
Hypothesis H0 : SNo x
Hypothesis H1 : y ∈ setexp (SNoS_ ω) ω
Hypothesis H3 : (∀u : set, u ∈ ω → ap y u < x)
Hypothesis H4 : (∀u : set, u ∈ ω → (∀v : set, v ∈ u → ap y v < ap y u))
Hypothesis H5 : (∀u : set, u ∈ ω → x < ap z u)
Hypothesis H6 : (∀u : set, u ∈ ω → (∀v : set, v ∈ u → ap z u < ap z v))
Hypothesis H7 : x = SNoCut (Repl ω (ap y)) (Repl ω (ap z))
Hypothesis H8 : SNo (- x)
Hypothesis H9 : (∀u : set, u ∈ ω → SNo (ap y u))
Hypothesis H10 : (∀u : set, u ∈ ω → SNo (ap z u))
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)
Hypothesis H12 : SNoLev x = ω
Hypothesis H13 : w ∈ SNoS_ ω
Hypothesis H14 : (∀u : set, u ∈ ω → abs_SNo (w + - x) < eps_ u)
Hypothesis H15 : SNoLev w ∈ ω
Hypothesis H16 : SNo w
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
Variable x : set
Variable y : set
Variable z : set
Hypothesis H0 : SNo x
Hypothesis H2 : z ∈ setexp (SNoS_ ω) ω
Hypothesis H3 : (∀w : set, w ∈ ω → ap y w < x)
Hypothesis H4 : (∀w : set, w ∈ ω → (∀u : set, u ∈ w → ap y u < ap y w))
Hypothesis H5 : (∀w : set, w ∈ ω → x < ap z w)
Hypothesis H6 : (∀w : set, w ∈ ω → (∀u : set, u ∈ w → ap z w < ap z u))
Hypothesis H7 : x = SNoCut (Repl ω (ap y)) (Repl ω (ap z))
Hypothesis H8 : SNo (- x)
Hypothesis H9 : (∀w : set, w ∈ ω → SNo (ap y w))
Hypothesis H10 : (∀w : set, w ∈ ω → SNo (ap z w))
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
Variable x : set
Variable y : set
Variable z : set
Hypothesis H0 : SNo x
Hypothesis H1 : y ∈ setexp (SNoS_ ω) ω
Hypothesis H2 : z ∈ setexp (SNoS_ ω) ω
Hypothesis H3 : (∀w : set, w ∈ ω → ap y w < x)
Hypothesis H4 : (∀w : set, w ∈ ω → (∀u : set, u ∈ w → ap y u < ap y w))
Hypothesis H5 : (∀w : set, w ∈ ω → x < ap z w)
Hypothesis H7 : x = SNoCut (Repl ω (ap y)) (Repl ω (ap z))
Hypothesis H8 : SNo (- x)
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
Variable x : set
Variable y : set
Hypothesis H1 : (∀z : set, z ∈ SNoS_ ω → (∀w : set, w ∈ ω → abs_SNo (z + - x) < eps_ w) → z = x)
Hypothesis H2 : SNo y
Hypothesis H3 : x < y
Hypothesis H4 : y ∈ SNoS_ ω
Hypothesis H5 : Empty < y + - x
Hypothesis H6 : ¬ (∃z : set, z ∈ ω ∧ (x + eps_ z) ≤ y)
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
Variable x : set
Variable y : set
Hypothesis H0 : SNo x
Hypothesis H2 : SNo y
Hypothesis H3 : x < y
Hypothesis H4 : y ∈ SNoS_ ω
Hypothesis H5 : Empty < y + - x
Hypothesis H6 : ¬ (∃z : set, z ∈ ω ∧ (x + eps_ z) ≤ y)
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
Variable x : set
Variable y : set
Variable z : set
Variable w : set
Hypothesis H1 : (∀u : set, u ∈ SNoS_ ω → (∀v : set, v ∈ ω → abs_SNo (u + - x) < eps_ v) → u = x)
Hypothesis H2 : (∀u : set, u ∈ ω → SNo (ap z u))
Hypothesis H3 : (∀u : set, u ∈ ω → (ap z u + - (eps_ u)) < x)
Hypothesis H4 : SNo (SNoCut (Repl ω (ap y)) (Repl ω (ap z)))
Hypothesis H5 : (∀u : set, u ∈ ω → SNoCut (Repl ω (ap y)) (Repl ω (ap z)) < ap z u)
Hypothesis H6 : SNo w
Hypothesis H7 : x < w
Hypothesis H8 : w ∈ SNoS_ ω
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
Variable x : set
Variable y : set
Variable z : set
Variable w : set
Hypothesis H0 : SNo x
Hypothesis H1 : (∀u : set, u ∈ SNoS_ ω → (∀v : set, v ∈ ω → abs_SNo (u + - x) < eps_ v) → u = x)
Hypothesis H3 : (∀u : set, u ∈ ω → SNo (ap y u))
Hypothesis H4 : SNo (SNoCut (Repl ω (ap y)) (Repl ω (ap z)))
Hypothesis H5 : (∀u : set, u ∈ ω → ap y u < SNoCut (Repl ω (ap y)) (Repl ω (ap z)))
Hypothesis H6 : SNo w
Hypothesis H7 : w < x
Hypothesis H8 : w ∈ SNoS_ ω
Hypothesis H9 : Empty < x + - w
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
Variable x : set
Variable y : set
Variable z : set
Variable w : set
Hypothesis H0 : SNo x
Hypothesis H1 : (∀u : set, u ∈ SNoS_ ω → (∀v : set, v ∈ ω → abs_SNo (u + - x) < eps_ v) → u = x)
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))
Hypothesis H3 : (∀u : set, u ∈ ω → SNo (ap y u))
Hypothesis H5 : (∀u : set, u ∈ ω → ap y u < SNoCut (Repl ω (ap y)) (Repl ω (ap z)))
Hypothesis H6 : SNo w
Hypothesis H7 : w < x
Hypothesis H8 : w ∈ SNoS_ ω
Hypothesis H9 : Empty < x + - w
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
Variable x : set
Variable y : set
Variable z : set
Variable w : set
Hypothesis H0 : SNo x
Hypothesis H1 : (∀u : set, u ∈ SNoS_ ω → (∀v : set, v ∈ ω → abs_SNo (u + - x) < eps_ v) → u = x)
Hypothesis H3 : (∀u : set, u ∈ ω → SNo (ap y u))
Hypothesis H4 : SNo (SNoCut (Repl ω (ap y)) (Repl ω (ap z)))
Hypothesis H5 : (∀u : set, u ∈ ω → ap y u < SNoCut (Repl ω (ap y)) (Repl ω (ap z)))
Hypothesis H6 : SNo w
Hypothesis H7 : w < x
Hypothesis H8 : w ∈ SNoS_ ω
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
Variable x : set
Variable P : prop
Variable y : set
Variable z : set
Hypothesis H0 : x ∈ real
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))
Hypothesis H2 : SNo x
Hypothesis H3 : (∀w : set, w ∈ SNoS_ ω → (∀u : set, u ∈ ω → abs_SNo (w + - x) < eps_ u) → w = x)
Hypothesis H4 : y ∈ setexp (SNoS_ ω) ω
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))
Hypothesis H6 : z ∈ setexp (SNoS_ ω) ω
Hypothesis H7 : (∀w : set, w ∈ ω → SNo (ap y w))
Hypothesis H8 : (∀w : set, w ∈ ω → SNo (ap z w))
Hypothesis H10 : (∀w : set, w ∈ ω → x < ap y w + eps_ w)
Hypothesis H11 : (∀w : set, w ∈ ω → (∀u : set, u ∈ w → ap y u < ap y w))
Hypothesis H12 : (∀w : set, w ∈ ω → (ap z w + - (eps_ w)) < x)
Hypothesis H13 : (∀w : set, w ∈ ω → x < ap z w)
Hypothesis H14 : (∀w : set, w ∈ ω → (∀u : set, u ∈ w → ap z w < ap z u))
Hypothesis H15 : SNoCutP (Repl ω (ap y)) (Repl ω (ap z))
Hypothesis H16 : SNo (SNoCut (Repl ω (ap y)) (Repl ω (ap z)))
Hypothesis H17 : (∀w : set, w ∈ ω → ap y w < SNoCut (Repl ω (ap y)) (Repl ω (ap z)))
Hypothesis H18 : (∀w : set, w ∈ ω → SNoCut (Repl ω (ap y)) (Repl ω (ap z)) < ap z w)
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
Variable x : set
Variable P : prop
Variable y : set
Variable z : set
Hypothesis H0 : x ∈ real
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))
Hypothesis H2 : SNo x
Hypothesis H3 : (∀w : set, w ∈ SNoS_ ω → (∀u : set, u ∈ ω → abs_SNo (w + - x) < eps_ u) → w = x)
Hypothesis H4 : y ∈ setexp (SNoS_ ω) ω
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))
Hypothesis H6 : z ∈ setexp (SNoS_ ω) ω
Hypothesis H7 : (∀w : set, w ∈ ω → SNo (ap y w))
Hypothesis H8 : (∀w : set, w ∈ ω → SNo (ap z w))
Hypothesis H9 : (∀w : set, w ∈ ω → ap y w < x)
Hypothesis H11 : (∀w : set, w ∈ ω → (∀u : set, u ∈ w → ap y u < ap y w))
Hypothesis H12 : (∀w : set, w ∈ ω → (ap z w + - (eps_ w)) < x)
Hypothesis H13 : (∀w : set, w ∈ ω → x < ap z w)
Hypothesis H14 : (∀w : set, w ∈ ω → (∀u : set, u ∈ w → ap z w < ap z u))
Hypothesis H15 : SNoCutP (Repl ω (ap y)) (Repl ω (ap z))
Hypothesis H16 : SNo (SNoCut (Repl ω (ap y)) (Repl ω (ap z)))
Hypothesis H17 : (∀w : set, w ∈ ω → ap y w < SNoCut (Repl ω (ap y)) (Repl ω (ap z)))
Hypothesis H18 : (∀w : set, w ∈ ω → SNoCut (Repl ω (ap y)) (Repl ω (ap z)) < ap z w)
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
Variable x : set
Variable P : prop
Variable y : set
Variable z : set
Hypothesis H0 : x ∈ real
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))
Hypothesis H3 : (∀w : set, w ∈ SNoS_ ω → (∀u : set, u ∈ ω → abs_SNo (w + - x) < eps_ u) → w = x)
Hypothesis H4 : y ∈ setexp (SNoS_ ω) ω
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))
Hypothesis H6 : z ∈ setexp (SNoS_ ω) ω
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))
Hypothesis H8 : (∀w : set, w ∈ ω → SNo (ap y w))
Hypothesis H9 : (∀w : set, w ∈ ω → SNo (ap z w))
Hypothesis H10 : (∀w : set, w ∈ ω → ap y w < x)
Hypothesis H11 : (∀w : set, w ∈ ω → x < ap y w + eps_ w)
Hypothesis H12 : (∀w : set, w ∈ ω → (∀u : set, u ∈ w → ap y u < ap y w))
Hypothesis H13 : (∀w : set, w ∈ ω → (ap z w + - (eps_ w)) < x)
Hypothesis H14 : (∀w : set, w ∈ ω → x < ap z w)
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
Variable x : set
Variable P : prop
Variable y : set
Variable z : set
Hypothesis H0 : x ∈ real
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))
Hypothesis H2 : SNo x
Hypothesis H4 : y ∈ setexp (SNoS_ ω) ω
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))
Hypothesis H6 : z ∈ setexp (SNoS_ ω) ω
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))
Hypothesis H8 : (∀w : set, w ∈ ω → SNo (ap y w))
Hypothesis H9 : (∀w : set, w ∈ ω → SNo (ap z w))
Hypothesis H10 : (∀w : set, w ∈ ω → ap y w < x)
Hypothesis H11 : (∀w : set, w ∈ ω → x < ap y w + eps_ w)
Hypothesis H12 : (∀w : set, w ∈ ω → (∀u : set, u ∈ w → ap y u < ap y w))
Hypothesis H13 : (∀w : set, w ∈ ω → (ap z w + - (eps_ w)) < x)
Hypothesis H14 : (∀w : set, w ∈ ω → x < ap z w)
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
Variable x : set
Variable P : prop
Variable y : set
Variable z : set
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))
Hypothesis H2 : SNo x
Hypothesis H3 : (∀w : set, w ∈ SNoS_ ω → (∀u : set, u ∈ ω → abs_SNo (w + - x) < eps_ u) → w = x)
Hypothesis H4 : y ∈ setexp (SNoS_ ω) ω
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))
Hypothesis H6 : z ∈ setexp (SNoS_ ω) ω
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))
Hypothesis H8 : (∀w : set, w ∈ ω → SNo (ap y w))
Hypothesis H9 : (∀w : set, w ∈ ω → SNo (ap z w))
Hypothesis H10 : (∀w : set, w ∈ ω → ap y w < x)
Hypothesis H11 : (∀w : set, w ∈ ω → x < ap y w + eps_ w)
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
Variable x : set
Variable P : prop
Variable y : set
Variable z : set
Hypothesis H0 : x ∈ real
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))
Hypothesis H2 : SNo x
Hypothesis H3 : (∀w : set, w ∈ SNoS_ ω → (∀u : set, u ∈ ω → abs_SNo (w + - x) < eps_ u) → w = x)
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))
Hypothesis H6 : z ∈ setexp (SNoS_ ω) ω
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))
Hypothesis H8 : (∀w : set, w ∈ ω → SNo (ap y w))
Hypothesis H9 : (∀w : set, w ∈ ω → SNo (ap z w))
Hypothesis H10 : (∀w : set, w ∈ ω → ap y w < x)
Hypothesis H11 : (∀w : set, w ∈ ω → x < ap y w + eps_ w)
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
Variable x : set
Variable P : prop
Variable y : set
Variable z : set
Hypothesis H0 : x ∈ real
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))
Hypothesis H2 : SNo x
Hypothesis H3 : (∀w : set, w ∈ SNoS_ ω → (∀u : set, u ∈ ω → abs_SNo (w + - x) < eps_ u) → w = x)
Hypothesis H4 : y ∈ setexp (SNoS_ ω) ω
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))
Hypothesis H6 : z ∈ setexp (SNoS_ ω) ω
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))
Hypothesis H8 : (∀w : set, w ∈ ω → SNo (ap y w))
Hypothesis H9 : (∀w : set, w ∈ ω → SNo (ap z w))
Hypothesis H11 : (∀w : set, w ∈ ω → x < ap y w + eps_ w)
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
Variable x : set
Variable P : prop
Variable y : set
Variable z : set
Hypothesis H0 : x ∈ real
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))
Hypothesis H3 : (∀w : set, w ∈ SNoS_ ω → (∀u : set, u ∈ ω → abs_SNo (w + - x) < eps_ u) → w = x)
Hypothesis H4 : y ∈ setexp (SNoS_ ω) ω
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))
Hypothesis H6 : z ∈ setexp (SNoS_ ω) ω
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))
Hypothesis H8 : (∀w : set, w ∈ ω → SNo (ap y w))
Hypothesis H9 : (∀w : set, w ∈ ω → SNo (ap z w))
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
Variable x : set
Variable P : prop
Variable y : set
Variable z : set
Hypothesis H0 : x ∈ real
Hypothesis H2 : SNo x
Hypothesis H3 : (∀w : set, w ∈ SNoS_ ω → (∀u : set, u ∈ ω → abs_SNo (w + - x) < eps_ u) → w = x)
Hypothesis H4 : y ∈ setexp (SNoS_ ω) ω
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))
Hypothesis H6 : z ∈ setexp (SNoS_ ω) ω
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))
Hypothesis H8 : (∀w : set, w ∈ ω → SNo (ap y w))
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
Variable x : set
Variable y : set
Variable z : set
Variable w : set
Variable u : set
Hypothesis H0 : (∀v : set, v ∈ ω → x < ap z v + eps_ v)
Hypothesis H1 : (∀v : set, v ∈ ω → y < ap w v + eps_ v)
Hypothesis H2 : SNo x
Hypothesis H3 : SNo y
Hypothesis H4 : (∀v : set, v ∈ ω → SNo (ap z (ordsucc v)))
Hypothesis H5 : (∀v : set, v ∈ ω → SNo (ap w (ordsucc v)))
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
Variable x : set
Variable y : set
Variable z : set
Variable w : set
Hypothesis H0 : (∀u : set, u ∈ ω → (∀v : set, v ∈ u → ap x v < ap x u))
Hypothesis H1 : (∀u : set, u ∈ ω → (∀v : set, v ∈ u → ap y v < ap y u))
Hypothesis H2 : (∀u : set, u ∈ ω → SNo (ap x (ordsucc u)))
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))
Hypothesis H5 : z ∈ ω
Hypothesis H6 : w ∈ z
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
Variable x : set
Variable y : set
Variable z : set
Variable w : set
Hypothesis H0 : (∀u : set, u ∈ ω → (∀v : set, v ∈ u → ap x u < ap x v))
Hypothesis H1 : (∀u : set, u ∈ ω → (∀v : set, v ∈ u → ap y u < ap y v))
Hypothesis H2 : (∀u : set, u ∈ ω → SNo (ap x (ordsucc u)))
Hypothesis H3 : (∀u : set, u ∈ ω → SNo (ap y (ordsucc u)))
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))
Hypothesis H6 : w ∈ z
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
Variable x : set
Variable y : set
Variable z : set
Variable w : set
Variable u : set
Hypothesis H0 : SNo x
Hypothesis H1 : SNo y
Hypothesis H2 : SNo (x + y)
Hypothesis H3 : (∀v : set, v ∈ SNoS_ ω → (∀x2 : set, x2 ∈ ω → abs_SNo (v + - y) < eps_ x2) → v = y)
Hypothesis H4 : (∀v : set, v ∈ ω → SNo (ap (Sigma ω (λx2 : set ⇒ ap z (ordsucc x2) + ap w (ordsucc x2))) v))
Hypothesis H5 : (∀v : set, v ∈ ω → (ap (Sigma ω (λx2 : set ⇒ ap z (ordsucc x2) + ap w (ordsucc x2))) v + - (eps_ v)) < x + y)
Hypothesis H7 : y < u
Hypothesis H8 : u ∈ SNoS_ ω
Hypothesis H9 : ¬ (∃v : set, v ∈ ω ∧ ap (Sigma ω (λx2 : set ⇒ ap z (ordsucc x2) + ap w (ordsucc x2))) v ≤ x + u)
Hypothesis H10 : Empty < u + - y
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
Variable x : set
Variable y : set
Variable z : set
Variable w : set
Variable u : set
Hypothesis H0 : SNo x
Hypothesis H1 : SNo y
Hypothesis H2 : SNo (x + y)
Hypothesis H3 : (∀v : set, v ∈ SNoS_ ω → (∀x2 : set, x2 ∈ ω → abs_SNo (v + - y) < eps_ x2) → v = y)
Hypothesis H4 : (∀v : set, v ∈ ω → SNo (ap (Sigma ω (λx2 : set ⇒ ap z (ordsucc x2) + ap w (ordsucc x2))) v))
Hypothesis H5 : (∀v : set, v ∈ ω → (ap (Sigma ω (λx2 : set ⇒ ap z (ordsucc x2) + ap w (ordsucc x2))) v + - (eps_ v)) < x + y)
Hypothesis H6 : SNo u
Hypothesis H7 : y < u
Hypothesis H8 : u ∈ SNoS_ ω
Hypothesis H9 : ¬ (∃v : set, v ∈ ω ∧ ap (Sigma ω (λx2 : set ⇒ ap z (ordsucc x2) + ap w (ordsucc x2))) v ≤ x + u)
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
Variable x : set
Variable y : set
Variable z : set
Variable w : set
Variable u : set
Hypothesis H1 : SNo y
Hypothesis H2 : SNo (x + y)
Hypothesis H3 : (∀v : set, v ∈ SNoS_ ω → (∀x2 : set, x2 ∈ ω → abs_SNo (v + - y) < eps_ x2) → v = y)
Hypothesis H4 : (∀v : set, v ∈ ω → SNo (ap (Sigma ω (λx2 : set ⇒ ap z (ordsucc x2) + ap w (ordsucc x2))) v))
Hypothesis H5 : (∀v : set, v ∈ ω → (ap (Sigma ω (λx2 : set ⇒ ap z (ordsucc x2) + ap w (ordsucc x2))) v + - (eps_ v)) < x + y)
Hypothesis H6 : SNo u
Hypothesis H7 : y < u
Hypothesis H8 : u ∈ SNoS_ ω
Hypothesis H9 : ¬ (∃v : set, v ∈ ω ∧ ap (Sigma ω (λx2 : set ⇒ ap z (ordsucc x2) + ap w (ordsucc x2))) v ≤ x + u)
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
Variable x : set
Variable y : set
Variable z : set
Variable w : set
Variable u : set
Variable v : set
Variable x2 : set
Hypothesis H1 : SNo y
Hypothesis H2 : SNo (x + y)
Hypothesis H3 : (∀y2 : set, y2 ∈ SNoS_ ω → (∀z2 : set, z2 ∈ ω → abs_SNo (y2 + - y) < eps_ z2) → y2 = y)
Hypothesis H4 : (∀y2 : set, y2 ∈ ω → SNo (ap (Sigma ω (λz2 : set ⇒ ap w (ordsucc z2) + ap v (ordsucc z2))) y2))
Hypothesis H5 : (∀y2 : set, y2 ∈ ω → (ap (Sigma ω (λz2 : set ⇒ ap w (ordsucc z2) + ap v (ordsucc z2))) y2 + - (eps_ y2)) < x + y)
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))))))
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)
Hypothesis H8 : SNo x2
Hypothesis H9 : y < x2
Hypothesis H10 : x2 ∈ SNoS_ ω
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
Variable x : set
Variable y : set
Variable z : set
Variable w : set
Variable u : set
Variable v : set
Variable x2 : set
Hypothesis H0 : SNo x
Hypothesis H1 : SNo y
Hypothesis H2 : SNo (x + y)
Hypothesis H3 : (∀y2 : set, y2 ∈ SNoS_ ω → (∀z2 : set, z2 ∈ ω → abs_SNo (y2 + - y) < eps_ z2) → y2 = y)
Hypothesis H4 : (∀y2 : set, y2 ∈ ω → SNo (ap (Sigma ω (λz2 : set ⇒ ap w (ordsucc z2) + ap v (ordsucc z2))) y2))
Hypothesis H5 : (∀y2 : set, y2 ∈ ω → (ap (Sigma ω (λz2 : set ⇒ ap w (ordsucc z2) + ap v (ordsucc z2))) y2 + - (eps_ y2)) < x + y)
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)
Hypothesis H8 : SNo x2
Hypothesis H9 : y < x2
Hypothesis H10 : x2 ∈ SNoS_ ω
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
Variable x : set
Variable y : set
Variable z : set
Variable w : set
Variable u : set
Variable v : set
Variable x2 : set
Hypothesis H0 : y ∈ real
Hypothesis H1 : SNo x
Hypothesis H3 : SNo (x + y)
Hypothesis H4 : (∀y2 : set, y2 ∈ SNoS_ ω → (∀z2 : set, z2 ∈ ω → abs_SNo (y2 + - y) < eps_ z2) → y2 = y)
Hypothesis H5 : (∀y2 : set, y2 ∈ ω → SNo (ap (Sigma ω (λz2 : set ⇒ ap w (ordsucc z2) + ap v (ordsucc z2))) y2))
Hypothesis H6 : (∀y2 : set, y2 ∈ ω → (ap (Sigma ω (λz2 : set ⇒ ap w (ordsucc z2) + ap v (ordsucc z2))) y2 + - (eps_ y2)) < x + y)
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))))))
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)
Hypothesis H9 : SNo x2
Hypothesis H10 : SNoLev x2 ∈ SNoLev y
Hypothesis H11 : y < x2
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
Variable x : set
Variable y : set
Variable z : set
Variable w : set
Variable u : set
Hypothesis H0 : SNo x
Hypothesis H1 : SNo y
Hypothesis H2 : SNo (x + y)
Hypothesis H3 : (∀v : set, v ∈ SNoS_ ω → (∀x2 : set, x2 ∈ ω → abs_SNo (v + - x) < eps_ x2) → v = x)
Hypothesis H4 : (∀v : set, v ∈ ω → SNo (ap (Sigma ω (λx2 : set ⇒ ap z (ordsucc x2) + ap w (ordsucc x2))) v))
Hypothesis H5 : (∀v : set, v ∈ ω → (ap (Sigma ω (λx2 : set ⇒ ap z (ordsucc x2) + ap w (ordsucc x2))) v + - (eps_ v)) < x + y)
Hypothesis H6 : SNo u
Hypothesis H8 : u ∈ SNoS_ ω
Hypothesis H9 : ¬ (∃v : set, v ∈ ω ∧ ap (Sigma ω (λx2 : set ⇒ ap z (ordsucc x2) + ap w (ordsucc x2))) v ≤ u + y)
Hypothesis H10 : Empty < u + - x
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
Variable x : set
Variable y : set
Variable z : set
Variable w : set
Variable u : set
Variable v : set
Variable x2 : set
Hypothesis H0 : x ∈ real
Hypothesis H1 : SNo x
Hypothesis H3 : SNo (x + y)
Hypothesis H4 : (∀y2 : set, y2 ∈ SNoS_ ω → (∀z2 : set, z2 ∈ ω → abs_SNo (y2 + - x) < eps_ z2) → y2 = x)
Hypothesis H5 : (∀y2 : set, y2 ∈ ω → SNo (ap (Sigma ω (λz2 : set ⇒ ap w (ordsucc z2) + ap v (ordsucc z2))) y2))
Hypothesis H6 : (∀y2 : set, y2 ∈ ω → (ap (Sigma ω (λz2 : set ⇒ ap w (ordsucc z2) + ap v (ordsucc z2))) y2 + - (eps_ y2)) < x + y)
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))))))
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)
Hypothesis H9 : SNo x2
Hypothesis H10 : SNoLev x2 ∈ SNoLev x
Hypothesis H11 : x < x2
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
Variable x : set
Variable y : set
Variable z : set
Variable w : set
Variable u : set
Variable v : set
Variable x2 : set
Hypothesis H0 : x ∈ real
Hypothesis H1 : SNo x
Hypothesis H2 : SNo y
Hypothesis H4 : (∀y2 : set, y2 ∈ SNoS_ ω → (∀z2 : set, z2 ∈ ω → abs_SNo (y2 + - x) < eps_ z2) → y2 = x)
Hypothesis H5 : (∀y2 : set, y2 ∈ ω → SNo (ap (Sigma ω (λz2 : set ⇒ ap w (ordsucc z2) + ap v (ordsucc z2))) y2))
Hypothesis H6 : (∀y2 : set, y2 ∈ ω → (ap (Sigma ω (λz2 : set ⇒ ap w (ordsucc z2) + ap v (ordsucc z2))) y2 + - (eps_ y2)) < x + y)
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))))))
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)
Hypothesis H9 : SNo x2
Hypothesis H10 : SNoLev x2 ∈ SNoLev x
Hypothesis H11 : x < x2
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
Variable x : set
Variable y : set
Variable z : set
Variable w : set
Variable u : set
Hypothesis H0 : SNo x
Hypothesis H1 : SNo y
Hypothesis H3 : (∀v : set, v ∈ SNoS_ ω → (∀x2 : set, x2 ∈ ω → abs_SNo (v + - x) < eps_ x2) → v = x)
Hypothesis H4 : (∀v : set, v ∈ ω → SNo (ap (Sigma ω (λx2 : set ⇒ ap z (ordsucc x2) + ap w (ordsucc x2))) v))
Hypothesis H5 : (∀v : set, v ∈ ω → (x + y) < ap (Sigma ω (λx2 : set ⇒ ap z (ordsucc x2) + ap w (ordsucc x2))) v + eps_ v)
Hypothesis H6 : SNo u
Hypothesis H7 : u < x
Hypothesis H8 : u ∈ SNoS_ ω
Hypothesis H9 : ¬ (∃v : set, v ∈ ω ∧ (u + y) ≤ ap (Sigma ω (λx2 : set ⇒ ap z (ordsucc x2) + ap w (ordsucc x2))) v)
Hypothesis H10 : Empty < x + - u
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
Variable x : set
Variable y : set
Variable z : set
Variable w : set
Variable u : set
Hypothesis H0 : SNo x
Hypothesis H1 : SNo y
Hypothesis H2 : SNo (x + y)
Hypothesis H3 : (∀v : set, v ∈ SNoS_ ω → (∀x2 : set, x2 ∈ ω → abs_SNo (v + - x) < eps_ x2) → v = x)
Hypothesis H4 : (∀v : set, v ∈ ω → SNo (ap (Sigma ω (λx2 : set ⇒ ap z (ordsucc x2) + ap w (ordsucc x2))) v))
Hypothesis H5 : (∀v : set, v ∈ ω → (x + y) < ap (Sigma ω (λx2 : set ⇒ ap z (ordsucc x2) + ap w (ordsucc x2))) v + eps_ v)
Hypothesis H7 : u < x
Hypothesis H8 : u ∈ SNoS_ ω
Hypothesis H9 : ¬ (∃v : set, v ∈ ω ∧ (u + y) ≤ ap (Sigma ω (λx2 : set ⇒ ap z (ordsucc x2) + ap w (ordsucc x2))) v)
Hypothesis H10 : Empty < x + - u
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
Variable x : set
Variable y : set
Variable z : set
Variable w : set
Variable u : set
Variable v : set
Hypothesis H0 : x ∈ real
Hypothesis H1 : y ∈ real
Hypothesis H2 : SNo x
Hypothesis H3 : SNo y
Hypothesis H4 : SNo (x + y)
Hypothesis H5 : (∀x2 : set, x2 ∈ SNoS_ ω → (∀y2 : set, y2 ∈ ω → abs_SNo (x2 + - x) < eps_ y2) → x2 = x)
Hypothesis H7 : (∀x2 : set, x2 ∈ ω → SNo (ap (Sigma ω (λy2 : set ⇒ ap z (ordsucc y2) + ap u (ordsucc y2))) x2))
Hypothesis H8 : (∀x2 : set, x2 ∈ ω → SNo (ap (Sigma ω (λy2 : set ⇒ ap w (ordsucc y2) + ap v (ordsucc y2))) x2))
Hypothesis H9 : Sigma ω (λx2 : set ⇒ ap z (ordsucc x2) + ap u (ordsucc x2)) ∈ setexp (SNoS_ ω) ω
Hypothesis H10 : Sigma ω (λx2 : set ⇒ ap w (ordsucc x2) + ap v (ordsucc x2)) ∈ setexp (SNoS_ ω) ω
Hypothesis H11 : (∀x2 : set, x2 ∈ ω → ap (Sigma ω (λy2 : set ⇒ ap z (ordsucc y2) + ap u (ordsucc y2))) x2 < x + y)
Hypothesis H12 : (∀x2 : set, x2 ∈ ω → (x + y) < ap (Sigma ω (λy2 : set ⇒ ap z (ordsucc y2) + ap u (ordsucc y2))) x2 + eps_ x2)
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))
Hypothesis H14 : (∀x2 : set, x2 ∈ ω → (ap (Sigma ω (λy2 : set ⇒ ap w (ordsucc y2) + ap v (ordsucc y2))) x2 + - (eps_ x2)) < x + y)
Hypothesis H15 : (∀x2 : set, x2 ∈ ω → (x + y) < ap (Sigma ω (λy2 : set ⇒ ap w (ordsucc y2) + ap v (ordsucc y2))) x2)
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))
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)))))
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))))))
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))))))
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)
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
Variable x : set
Variable y : set
Variable z : set
Variable w : set
Variable u : set
Variable v : set
Hypothesis H0 : x ∈ real
Hypothesis H1 : y ∈ real
Hypothesis H2 : SNo x
Hypothesis H3 : SNo y
Hypothesis H4 : SNo (x + y)
Hypothesis H5 : (∀x2 : set, x2 ∈ SNoS_ ω → (∀y2 : set, y2 ∈ ω → abs_SNo (x2 + - x) < eps_ y2) → x2 = x)
Hypothesis H6 : (∀x2 : set, x2 ∈ SNoS_ ω → (∀y2 : set, y2 ∈ ω → abs_SNo (x2 + - y) < eps_ y2) → x2 = y)
Hypothesis H7 : (∀x2 : set, x2 ∈ ω → SNo (ap (Sigma ω (λy2 : set ⇒ ap z (ordsucc y2) + ap u (ordsucc y2))) x2))
Hypothesis H9 : Sigma ω (λx2 : set ⇒ ap z (ordsucc x2) + ap u (ordsucc x2)) ∈ setexp (SNoS_ ω) ω
Hypothesis H10 : Sigma ω (λx2 : set ⇒ ap w (ordsucc x2) + ap v (ordsucc x2)) ∈ setexp (SNoS_ ω) ω
Hypothesis H11 : (∀x2 : set, x2 ∈ ω → ap (Sigma ω (λy2 : set ⇒ ap z (ordsucc y2) + ap u (ordsucc y2))) x2 < x + y)
Hypothesis H12 : (∀x2 : set, x2 ∈ ω → (x + y) < ap (Sigma ω (λy2 : set ⇒ ap z (ordsucc y2) + ap u (ordsucc y2))) x2 + eps_ x2)
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))
Hypothesis H14 : (∀x2 : set, x2 ∈ ω → (ap (Sigma ω (λy2 : set ⇒ ap w (ordsucc y2) + ap v (ordsucc y2))) x2 + - (eps_ x2)) < x + y)
Hypothesis H15 : (∀x2 : set, x2 ∈ ω → (x + y) < ap (Sigma ω (λy2 : set ⇒ ap w (ordsucc y2) + ap v (ordsucc y2))) x2)
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))
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)))))
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))))))
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))))))
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)
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
Variable x : set
Variable y : set
Variable z : set
Variable w : set
Variable u : set
Variable v : set
Hypothesis H1 : y ∈ real
Hypothesis H2 : SNo x
Hypothesis H3 : SNo y
Hypothesis H4 : SNo (x + y)
Hypothesis H5 : (∀x2 : set, x2 ∈ SNoS_ ω → (∀y2 : set, y2 ∈ ω → abs_SNo (x2 + - x) < eps_ y2) → x2 = x)
Hypothesis H6 : (∀x2 : set, x2 ∈ SNoS_ ω → (∀y2 : set, y2 ∈ ω → abs_SNo (x2 + - y) < eps_ y2) → x2 = y)
Hypothesis H7 : (∀x2 : set, x2 ∈ ω → SNo (ap (Sigma ω (λy2 : set ⇒ ap z (ordsucc y2) + ap u (ordsucc y2))) x2))
Hypothesis H8 : (∀x2 : set, x2 ∈ ω → SNo (ap (Sigma ω (λy2 : set ⇒ ap w (ordsucc y2) + ap v (ordsucc y2))) x2))
Hypothesis H9 : Sigma ω (λx2 : set ⇒ ap z (ordsucc x2) + ap u (ordsucc x2)) ∈ setexp (SNoS_ ω) ω
Hypothesis H10 : Sigma ω (λx2 : set ⇒ ap w (ordsucc x2) + ap v (ordsucc x2)) ∈ setexp (SNoS_ ω) ω
Hypothesis H11 : (∀x2 : set, x2 ∈ ω → ap (Sigma ω (λy2 : set ⇒ ap z (ordsucc y2) + ap u (ordsucc y2))) x2 < x + y)
Hypothesis H12 : (∀x2 : set, x2 ∈ ω → (x + y) < ap (Sigma ω (λy2 : set ⇒ ap z (ordsucc y2) + ap u (ordsucc y2))) x2 + eps_ x2)
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))
Hypothesis H14 : (∀x2 : set, x2 ∈ ω → (ap (Sigma ω (λy2 : set ⇒ ap w (ordsucc y2) + ap v (ordsucc y2))) x2 + - (eps_ x2)) < x + y)
Hypothesis H15 : (∀x2 : set, x2 ∈ ω → (x + y) < ap (Sigma ω (λy2 : set ⇒ ap w (ordsucc y2) + ap v (ordsucc y2))) x2)
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))
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
Variable x : set
Variable y : set
Variable z : set
Variable w : set
Variable u : set
Variable v : set
Hypothesis H0 : x ∈ real
Hypothesis H1 : y ∈ real
Hypothesis H2 : SNo x
Hypothesis H3 : SNo y
Hypothesis H4 : SNo (x + y)
Hypothesis H5 : (∀x2 : set, x2 ∈ SNoS_ ω → (∀y2 : set, y2 ∈ ω → abs_SNo (x2 + - x) < eps_ y2) → x2 = x)
Hypothesis H6 : (∀x2 : set, x2 ∈ SNoS_ ω → (∀y2 : set, y2 ∈ ω → abs_SNo (x2 + - y) < eps_ y2) → x2 = y)
Hypothesis H7 : (∀x2 : set, x2 ∈ ω → SNo (ap (Sigma ω (λy2 : set ⇒ ap z (ordsucc y2) + ap u (ordsucc y2))) x2))
Hypothesis H8 : (∀x2 : set, x2 ∈ ω → SNo (ap (Sigma ω (λy2 : set ⇒ ap w (ordsucc y2) + ap v (ordsucc y2))) x2))
Hypothesis H9 : Sigma ω (λx2 : set ⇒ ap z (ordsucc x2) + ap u (ordsucc x2)) ∈ setexp (SNoS_ ω) ω
Hypothesis H10 : Sigma ω (λx2 : set ⇒ ap w (ordsucc x2) + ap v (ordsucc x2)) ∈ setexp (SNoS_ ω) ω
Hypothesis H11 : (∀x2 : set, x2 ∈ ω → ap (Sigma ω (λy2 : set ⇒ ap z (ordsucc y2) + ap u (ordsucc y2))) x2 < x + y)
Hypothesis H12 : (∀x2 : set, x2 ∈ ω → (x + y) < ap (Sigma ω (λy2 : set ⇒ ap z (ordsucc y2) + ap u (ordsucc y2))) x2 + eps_ x2)
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))
Hypothesis H15 : (∀x2 : set, x2 ∈ ω → (x + y) < ap (Sigma ω (λy2 : set ⇒ ap w (ordsucc y2) + ap v (ordsucc y2))) x2)
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))
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
Variable x : set
Variable y : set
Variable z : set
Variable w : set
Variable u : set
Variable v : set
Hypothesis H0 : x ∈ real
Hypothesis H1 : y ∈ real
Hypothesis H2 : (∀x2 : set, x2 ∈ ω → x < ap w x2)
Hypothesis H3 : (∀x2 : set, x2 ∈ ω → (∀y2 : set, y2 ∈ x2 → ap w x2 < ap w y2))
Hypothesis H4 : (∀x2 : set, x2 ∈ ω → y < ap v x2)
Hypothesis H5 : (∀x2 : set, x2 ∈ ω → (∀y2 : set, y2 ∈ x2 → ap v x2 < ap v y2))
Hypothesis H6 : SNo x
Hypothesis H7 : SNo y
Hypothesis H8 : SNo (x + y)
Hypothesis H9 : (∀x2 : set, x2 ∈ SNoS_ ω → (∀y2 : set, y2 ∈ ω → abs_SNo (x2 + - x) < eps_ y2) → x2 = x)
Hypothesis H10 : (∀x2 : set, x2 ∈ SNoS_ ω → (∀y2 : set, y2 ∈ ω → abs_SNo (x2 + - y) < eps_ y2) → x2 = y)
Hypothesis H11 : (∀x2 : set, x2 ∈ ω → SNo (ap w (ordsucc x2)))
Hypothesis H12 : (∀x2 : set, x2 ∈ ω → SNo (ap v (ordsucc x2)))
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))
Hypothesis H14 : (∀x2 : set, x2 ∈ ω → SNo (ap (Sigma ω (λy2 : set ⇒ ap z (ordsucc y2) + ap u (ordsucc y2))) x2))
Hypothesis H15 : (∀x2 : set, x2 ∈ ω → SNo (ap (Sigma ω (λy2 : set ⇒ ap w (ordsucc y2) + ap v (ordsucc y2))) x2))
Hypothesis H16 : Sigma ω (λx2 : set ⇒ ap z (ordsucc x2) + ap u (ordsucc x2)) ∈ setexp (SNoS_ ω) ω
Hypothesis H17 : Sigma ω (λx2 : set ⇒ ap w (ordsucc x2) + ap v (ordsucc x2)) ∈ setexp (SNoS_ ω) ω
Hypothesis H19 : (∀x2 : set, x2 ∈ ω → (x + y) < ap (Sigma ω (λy2 : set ⇒ ap z (ordsucc y2) + ap u (ordsucc y2))) x2 + eps_ x2)
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))
Hypothesis H21 : (∀x2 : set, x2 ∈ ω → (ap (Sigma ω (λy2 : set ⇒ ap w (ordsucc y2) + ap v (ordsucc y2))) x2 + - (eps_ x2)) < x + y)
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
Variable x : set
Variable y : set
Variable z : set
Variable w : set
Variable u : set
Variable v : set
Hypothesis H0 : x ∈ real
Hypothesis H1 : y ∈ real
Hypothesis H2 : (∀x2 : set, x2 ∈ ω → ap z x2 < x)
Hypothesis H3 : (∀x2 : set, x2 ∈ ω → x < ap z x2 + eps_ x2)
Hypothesis H4 : (∀x2 : set, x2 ∈ ω → (∀y2 : set, y2 ∈ x2 → ap z y2 < ap z x2))
Hypothesis H5 : (∀x2 : set, x2 ∈ ω → (ap w x2 + - (eps_ x2)) < x)
Hypothesis H6 : (∀x2 : set, x2 ∈ ω → x < ap w x2)
Hypothesis H7 : (∀x2 : set, x2 ∈ ω → (∀y2 : set, y2 ∈ x2 → ap w x2 < ap w y2))
Hypothesis H9 : (∀x2 : set, x2 ∈ ω → y < ap u x2 + eps_ x2)
Hypothesis H10 : (∀x2 : set, x2 ∈ ω → (∀y2 : set, y2 ∈ x2 → ap u y2 < ap u x2))
Hypothesis H11 : (∀x2 : set, x2 ∈ ω → (ap v x2 + - (eps_ x2)) < y)
Hypothesis H12 : (∀x2 : set, x2 ∈ ω → y < ap v x2)
Hypothesis H13 : (∀x2 : set, x2 ∈ ω → (∀y2 : set, y2 ∈ x2 → ap v x2 < ap v y2))
Hypothesis H14 : SNo x
Hypothesis H15 : SNo y
Hypothesis H16 : SNo (x + y)
Hypothesis H17 : (∀x2 : set, x2 ∈ SNoS_ ω → (∀y2 : set, y2 ∈ ω → abs_SNo (x2 + - x) < eps_ y2) → x2 = x)
Hypothesis H18 : (∀x2 : set, x2 ∈ SNoS_ ω → (∀y2 : set, y2 ∈ ω → abs_SNo (x2 + - y) < eps_ y2) → x2 = y)
Hypothesis H19 : (∀x2 : set, x2 ∈ ω → SNo (ap z (ordsucc x2)))
Hypothesis H20 : (∀x2 : set, x2 ∈ ω → SNo (ap u (ordsucc x2)))
Hypothesis H21 : (∀x2 : set, x2 ∈ ω → SNo (ap w (ordsucc x2)))
Hypothesis H22 : (∀x2 : set, x2 ∈ ω → SNo (ap v (ordsucc x2)))
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))
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))
Hypothesis H25 : (∀x2 : set, x2 ∈ ω → SNo (ap (Sigma ω (λy2 : set ⇒ ap z (ordsucc y2) + ap u (ordsucc y2))) x2))
Hypothesis H26 : (∀x2 : set, x2 ∈ ω → SNo (ap (Sigma ω (λy2 : set ⇒ ap w (ordsucc y2) + ap v (ordsucc y2))) x2))
Hypothesis H27 : Sigma ω (λx2 : set ⇒ ap z (ordsucc x2) + ap u (ordsucc x2)) ∈ setexp (SNoS_ ω) ω
Hypothesis H28 : Sigma ω (λx2 : set ⇒ ap w (ordsucc x2) + ap v (ordsucc x2)) ∈ setexp (SNoS_ ω) ω
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
Variable x : set
Variable y : set
Variable z : set
Variable w : set
Variable u : set
Variable v : set
Hypothesis H0 : x ∈ real
Hypothesis H1 : y ∈ real
Hypothesis H2 : (∀x2 : set, x2 ∈ ω → ap z x2 < x)
Hypothesis H3 : (∀x2 : set, x2 ∈ ω → x < ap z x2 + eps_ x2)
Hypothesis H4 : (∀x2 : set, x2 ∈ ω → (∀y2 : set, y2 ∈ x2 → ap z y2 < ap z x2))
Hypothesis H5 : (∀x2 : set, x2 ∈ ω → (ap w x2 + - (eps_ x2)) < x)
Hypothesis H6 : (∀x2 : set, x2 ∈ ω → x < ap w x2)
Hypothesis H7 : (∀x2 : set, x2 ∈ ω → (∀y2 : set, y2 ∈ x2 → ap w x2 < ap w y2))
Hypothesis H8 : (∀x2 : set, x2 ∈ ω → ap u x2 < y)
Hypothesis H9 : (∀x2 : set, x2 ∈ ω → y < ap u x2 + eps_ x2)
Hypothesis H10 : (∀x2 : set, x2 ∈ ω → (∀y2 : set, y2 ∈ x2 → ap u y2 < ap u x2))
Hypothesis H11 : (∀x2 : set, x2 ∈ ω → (ap v x2 + - (eps_ x2)) < y)
Hypothesis H12 : (∀x2 : set, x2 ∈ ω → y < ap v x2)
Hypothesis H13 : (∀x2 : set, x2 ∈ ω → (∀y2 : set, y2 ∈ x2 → ap v x2 < ap v y2))
Hypothesis H14 : SNo x
Hypothesis H15 : SNo y
Hypothesis H16 : SNo (x + y)
Hypothesis H17 : (∀x2 : set, x2 ∈ SNoS_ ω → (∀y2 : set, y2 ∈ ω → abs_SNo (x2 + - x) < eps_ y2) → x2 = x)
Hypothesis H18 : (∀x2 : set, x2 ∈ SNoS_ ω → (∀y2 : set, y2 ∈ ω → abs_SNo (x2 + - y) < eps_ y2) → x2 = y)
Hypothesis H19 : (∀x2 : set, x2 ∈ ω → SNo (ap z (ordsucc x2)))
Hypothesis H20 : (∀x2 : set, x2 ∈ ω → SNo (ap u (ordsucc x2)))
Hypothesis H21 : (∀x2 : set, x2 ∈ ω → SNo (ap w (ordsucc x2)))
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))
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))
Hypothesis H25 : (∀x2 : set, x2 ∈ ω → SNo (ap (Sigma ω (λy2 : set ⇒ ap z (ordsucc y2) + ap u (ordsucc y2))) x2))
Hypothesis H26 : (∀x2 : set, x2 ∈ ω → SNo (ap (Sigma ω (λy2 : set ⇒ ap w (ordsucc y2) + ap v (ordsucc y2))) x2))
Hypothesis H27 : Sigma ω (λx2 : set ⇒ ap z (ordsucc x2) + ap u (ordsucc x2)) ∈ setexp (SNoS_ ω) ω
Hypothesis H28 : Sigma ω (λx2 : set ⇒ ap w (ordsucc x2) + ap v (ordsucc x2)) ∈ setexp (SNoS_ ω) ω
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
Variable x : set
Variable y : set
Variable z : set
Variable w : set
Variable u : set
Variable v : set
Hypothesis H0 : x ∈ real
Hypothesis H1 : y ∈ real
Hypothesis H2 : (∀x2 : set, x2 ∈ ω → ap z x2 < x)
Hypothesis H4 : (∀x2 : set, x2 ∈ ω → (∀y2 : set, y2 ∈ x2 → ap z y2 < ap z x2))
Hypothesis H5 : (∀x2 : set, x2 ∈ ω → (ap w x2 + - (eps_ x2)) < x)
Hypothesis H6 : (∀x2 : set, x2 ∈ ω → x < ap w x2)
Hypothesis H7 : (∀x2 : set, x2 ∈ ω → (∀y2 : set, y2 ∈ x2 → ap w x2 < ap w y2))
Hypothesis H8 : (∀x2 : set, x2 ∈ ω → ap u x2 < y)
Hypothesis H9 : (∀x2 : set, x2 ∈ ω → y < ap u x2 + eps_ x2)
Hypothesis H10 : (∀x2 : set, x2 ∈ ω → (∀y2 : set, y2 ∈ x2 → ap u y2 < ap u x2))
Hypothesis H11 : (∀x2 : set, x2 ∈ ω → (ap v x2 + - (eps_ x2)) < y)
Hypothesis H12 : (∀x2 : set, x2 ∈ ω → y < ap v x2)
Hypothesis H13 : (∀x2 : set, x2 ∈ ω → (∀y2 : set, y2 ∈ x2 → ap v x2 < ap v y2))
Hypothesis H14 : SNo x
Hypothesis H15 : SNo y
Hypothesis H16 : SNo (x + y)
Hypothesis H17 : (∀x2 : set, x2 ∈ SNoS_ ω → (∀y2 : set, y2 ∈ ω → abs_SNo (x2 + - x) < eps_ y2) → x2 = x)
Hypothesis H18 : (∀x2 : set, x2 ∈ SNoS_ ω → (∀y2 : set, y2 ∈ ω → abs_SNo (x2 + - y) < eps_ y2) → x2 = y)
Hypothesis H19 : (∀x2 : set, x2 ∈ ω → SNo (ap z (ordsucc x2)))
Hypothesis H20 : (∀x2 : set, x2 ∈ ω → SNo (ap u (ordsucc x2)))
Hypothesis H21 : (∀x2 : set, x2 ∈ ω → ap w (ordsucc x2) ∈ SNoS_ ω)
Hypothesis H22 : (∀x2 : set, x2 ∈ ω → SNo (ap w (ordsucc x2)))
Hypothesis H23 : (∀x2 : set, x2 ∈ ω → ap v (ordsucc x2) ∈ SNoS_ ω)
Hypothesis H24 : (∀x2 : set, x2 ∈ ω → SNo (ap v (ordsucc x2)))
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))
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))
Hypothesis H27 : (∀x2 : set, x2 ∈ ω → SNo (ap (Sigma ω (λy2 : set ⇒ ap z (ordsucc y2) + ap u (ordsucc y2))) x2))
Hypothesis H28 : (∀x2 : set, x2 ∈ ω → SNo (ap (Sigma ω (λy2 : set ⇒ ap w (ordsucc y2) + ap v (ordsucc y2))) x2))
Hypothesis H29 : Sigma ω (λx2 : set ⇒ ap z (ordsucc x2) + ap u (ordsucc x2)) ∈ setexp (SNoS_ ω) ω
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
Variable x : set
Variable y : set
Variable z : set
Variable w : set
Variable u : set
Variable v : set
Hypothesis H0 : x ∈ real
Hypothesis H1 : y ∈ real
Hypothesis H2 : (∀x2 : set, x2 ∈ ω → ap z x2 < x)
Hypothesis H3 : (∀x2 : set, x2 ∈ ω → x < ap z x2 + eps_ x2)
Hypothesis H4 : (∀x2 : set, x2 ∈ ω → (∀y2 : set, y2 ∈ x2 → ap z y2 < ap z x2))
Hypothesis H5 : (∀x2 : set, x2 ∈ ω → (ap w x2 + - (eps_ x2)) < x)
Hypothesis H6 : (∀x2 : set, x2 ∈ ω → x < ap w x2)
Hypothesis H8 : (∀x2 : set, x2 ∈ ω → ap u x2 < y)
Hypothesis H9 : (∀x2 : set, x2 ∈ ω → y < ap u x2 + eps_ x2)
Hypothesis H10 : (∀x2 : set, x2 ∈ ω → (∀y2 : set, y2 ∈ x2 → ap u y2 < ap u x2))
Hypothesis H11 : (∀x2 : set, x2 ∈ ω → (ap v x2 + - (eps_ x2)) < y)
Hypothesis H12 : (∀x2 : set, x2 ∈ ω → y < ap v x2)
Hypothesis H13 : (∀x2 : set, x2 ∈ ω → (∀y2 : set, y2 ∈ x2 → ap v x2 < ap v y2))
Hypothesis H14 : SNo x
Hypothesis H15 : SNo y
Hypothesis H16 : SNo (x + y)
Hypothesis H17 : (∀x2 : set, x2 ∈ SNoS_ ω → (∀y2 : set, y2 ∈ ω → abs_SNo (x2 + - x) < eps_ y2) → x2 = x)
Hypothesis H18 : (∀x2 : set, x2 ∈ SNoS_ ω → (∀y2 : set, y2 ∈ ω → abs_SNo (x2 + - y) < eps_ y2) → x2 = y)
Hypothesis H19 : (∀x2 : set, x2 ∈ ω → SNo (ap z (ordsucc x2)))
Hypothesis H20 : (∀x2 : set, x2 ∈ ω → SNo (ap u (ordsucc x2)))
Hypothesis H21 : (∀x2 : set, x2 ∈ ω → ap w (ordsucc x2) ∈ SNoS_ ω)
Hypothesis H22 : (∀x2 : set, x2 ∈ ω → SNo (ap w (ordsucc x2)))
Hypothesis H23 : (∀x2 : set, x2 ∈ ω → ap v (ordsucc x2) ∈ SNoS_ ω)
Hypothesis H24 : (∀x2 : set, x2 ∈ ω → SNo (ap v (ordsucc x2)))
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))
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))
Hypothesis H27 : (∀x2 : set, x2 ∈ ω → SNo (ap (Sigma ω (λy2 : set ⇒ ap z (ordsucc y2) + ap u (ordsucc y2))) x2))
Hypothesis H28 : (∀x2 : set, x2 ∈ ω → SNo (ap (Sigma ω (λy2 : set ⇒ ap w (ordsucc y2) + ap v (ordsucc y2))) x2))
Hypothesis H29 : Sigma ω (λx2 : set ⇒ ap z (ordsucc x2) + ap u (ordsucc x2)) ∈ setexp (SNoS_ ω) ω
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
Variable x : set
Variable y : set
Variable z : set
Variable w : set
Variable u : set
Variable v : set
Hypothesis H0 : x ∈ real
Hypothesis H1 : y ∈ real
Hypothesis H2 : (∀x2 : set, x2 ∈ ω → ap z x2 < x)
Hypothesis H3 : (∀x2 : set, x2 ∈ ω → x < ap z x2 + eps_ x2)
Hypothesis H4 : (∀x2 : set, x2 ∈ ω → (∀y2 : set, y2 ∈ x2 → ap z y2 < ap z x2))
Hypothesis H5 : (∀x2 : set, x2 ∈ ω → (ap w x2 + - (eps_ x2)) < x)
Hypothesis H6 : (∀x2 : set, x2 ∈ ω → x < ap w x2)
Hypothesis H7 : (∀x2 : set, x2 ∈ ω → (∀y2 : set, y2 ∈ x2 → ap w x2 < ap w y2))
Hypothesis H8 : (∀x2 : set, x2 ∈ ω → ap u x2 < y)
Hypothesis H9 : (∀x2 : set, x2 ∈ ω → y < ap u x2 + eps_ x2)
Hypothesis H10 : (∀x2 : set, x2 ∈ ω → (∀y2 : set, y2 ∈ x2 → ap u y2 < ap u x2))
Hypothesis H11 : (∀x2 : set, x2 ∈ ω → (ap v x2 + - (eps_ x2)) < y)
Hypothesis H12 : (∀x2 : set, x2 ∈ ω → y < ap v x2)
Hypothesis H13 : (∀x2 : set, x2 ∈ ω → (∀y2 : set, y2 ∈ x2 → ap v x2 < ap v y2))
Hypothesis H14 : SNo x
Hypothesis H15 : SNo y
Hypothesis H16 : SNo (x + y)
Hypothesis H17 : (∀x2 : set, x2 ∈ SNoS_ ω → (∀y2 : set, y2 ∈ ω → abs_SNo (x2 + - x) < eps_ y2) → x2 = x)
Hypothesis H18 : (∀x2 : set, x2 ∈ SNoS_ ω → (∀y2 : set, y2 ∈ ω → abs_SNo (x2 + - y) < eps_ y2) → x2 = y)
Hypothesis H19 : (∀x2 : set, x2 ∈ ω → SNo (ap z (ordsucc x2)))
Hypothesis H20 : (∀x2 : set, x2 ∈ ω → SNo (ap u (ordsucc x2)))
Hypothesis H21 : (∀x2 : set, x2 ∈ ω → ap w (ordsucc x2) ∈ SNoS_ ω)
Hypothesis H22 : (∀x2 : set, x2 ∈ ω → SNo (ap w (ordsucc x2)))
Hypothesis H23 : (∀x2 : set, x2 ∈ ω → ap v (ordsucc x2) ∈ SNoS_ ω)
Hypothesis H24 : (∀x2 : set, x2 ∈ ω → SNo (ap v (ordsucc x2)))
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))
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))
Hypothesis H27 : (∀x2 : set, x2 ∈ ω → SNo (ap (Sigma ω (λy2 : set ⇒ ap z (ordsucc y2) + ap u (ordsucc y2))) x2))
Hypothesis H29 : Sigma ω (λx2 : set ⇒ ap z (ordsucc x2) + ap u (ordsucc x2)) ∈ setexp (SNoS_ ω) ω
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
Variable x : set
Variable y : set
Variable z : set
Variable w : set
Variable u : set
Variable v : set
Hypothesis H0 : x ∈ real
Hypothesis H1 : y ∈ real
Hypothesis H2 : (∀x2 : set, x2 ∈ ω → ap z x2 < x)
Hypothesis H3 : (∀x2 : set, x2 ∈ ω → x < ap z x2 + eps_ x2)
Hypothesis H4 : (∀x2 : set, x2 ∈ ω → (∀y2 : set, y2 ∈ x2 → ap z y2 < ap z x2))
Hypothesis H5 : (∀x2 : set, x2 ∈ ω → (ap w x2 + - (eps_ x2)) < x)
Hypothesis H6 : (∀x2 : set, x2 ∈ ω → x < ap w x2)
Hypothesis H7 : (∀x2 : set, x2 ∈ ω → (∀y2 : set, y2 ∈ x2 → ap w x2 < ap w y2))
Hypothesis H8 : (∀x2 : set, x2 ∈ ω → ap u x2 < y)
Hypothesis H9 : (∀x2 : set, x2 ∈ ω → y < ap u x2 + eps_ x2)
Hypothesis H10 : (∀x2 : set, x2 ∈ ω → (∀y2 : set, y2 ∈ x2 → ap u y2 < ap u x2))
Hypothesis H11 : (∀x2 : set, x2 ∈ ω → (ap v x2 + - (eps_ x2)) < y)
Hypothesis H12 : (∀x2 : set, x2 ∈ ω → y < ap v x2)
Hypothesis H13 : (∀x2 : set, x2 ∈ ω → (∀y2 : set, y2 ∈ x2 → ap v x2 < ap v y2))
Hypothesis H14 : SNo x
Hypothesis H15 : SNo y
Hypothesis H16 : SNo (x + y)
Hypothesis H17 : (∀x2 : set, x2 ∈ SNoS_ ω → (∀y2 : set, y2 ∈ ω → abs_SNo (x2 + - x) < eps_ y2) → x2 = x)
Hypothesis H18 : (∀x2 : set, x2 ∈ SNoS_ ω → (∀y2 : set, y2 ∈ ω → abs_SNo (x2 + - y) < eps_ y2) → x2 = y)
Hypothesis H19 : (∀x2 : set, x2 ∈ ω → ap z (ordsucc x2) ∈ SNoS_ ω)
Hypothesis H20 : (∀x2 : set, x2 ∈ ω → SNo (ap z (ordsucc x2)))
Hypothesis H21 : (∀x2 : set, x2 ∈ ω → ap u (ordsucc x2) ∈ SNoS_ ω)
Hypothesis H22 : (∀x2 : set, x2 ∈ ω → SNo (ap u (ordsucc x2)))
Hypothesis H23 : (∀x2 : set, x2 ∈ ω → ap w (ordsucc x2) ∈ SNoS_ ω)
Hypothesis H25 : (∀x2 : set, x2 ∈ ω → ap v (ordsucc x2) ∈ SNoS_ ω)
Hypothesis H26 : (∀x2 : set, x2 ∈ ω → SNo (ap v (ordsucc x2)))
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))
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))
Hypothesis H29 : (∀x2 : set, x2 ∈ ω → SNo (ap (Sigma ω (λy2 : set ⇒ ap z (ordsucc y2) + ap u (ordsucc y2))) x2))
Hypothesis H30 : (∀x2 : set, x2 ∈ ω → SNo (ap (Sigma ω (λy2 : set ⇒ ap w (ordsucc y2) + ap v (ordsucc y2))) x2))
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
Variable x : set
Variable y : set
Variable z : set
Variable w : set
Variable u : set
Variable v : set
Hypothesis H0 : x ∈ real
Hypothesis H1 : y ∈ real
Hypothesis H2 : (∀x2 : set, x2 ∈ ω → ap z x2 < x)
Hypothesis H3 : (∀x2 : set, x2 ∈ ω → x < ap z x2 + eps_ x2)
Hypothesis H4 : (∀x2 : set, x2 ∈ ω → (∀y2 : set, y2 ∈ x2 → ap z y2 < ap z x2))
Hypothesis H5 : (∀x2 : set, x2 ∈ ω → (ap w x2 + - (eps_ x2)) < x)
Hypothesis H6 : (∀x2 : set, x2 ∈ ω → x < ap w x2)
Hypothesis H7 : (∀x2 : set, x2 ∈ ω → (∀y2 : set, y2 ∈ x2 → ap w x2 < ap w y2))
Hypothesis H8 : (∀x2 : set, x2 ∈ ω → ap u x2 < y)
Hypothesis H9 : (∀x2 : set, x2 ∈ ω → y < ap u x2 + eps_ x2)
Hypothesis H10 : (∀x2 : set, x2 ∈ ω → (∀y2 : set, y2 ∈ x2 → ap u y2 < ap u x2))
Hypothesis H11 : (∀x2 : set, x2 ∈ ω → (ap v x2 + - (eps_ x2)) < y)
Hypothesis H12 : (∀x2 : set, x2 ∈ ω → y < ap v x2)
Hypothesis H13 : (∀x2 : set, x2 ∈ ω → (∀y2 : set, y2 ∈ x2 → ap v x2 < ap v y2))
Hypothesis H14 : SNo x
Hypothesis H15 : SNo y
Hypothesis H16 : SNo (x + y)
Hypothesis H18 : (∀x2 : set, x2 ∈ SNoS_ ω → (∀y2 : set, y2 ∈ ω → abs_SNo (x2 + - y) < eps_ y2) → x2 = y)
Hypothesis H19 : (∀x2 : set, x2 ∈ ω → ap z (ordsucc x2) ∈ SNoS_ ω)
Hypothesis H20 : (∀x2 : set, x2 ∈ ω → SNo (ap z (ordsucc x2)))
Hypothesis H21 : (∀x2 : set, x2 ∈ ω → ap u (ordsucc x2) ∈ SNoS_ ω)
Hypothesis H22 : (∀x2 : set, x2 ∈ ω → SNo (ap u (ordsucc x2)))
Hypothesis H23 : (∀x2 : set, x2 ∈ ω → ap w (ordsucc x2) ∈ SNoS_ ω)
Hypothesis H24 : (∀x2 : set, x2 ∈ ω → SNo (ap w (ordsucc x2)))
Hypothesis H25 : (∀x2 : set, x2 ∈ ω → ap v (ordsucc x2) ∈ SNoS_ ω)
Hypothesis H26 : (∀x2 : set, x2 ∈ ω → SNo (ap v (ordsucc x2)))
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))
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))
Hypothesis H29 : (∀x2 : set, x2 ∈ ω → SNo (ap (Sigma ω (λy2 : set ⇒ ap z (ordsucc y2) + ap u (ordsucc y2))) x2))
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
Variable x : set
Variable y : set
Variable z : set
Variable w : set
Variable u : set
Variable v : set
Hypothesis H0 : x ∈ real
Hypothesis H1 : y ∈ real
Hypothesis H3 : (∀x2 : set, x2 ∈ ω → x < ap z x2 + eps_ x2)
Hypothesis H4 : (∀x2 : set, x2 ∈ ω → (∀y2 : set, y2 ∈ x2 → ap z y2 < ap z x2))
Hypothesis H5 : (∀x2 : set, x2 ∈ ω → (ap w x2 + - (eps_ x2)) < x)
Hypothesis H6 : (∀x2 : set, x2 ∈ ω → x < ap w x2)
Hypothesis H7 : (∀x2 : set, x2 ∈ ω → (∀y2 : set, y2 ∈ x2 → ap w x2 < ap w y2))
Hypothesis H8 : (∀x2 : set, x2 ∈ ω → ap u x2 < y)
Hypothesis H9 : (∀x2 : set, x2 ∈ ω → y < ap u x2 + eps_ x2)
Hypothesis H10 : (∀x2 : set, x2 ∈ ω → (∀y2 : set, y2 ∈ x2 → ap u y2 < ap u x2))
Hypothesis H11 : (∀x2 : set, x2 ∈ ω → (ap v x2 + - (eps_ x2)) < y)
Hypothesis H12 : (∀x2 : set, x2 ∈ ω → y < ap v x2)
Hypothesis H13 : (∀x2 : set, x2 ∈ ω → (∀y2 : set, y2 ∈ x2 → ap v x2 < ap v y2))
Hypothesis H14 : SNo x
Hypothesis H15 : SNo y
Hypothesis H16 : SNo (x + y)
Hypothesis H17 : (∀x2 : set, x2 ∈ SNoS_ ω → (∀y2 : set, y2 ∈ ω → abs_SNo (x2 + - x) < eps_ y2) → x2 = x)
Hypothesis H18 : (∀x2 : set, x2 ∈ SNoS_ ω → (∀y2 : set, y2 ∈ ω → abs_SNo (x2 + - y) < eps_ y2) → x2 = y)
Hypothesis H19 : (∀x2 : set, x2 ∈ ω → ap z (ordsucc x2) ∈ SNoS_ ω)
Hypothesis H20 : (∀x2 : set, x2 ∈ ω → SNo (ap z (ordsucc x2)))
Hypothesis H21 : (∀x2 : set, x2 ∈ ω → ap u (ordsucc x2) ∈ SNoS_ ω)
Hypothesis H22 : (∀x2 : set, x2 ∈ ω → SNo (ap u (ordsucc x2)))
Hypothesis H23 : (∀x2 : set, x2 ∈ ω → ap w (ordsucc x2) ∈ SNoS_ ω)
Hypothesis H24 : (∀x2 : set, x2 ∈ ω → SNo (ap w (ordsucc x2)))
Hypothesis H25 : (∀x2 : set, x2 ∈ ω → ap v (ordsucc x2) ∈ SNoS_ ω)
Hypothesis H26 : (∀x2 : set, x2 ∈ ω → SNo (ap v (ordsucc x2)))
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))
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))
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
Variable x : set
Variable y : set
Variable z : set
Variable w : set
Variable u : set
Variable v : set
Hypothesis H0 : x ∈ real
Hypothesis H1 : y ∈ real
Hypothesis H2 : (∀x2 : set, x2 ∈ ω → ap z x2 < x)
Hypothesis H3 : (∀x2 : set, x2 ∈ ω → x < ap z x2 + eps_ x2)
Hypothesis H4 : (∀x2 : set, x2 ∈ ω → (∀y2 : set, y2 ∈ x2 → ap z y2 < ap z x2))
Hypothesis H5 : (∀x2 : set, x2 ∈ ω → (ap w x2 + - (eps_ x2)) < x)
Hypothesis H6 : (∀x2 : set, x2 ∈ ω → x < ap w x2)
Hypothesis H7 : (∀x2 : set, x2 ∈ ω → (∀y2 : set, y2 ∈ x2 → ap w x2 < ap w y2))
Hypothesis H8 : (∀x2 : set, x2 ∈ ω → ap u x2 < y)
Hypothesis H9 : (∀x2 : set, x2 ∈ ω → y < ap u x2 + eps_ x2)
Hypothesis H10 : (∀x2 : set, x2 ∈ ω → (∀y2 : set, y2 ∈ x2 → ap u y2 < ap u x2))
Hypothesis H12 : (∀x2 : set, x2 ∈ ω → y < ap v x2)
Hypothesis H13 : (∀x2 : set, x2 ∈ ω → (∀y2 : set, y2 ∈ x2 → ap v x2 < ap v y2))
Hypothesis H14 : SNo x
Hypothesis H15 : SNo y
Hypothesis H16 : SNo (x + y)
Hypothesis H17 : (∀x2 : set, x2 ∈ SNoS_ ω → (∀y2 : set, y2 ∈ ω → abs_SNo (x2 + - x) < eps_ y2) → x2 = x)
Hypothesis H18 : (∀x2 : set, x2 ∈ SNoS_ ω → (∀y2 : set, y2 ∈ ω → abs_SNo (x2 + - y) < eps_ y2) → x2 = y)
Hypothesis H19 : (∀x2 : set, x2 ∈ ω → ap z (ordsucc x2) ∈ SNoS_ ω)
Hypothesis H20 : (∀x2 : set, x2 ∈ ω → SNo (ap z (ordsucc x2)))
Hypothesis H21 : (∀x2 : set, x2 ∈ ω → ap u (ordsucc x2) ∈ SNoS_ ω)
Hypothesis H22 : (∀x2 : set, x2 ∈ ω → SNo (ap u (ordsucc x2)))
Hypothesis H23 : (∀x2 : set, x2 ∈ ω → ap w (ordsucc x2) ∈ SNoS_ ω)
Hypothesis H24 : (∀x2 : set, x2 ∈ ω → SNo (ap w (ordsucc x2)))
Hypothesis H25 : (∀x2 : set, x2 ∈ ω → ap v (ordsucc x2) ∈ SNoS_ ω)
Hypothesis H26 : (∀x2 : set, x2 ∈ ω → SNo (ap v (ordsucc x2)))
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))
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))
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
Variable x : set
Variable y : set
Variable z : set
Variable w : set
Variable u : set
Variable v : set
Hypothesis H0 : x ∈ real
Hypothesis H1 : y ∈ real
Hypothesis H2 : (∀x2 : set, x2 ∈ ω → ap z x2 < x)
Hypothesis H3 : (∀x2 : set, x2 ∈ ω → x < ap z x2 + eps_ x2)
Hypothesis H5 : (∀x2 : set, x2 ∈ ω → (ap w x2 + - (eps_ x2)) < x)
Hypothesis H6 : (∀x2 : set, x2 ∈ ω → x < ap w x2)
Hypothesis H7 : (∀x2 : set, x2 ∈ ω → (∀y2 : set, y2 ∈ x2 → ap w x2 < ap w y2))
Hypothesis H8 : (∀x2 : set, x2 ∈ ω → ap u x2 < y)
Hypothesis H9 : (∀x2 : set, x2 ∈ ω → y < ap u x2 + eps_ x2)
Hypothesis H10 : (∀x2 : set, x2 ∈ ω → (∀y2 : set, y2 ∈ x2 → ap u y2 < ap u x2))
Hypothesis H11 : (∀x2 : set, x2 ∈ ω → (ap v x2 + - (eps_ x2)) < y)
Hypothesis H12 : (∀x2 : set, x2 ∈ ω → y < ap v x2)
Hypothesis H13 : (∀x2 : set, x2 ∈ ω → (∀y2 : set, y2 ∈ x2 → ap v x2 < ap v y2))
Hypothesis H14 : SNo x
Hypothesis H15 : SNo y
Hypothesis H16 : SNo (x + y)
Hypothesis H17 : (∀x2 : set, x2 ∈ SNoS_ ω → (∀y2 : set, y2 ∈ ω → abs_SNo (x2 + - x) < eps_ y2) → x2 = x)
Hypothesis H18 : (∀x2 : set, x2 ∈ SNoS_ ω → (∀y2 : set, y2 ∈ ω → abs_SNo (x2 + - y) < eps_ y2) → x2 = y)
Hypothesis H19 : (∀x2 : set, x2 ∈ ω → ap z (ordsucc x2) ∈ SNoS_ ω)
Hypothesis H20 : (∀x2 : set, x2 ∈ ω → SNo (ap z (ordsucc x2)))
Hypothesis H21 : (∀x2 : set, x2 ∈ ω → ap u (ordsucc x2) ∈ SNoS_ ω)
Hypothesis H22 : (∀x2 : set, x2 ∈ ω → SNo (ap u (ordsucc x2)))
Hypothesis H23 : (∀x2 : set, x2 ∈ ω → ap w (ordsucc x2) ∈ SNoS_ ω)
Hypothesis H24 : (∀x2 : set, x2 ∈ ω → SNo (ap w (ordsucc x2)))
Hypothesis H25 : (∀x2 : set, x2 ∈ ω → ap v (ordsucc x2) ∈ SNoS_ ω)
Hypothesis H26 : (∀x2 : set, x2 ∈ ω → SNo (ap v (ordsucc x2)))
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))
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
Variable x : set
Variable y : set
Variable z : set
Variable w : set
Variable u : set
Variable v : set
Hypothesis H0 : x ∈ real
Hypothesis H1 : y ∈ real
Hypothesis H2 : (∀x2 : set, x2 ∈ ω → ap z x2 < x)
Hypothesis H3 : (∀x2 : set, x2 ∈ ω → x < ap z x2 + eps_ x2)
Hypothesis H4 : (∀x2 : set, x2 ∈ ω → (∀y2 : set, y2 ∈ x2 → ap z y2 < ap z x2))
Hypothesis H5 : (∀x2 : set, x2 ∈ ω → (ap w x2 + - (eps_ x2)) < x)
Hypothesis H6 : (∀x2 : set, x2 ∈ ω → x < ap w x2)
Hypothesis H7 : (∀x2 : set, x2 ∈ ω → (∀y2 : set, y2 ∈ x2 → ap w x2 < ap w y2))
Hypothesis H8 : (∀x2 : set, x2 ∈ ω → ap u x2 < y)
Hypothesis H9 : (∀x2 : set, x2 ∈ ω → y < ap u x2 + eps_ x2)
Hypothesis H10 : (∀x2 : set, x2 ∈ ω → (∀y2 : set, y2 ∈ x2 → ap u y2 < ap u x2))
Hypothesis H12 : (∀x2 : set, x2 ∈ ω → y < ap v x2)
Hypothesis H13 : (∀x2 : set, x2 ∈ ω → (∀y2 : set, y2 ∈ x2 → ap v x2 < ap v y2))
Hypothesis H14 : SNo x
Hypothesis H15 : SNo y
Hypothesis H16 : SNo (x + y)
Hypothesis H17 : (∀x2 : set, x2 ∈ SNoS_ ω → (∀y2 : set, y2 ∈ ω → abs_SNo (x2 + - x) < eps_ y2) → x2 = x)
Hypothesis H18 : (∀x2 : set, x2 ∈ SNoS_ ω → (∀y2 : set, y2 ∈ ω → abs_SNo (x2 + - y) < eps_ y2) → x2 = y)
Hypothesis H19 : (∀x2 : set, x2 ∈ ω → ap z (ordsucc x2) ∈ SNoS_ ω)
Hypothesis H20 : (∀x2 : set, x2 ∈ ω → SNo (ap z (ordsucc x2)))
Hypothesis H21 : (∀x2 : set, x2 ∈ ω → ap u (ordsucc x2) ∈ SNoS_ ω)
Hypothesis H22 : (∀x2 : set, x2 ∈ ω → SNo (ap u (ordsucc x2)))
Hypothesis H23 : (∀x2 : set, x2 ∈ ω → ap w (ordsucc x2) ∈ SNoS_ ω)
Hypothesis H24 : (∀x2 : set, x2 ∈ ω → SNo (ap w (ordsucc x2)))
Hypothesis H25 : (∀x2 : set, x2 ∈ ω → ap v (ordsucc x2) ∈ SNoS_ ω)
Hypothesis H26 : (∀x2 : set, x2 ∈ ω → SNo (ap v (ordsucc x2)))
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
Variable x : set
Variable y : set
Variable z : set
Variable w : set
Variable u : set
Variable v : set
Hypothesis H0 : x ∈ real
Hypothesis H1 : y ∈ real
Hypothesis H2 : (∀x2 : set, x2 ∈ ω → ap z x2 < x)
Hypothesis H3 : (∀x2 : set, x2 ∈ ω → x < ap z x2 + eps_ x2)
Hypothesis H4 : (∀x2 : set, x2 ∈ ω → (∀y2 : set, y2 ∈ x2 → ap z y2 < ap z x2))
Hypothesis H5 : (∀x2 : set, x2 ∈ ω → (ap w x2 + - (eps_ x2)) < x)
Hypothesis H6 : (∀x2 : set, x2 ∈ ω → x < ap w x2)
Hypothesis H7 : (∀x2 : set, x2 ∈ ω → (∀y2 : set, y2 ∈ x2 → ap w x2 < ap w y2))
Hypothesis H8 : (∀x2 : set, x2 ∈ ω → ap u x2 < y)
Hypothesis H9 : (∀x2 : set, x2 ∈ ω → y < ap u x2 + eps_ x2)
Hypothesis H10 : (∀x2 : set, x2 ∈ ω → (∀y2 : set, y2 ∈ x2 → ap u y2 < ap u x2))
Hypothesis H11 : (∀x2 : set, x2 ∈ ω → (ap v x2 + - (eps_ x2)) < y)
Hypothesis H12 : (∀x2 : set, x2 ∈ ω → y < ap v x2)
Hypothesis H14 : SNo x
Hypothesis H15 : SNo y
Hypothesis H16 : SNo (x + y)
Hypothesis H17 : (∀x2 : set, x2 ∈ SNoS_ ω → (∀y2 : set, y2 ∈ ω → abs_SNo (x2 + - x) < eps_ y2) → x2 = x)
Hypothesis H18 : (∀x2 : set, x2 ∈ SNoS_ ω → (∀y2 : set, y2 ∈ ω → abs_SNo (x2 + - y) < eps_ y2) → x2 = y)
Hypothesis H19 : (∀x2 : set, x2 ∈ ω → ap z (ordsucc x2) ∈ SNoS_ ω)
Hypothesis H20 : (∀x2 : set, x2 ∈ ω → SNo (ap z (ordsucc x2)))
Hypothesis H21 : (∀x2 : set, x2 ∈ ω → ap u (ordsucc x2) ∈ SNoS_ ω)
Hypothesis H22 : (∀x2 : set, x2 ∈ ω → SNo (ap u (ordsucc x2)))
Hypothesis H23 : (∀x2 : set, x2 ∈ ω → ap w (ordsucc x2) ∈ SNoS_ ω)
Hypothesis H24 : (∀x2 : set, x2 ∈ ω → SNo (ap w (ordsucc x2)))
Hypothesis H25 : (∀x2 : set, x2 ∈ ω → ap v (ordsucc x2) ∈ SNoS_ ω)
Hypothesis H26 : (∀x2 : set, x2 ∈ ω → SNo (ap v (ordsucc x2)))
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
Variable x : set
Variable y : set
Variable z : set
Variable w : set
Variable u : set
Variable v : set
Hypothesis H0 : x ∈ real
Hypothesis H1 : y ∈ real
Hypothesis H2 : (∀x2 : set, x2 ∈ ω → ap z x2 < x)
Hypothesis H3 : (∀x2 : set, x2 ∈ ω → x < ap z x2 + eps_ x2)
Hypothesis H4 : (∀x2 : set, x2 ∈ ω → (∀y2 : set, y2 ∈ x2 → ap z y2 < ap z x2))
Hypothesis H5 : (∀x2 : set, x2 ∈ ω → (ap w x2 + - (eps_ x2)) < x)
Hypothesis H6 : (∀x2 : set, x2 ∈ ω → x < ap w x2)
Hypothesis H7 : (∀x2 : set, x2 ∈ ω → (∀y2 : set, y2 ∈ x2 → ap w x2 < ap w y2))
Hypothesis H8 : (∀x2 : set, x2 ∈ ω → ap u x2 < y)
Hypothesis H9 : (∀x2 : set, x2 ∈ ω → y < ap u x2 + eps_ x2)
Hypothesis H10 : (∀x2 : set, x2 ∈ ω → (∀y2 : set, y2 ∈ x2 → ap u y2 < ap u x2))
Hypothesis H11 : (∀x2 : set, x2 ∈ ω → (ap v x2 + - (eps_ x2)) < y)
Hypothesis H13 : (∀x2 : set, x2 ∈ ω → (∀y2 : set, y2 ∈ x2 → ap v x2 < ap v y2))
Hypothesis H14 : SNo x
Hypothesis H15 : SNo y
Hypothesis H16 : SNo (x + y)
Hypothesis H17 : (∀x2 : set, x2 ∈ SNoS_ ω → (∀y2 : set, y2 ∈ ω → abs_SNo (x2 + - x) < eps_ y2) → x2 = x)
Hypothesis H18 : (∀x2 : set, x2 ∈ SNoS_ ω → (∀y2 : set, y2 ∈ ω → abs_SNo (x2 + - y) < eps_ y2) → x2 = y)
Hypothesis H19 : (∀x2 : set, x2 ∈ ω → ap z (ordsucc x2) ∈ SNoS_ ω)
Hypothesis H20 : (∀x2 : set, x2 ∈ ω → SNo (ap z (ordsucc x2)))
Hypothesis H21 : (∀x2 : set, x2 ∈ ω → ap u (ordsucc x2) ∈ SNoS_ ω)
Hypothesis H22 : (∀x2 : set, x2 ∈ ω → SNo (ap u (ordsucc x2)))
Hypothesis H23 : (∀x2 : set, x2 ∈ ω → ap w (ordsucc x2) ∈ SNoS_ ω)
Hypothesis H24 : (∀x2 : set, x2 ∈ ω → SNo (ap w (ordsucc x2)))
Hypothesis H25 : (∀x2 : set, x2 ∈ ω → ap v (ordsucc x2) ∈ SNoS_ ω)
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
Variable x : set
Variable y : set
Variable z : set
Variable w : set
Variable u : set
Variable v : set
Hypothesis H0 : x ∈ real
Hypothesis H1 : y ∈ real
Hypothesis H2 : (∀x2 : set, x2 ∈ ω → ap z x2 < x)
Hypothesis H3 : (∀x2 : set, x2 ∈ ω → x < ap z x2 + eps_ x2)
Hypothesis H4 : (∀x2 : set, x2 ∈ ω → (∀y2 : set, y2 ∈ x2 → ap z y2 < ap z x2))
Hypothesis H5 : (∀x2 : set, x2 ∈ ω → (ap w x2 + - (eps_ x2)) < x)
Hypothesis H6 : (∀x2 : set, x2 ∈ ω → x < ap w x2)
Hypothesis H7 : (∀x2 : set, x2 ∈ ω → (∀y2 : set, y2 ∈ x2 → ap w x2 < ap w y2))
Hypothesis H8 : (∀x2 : set, x2 ∈ ω → ap u x2 < y)
Hypothesis H9 : (∀x2 : set, x2 ∈ ω → y < ap u x2 + eps_ x2)
Hypothesis H10 : (∀x2 : set, x2 ∈ ω → (∀y2 : set, y2 ∈ x2 → ap u y2 < ap u x2))
Hypothesis H11 : (∀x2 : set, x2 ∈ ω → (ap v x2 + - (eps_ x2)) < y)
Hypothesis H12 : (∀x2 : set, x2 ∈ ω → y < ap v x2)
Hypothesis H13 : (∀x2 : set, x2 ∈ ω → (∀y2 : set, y2 ∈ x2 → ap v x2 < ap v y2))
Hypothesis H14 : SNo x
Hypothesis H16 : SNo (x + y)
Hypothesis H17 : (∀x2 : set, x2 ∈ SNoS_ ω → (∀y2 : set, y2 ∈ ω → abs_SNo (x2 + - x) < eps_ y2) → x2 = x)
Hypothesis H18 : (∀x2 : set, x2 ∈ SNoS_ ω → (∀y2 : set, y2 ∈ ω → abs_SNo (x2 + - y) < eps_ y2) → x2 = y)
Hypothesis H19 : (∀x2 : set, x2 ∈ ω → ap z (ordsucc x2) ∈ SNoS_ ω)
Hypothesis H20 : (∀x2 : set, x2 ∈ ω → SNo (ap z (ordsucc x2)))
Hypothesis H21 : (∀x2 : set, x2 ∈ ω → ap u (ordsucc x2) ∈ SNoS_ ω)
Hypothesis H22 : (∀x2 : set, x2 ∈ ω → SNo (ap u (ordsucc x2)))
Hypothesis H23 : (∀x2 : set, x2 ∈ ω → ap w (ordsucc x2) ∈ SNoS_ ω)
Hypothesis H24 : (∀x2 : set, x2 ∈ ω → SNo (ap w (ordsucc x2)))
Hypothesis H25 : (∀x2 : set, x2 ∈ ω → ap v (ordsucc x2) ∈ SNoS_ ω)
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
Variable x : set
Variable y : set
Variable z : set
Variable w : set
Variable u : set
Variable v : set
Hypothesis H0 : x ∈ real
Hypothesis H1 : y ∈ real
Hypothesis H2 : (∀x2 : set, x2 ∈ ω → ap z x2 < x)
Hypothesis H3 : (∀x2 : set, x2 ∈ ω → x < ap z x2 + eps_ x2)
Hypothesis H4 : (∀x2 : set, x2 ∈ ω → (∀y2 : set, y2 ∈ x2 → ap z y2 < ap z x2))
Hypothesis H5 : (∀x2 : set, x2 ∈ ω → (ap w x2 + - (eps_ x2)) < x)
Hypothesis H6 : (∀x2 : set, x2 ∈ ω → x < ap w x2)
Hypothesis H7 : (∀x2 : set, x2 ∈ ω → (∀y2 : set, y2 ∈ x2 → ap w x2 < ap w y2))
Hypothesis H8 : v ∈ setexp (SNoS_ ω) ω
Hypothesis H10 : (∀x2 : set, x2 ∈ ω → y < ap u x2 + eps_ x2)
Hypothesis H11 : (∀x2 : set, x2 ∈ ω → (∀y2 : set, y2 ∈ x2 → ap u y2 < ap u x2))
Hypothesis H12 : (∀x2 : set, x2 ∈ ω → (ap v x2 + - (eps_ x2)) < y)
Hypothesis H13 : (∀x2 : set, x2 ∈ ω → y < ap v x2)
Hypothesis H14 : (∀x2 : set, x2 ∈ ω → (∀y2 : set, y2 ∈ x2 → ap v x2 < ap v y2))
Hypothesis H15 : SNo x
Hypothesis H16 : SNo y
Hypothesis H17 : SNo (x + y)
Hypothesis H18 : (∀x2 : set, x2 ∈ SNoS_ ω → (∀y2 : set, y2 ∈ ω → abs_SNo (x2 + - x) < eps_ y2) → x2 = x)
Hypothesis H19 : (∀x2 : set, x2 ∈ SNoS_ ω → (∀y2 : set, y2 ∈ ω → abs_SNo (x2 + - y) < eps_ y2) → x2 = y)
Hypothesis H20 : (∀x2 : set, x2 ∈ ω → ap z (ordsucc x2) ∈ SNoS_ ω)
Hypothesis H21 : (∀x2 : set, x2 ∈ ω → SNo (ap z (ordsucc x2)))
Hypothesis H22 : (∀x2 : set, x2 ∈ ω → ap u (ordsucc x2) ∈ SNoS_ ω)
Hypothesis H23 : (∀x2 : set, x2 ∈ ω → SNo (ap u (ordsucc x2)))
Hypothesis H24 : (∀x2 : set, x2 ∈ ω → ap w (ordsucc x2) ∈ SNoS_ ω)
Hypothesis H25 : (∀x2 : set, x2 ∈ ω → SNo (ap w (ordsucc x2)))
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
Variable x : set
Variable y : set
Variable z : set
Variable w : set
Variable u : set
Variable v : set
Hypothesis H0 : x ∈ real
Hypothesis H1 : y ∈ real
Hypothesis H3 : (∀x2 : set, x2 ∈ ω → ap z x2 < x)
Hypothesis H4 : (∀x2 : set, x2 ∈ ω → x < ap z x2 + eps_ x2)
Hypothesis H5 : (∀x2 : set, x2 ∈ ω → (∀y2 : set, y2 ∈ x2 → ap z y2 < ap z x2))
Hypothesis H6 : (∀x2 : set, x2 ∈ ω → (ap w x2 + - (eps_ x2)) < x)
Hypothesis H7 : (∀x2 : set, x2 ∈ ω → x < ap w x2)
Hypothesis H8 : (∀x2 : set, x2 ∈ ω → (∀y2 : set, y2 ∈ x2 → ap w x2 < ap w y2))
Hypothesis H9 : v ∈ setexp (SNoS_ ω) ω
Hypothesis H10 : (∀x2 : set, x2 ∈ ω → ap u x2 < y)
Hypothesis H11 : (∀x2 : set, x2 ∈ ω → y < ap u x2 + eps_ x2)
Hypothesis H12 : (∀x2 : set, x2 ∈ ω → (∀y2 : set, y2 ∈ x2 → ap u y2 < ap u x2))
Hypothesis H13 : (∀x2 : set, x2 ∈ ω → (ap v x2 + - (eps_ x2)) < y)
Hypothesis H14 : (∀x2 : set, x2 ∈ ω → y < ap v x2)
Hypothesis H15 : (∀x2 : set, x2 ∈ ω → (∀y2 : set, y2 ∈ x2 → ap v x2 < ap v y2))
Hypothesis H16 : SNo x
Hypothesis H17 : SNo y
Hypothesis H18 : SNo (x + y)
Hypothesis H19 : (∀x2 : set, x2 ∈ SNoS_ ω → (∀y2 : set, y2 ∈ ω → abs_SNo (x2 + - x) < eps_ y2) → x2 = x)
Hypothesis H20 : (∀x2 : set, x2 ∈ SNoS_ ω → (∀y2 : set, y2 ∈ ω → abs_SNo (x2 + - y) < eps_ y2) → x2 = y)
Hypothesis H21 : (∀x2 : set, x2 ∈ ω → ap z (ordsucc x2) ∈ SNoS_ ω)
Hypothesis H22 : (∀x2 : set, x2 ∈ ω → SNo (ap z (ordsucc x2)))
Hypothesis H23 : (∀x2 : set, x2 ∈ ω → ap u (ordsucc x2) ∈ SNoS_ ω)
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
Variable x : set
Variable y : set
Variable z : set
Variable w : set
Variable u : set
Variable v : set
Hypothesis H0 : x ∈ real
Hypothesis H1 : y ∈ real
Hypothesis H2 : w ∈ setexp (SNoS_ ω) ω
Hypothesis H3 : (∀x2 : set, x2 ∈ ω → ap z x2 < x)
Hypothesis H4 : (∀x2 : set, x2 ∈ ω → x < ap z x2 + eps_ x2)
Hypothesis H5 : (∀x2 : set, x2 ∈ ω → (∀y2 : set, y2 ∈ x2 → ap z y2 < ap z x2))
Hypothesis H6 : (∀x2 : set, x2 ∈ ω → (ap w x2 + - (eps_ x2)) < x)
Hypothesis H7 : (∀x2 : set, x2 ∈ ω → x < ap w x2)
Hypothesis H8 : (∀x2 : set, x2 ∈ ω → (∀y2 : set, y2 ∈ x2 → ap w x2 < ap w y2))
Hypothesis H9 : v ∈ setexp (SNoS_ ω) ω
Hypothesis H10 : (∀x2 : set, x2 ∈ ω → ap u x2 < y)
Hypothesis H11 : (∀x2 : set, x2 ∈ ω → y < ap u x2 + eps_ x2)
Hypothesis H12 : (∀x2 : set, x2 ∈ ω → (∀y2 : set, y2 ∈ x2 → ap u y2 < ap u x2))
Hypothesis H13 : (∀x2 : set, x2 ∈ ω → (ap v x2 + - (eps_ x2)) < y)
Hypothesis H14 : (∀x2 : set, x2 ∈ ω → y < ap v x2)
Hypothesis H15 : (∀x2 : set, x2 ∈ ω → (∀y2 : set, y2 ∈ x2 → ap v x2 < ap v y2))
Hypothesis H16 : SNo x
Hypothesis H17 : SNo y
Hypothesis H18 : SNo (x + y)
Hypothesis H20 : (∀x2 : set, x2 ∈ SNoS_ ω → (∀y2 : set, y2 ∈ ω → abs_SNo (x2 + - y) < eps_ y2) → x2 = y)
Hypothesis H21 : (∀x2 : set, x2 ∈ ω → ap z (ordsucc x2) ∈ SNoS_ ω)
Hypothesis H22 : (∀x2 : set, x2 ∈ ω → SNo (ap z (ordsucc x2)))
Hypothesis H23 : (∀x2 : set, x2 ∈ ω → ap u (ordsucc x2) ∈ SNoS_ ω)
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
Variable x : set
Variable y : set
Variable z : set
Variable w : set
Variable u : set
Variable v : set
Hypothesis H0 : x ∈ real
Hypothesis H1 : y ∈ real
Hypothesis H2 : w ∈ setexp (SNoS_ ω) ω
Hypothesis H3 : (∀x2 : set, x2 ∈ ω → ap z x2 < x)
Hypothesis H4 : (∀x2 : set, x2 ∈ ω → x < ap z x2 + eps_ x2)
Hypothesis H5 : (∀x2 : set, x2 ∈ ω → (∀y2 : set, y2 ∈ x2 → ap z y2 < ap z x2))
Hypothesis H6 : (∀x2 : set, x2 ∈ ω → (ap w x2 + - (eps_ x2)) < x)
Hypothesis H7 : (∀x2 : set, x2 ∈ ω → x < ap w x2)
Hypothesis H8 : (∀x2 : set, x2 ∈ ω → (∀y2 : set, y2 ∈ x2 → ap w x2 < ap w y2))
Hypothesis H10 : v ∈ setexp (SNoS_ ω) ω
Hypothesis H11 : (∀x2 : set, x2 ∈ ω → ap u x2 < y)
Hypothesis H12 : (∀x2 : set, x2 ∈ ω → y < ap u x2 + eps_ x2)
Hypothesis H13 : (∀x2 : set, x2 ∈ ω → (∀y2 : set, y2 ∈ x2 → ap u y2 < ap u x2))
Hypothesis H14 : (∀x2 : set, x2 ∈ ω → (ap v x2 + - (eps_ x2)) < y)
Hypothesis H15 : (∀x2 : set, x2 ∈ ω → y < ap v x2)
Hypothesis H16 : (∀x2 : set, x2 ∈ ω → (∀y2 : set, y2 ∈ x2 → ap v x2 < ap v y2))
Hypothesis H17 : SNo x
Hypothesis H18 : SNo y
Hypothesis H19 : SNo (x + y)
Hypothesis H20 : (∀x2 : set, x2 ∈ SNoS_ ω → (∀y2 : set, y2 ∈ ω → abs_SNo (x2 + - x) < eps_ y2) → x2 = x)
Hypothesis H21 : (∀x2 : set, x2 ∈ SNoS_ ω → (∀y2 : set, y2 ∈ ω → abs_SNo (x2 + - y) < eps_ y2) → x2 = y)
Hypothesis H22 : (∀x2 : set, x2 ∈ ω → ap z (ordsucc x2) ∈ SNoS_ ω)
Hypothesis H23 : (∀x2 : set, x2 ∈ ω → SNo (ap z (ordsucc x2)))
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
Variable x : set
Variable y : set
Variable z : set
Variable w : set
Variable u : set
Variable v : set
Hypothesis H0 : x ∈ real
Hypothesis H1 : y ∈ real
Hypothesis H2 : z ∈ setexp (SNoS_ ω) ω
Hypothesis H3 : w ∈ setexp (SNoS_ ω) ω
Hypothesis H4 : (∀x2 : set, x2 ∈ ω → ap z x2 < x)
Hypothesis H5 : (∀x2 : set, x2 ∈ ω → x < ap z x2 + eps_ x2)
Hypothesis H6 : (∀x2 : set, x2 ∈ ω → (∀y2 : set, y2 ∈ x2 → ap z y2 < ap z x2))
Hypothesis H7 : (∀x2 : set, x2 ∈ ω → (ap w x2 + - (eps_ x2)) < x)
Hypothesis H8 : (∀x2 : set, x2 ∈ ω → x < ap w x2)
Hypothesis H9 : (∀x2 : set, x2 ∈ ω → (∀y2 : set, y2 ∈ x2 → ap w x2 < ap w y2))
Hypothesis H11 : v ∈ setexp (SNoS_ ω) ω
Hypothesis H12 : (∀x2 : set, x2 ∈ ω → ap u x2 < y)
Hypothesis H13 : (∀x2 : set, x2 ∈ ω → y < ap u x2 + eps_ x2)
Hypothesis H14 : (∀x2 : set, x2 ∈ ω → (∀y2 : set, y2 ∈ x2 → ap u y2 < ap u x2))
Hypothesis H15 : (∀x2 : set, x2 ∈ ω → (ap v x2 + - (eps_ x2)) < y)
Hypothesis H16 : (∀x2 : set, x2 ∈ ω → y < ap v x2)
Hypothesis H17 : (∀x2 : set, x2 ∈ ω → (∀y2 : set, y2 ∈ x2 → ap v x2 < ap v y2))
Hypothesis H18 : SNo x
Hypothesis H19 : SNo y
Hypothesis H20 : SNo (x + y)
Hypothesis H21 : (∀x2 : set, x2 ∈ SNoS_ ω → (∀y2 : set, y2 ∈ ω → abs_SNo (x2 + - x) < eps_ y2) → x2 = x)
Hypothesis H22 : (∀x2 : set, x2 ∈ SNoS_ ω → (∀y2 : set, y2 ∈ ω → abs_SNo (x2 + - y) < eps_ y2) → x2 = y)
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
Variable x : set
Variable y : set
Variable z : set
Variable w : set
Variable u : set
Variable v : set
Hypothesis H0 : x ∈ real
Hypothesis H1 : y ∈ real
Hypothesis H2 : z ∈ setexp (SNoS_ ω) ω
Hypothesis H3 : w ∈ setexp (SNoS_ ω) ω
Hypothesis H4 : (∀x2 : set, x2 ∈ ω → ap z x2 < x)
Hypothesis H5 : (∀x2 : set, x2 ∈ ω → x < ap z x2 + eps_ x2)
Hypothesis H6 : (∀x2 : set, x2 ∈ ω → (∀y2 : set, y2 ∈ x2 → ap z y2 < ap z x2))
Hypothesis H8 : (∀x2 : set, x2 ∈ ω → x < ap w x2)
Hypothesis H9 : (∀x2 : set, x2 ∈ ω → (∀y2 : set, y2 ∈ x2 → ap w x2 < ap w y2))
Hypothesis H10 : u ∈ setexp (SNoS_ ω) ω
Hypothesis H11 : v ∈ setexp (SNoS_ ω) ω
Hypothesis H12 : (∀x2 : set, x2 ∈ ω → ap u x2 < y)
Hypothesis H13 : (∀x2 : set, x2 ∈ ω → y < ap u x2 + eps_ x2)
Hypothesis H14 : (∀x2 : set, x2 ∈ ω → (∀y2 : set, y2 ∈ x2 → ap u y2 < ap u x2))
Hypothesis H15 : (∀x2 : set, x2 ∈ ω → (ap v x2 + - (eps_ x2)) < y)
Hypothesis H16 : (∀x2 : set, x2 ∈ ω → y < ap v x2)
Hypothesis H17 : (∀x2 : set, x2 ∈ ω → (∀y2 : set, y2 ∈ x2 → ap v x2 < ap v y2))
Hypothesis H18 : SNo x
Hypothesis H19 : SNo y
Hypothesis H20 : SNo (x + y)
Hypothesis H21 : (∀x2 : set, x2 ∈ SNoS_ ω → (∀y2 : set, y2 ∈ ω → abs_SNo (x2 + - x) < eps_ y2) → x2 = x)
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
Variable x : set
Variable y : set
Variable z : set
Variable w : set
Variable u : set
Variable v : set
Hypothesis H0 : x ∈ real
Hypothesis H1 : y ∈ real
Hypothesis H2 : z ∈ setexp (SNoS_ ω) ω
Hypothesis H3 : w ∈ setexp (SNoS_ ω) ω
Hypothesis H4 : (∀x2 : set, x2 ∈ ω → ap z x2 < x)
Hypothesis H5 : (∀x2 : set, x2 ∈ ω → x < ap z x2 + eps_ x2)
Hypothesis H6 : (∀x2 : set, x2 ∈ ω → (∀y2 : set, y2 ∈ x2 → ap z y2 < ap z x2))
Hypothesis H7 : (∀x2 : set, x2 ∈ ω → (ap w x2 + - (eps_ x2)) < x)
Hypothesis H8 : (∀x2 : set, x2 ∈ ω → x < ap w x2)
Hypothesis H9 : (∀x2 : set, x2 ∈ ω → (∀y2 : set, y2 ∈ x2 → ap w x2 < ap w y2))
Hypothesis H10 : u ∈ setexp (SNoS_ ω) ω
Hypothesis H11 : v ∈ setexp (SNoS_ ω) ω
Hypothesis H12 : (∀x2 : set, x2 ∈ ω → ap u x2 < y)
Hypothesis H13 : (∀x2 : set, x2 ∈ ω → y < ap u x2 + eps_ x2)
Hypothesis H14 : (∀x2 : set, x2 ∈ ω → (∀y2 : set, y2 ∈ x2 → ap u y2 < ap u x2))
Hypothesis H15 : (∀x2 : set, x2 ∈ ω → (ap v x2 + - (eps_ x2)) < y)
Hypothesis H16 : (∀x2 : set, x2 ∈ ω → y < ap v x2)
Hypothesis H18 : SNo x
Hypothesis H19 : SNo y
Hypothesis H20 : SNo (x + y)
Hypothesis H21 : (∀x2 : set, x2 ∈ SNoS_ ω → (∀y2 : set, y2 ∈ ω → abs_SNo (x2 + - x) < eps_ y2) → x2 = x)
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
Variable x : set
Variable y : set
Variable z : set
Variable w : set
Variable u : set
Variable v : set
Hypothesis H0 : x ∈ real
Hypothesis H1 : y ∈ real
Hypothesis H2 : z ∈ setexp (SNoS_ ω) ω
Hypothesis H3 : w ∈ setexp (SNoS_ ω) ω
Hypothesis H4 : (∀x2 : set, x2 ∈ ω → ap z x2 < x)
Hypothesis H5 : (∀x2 : set, x2 ∈ ω → x < ap z x2 + eps_ x2)
Hypothesis H6 : (∀x2 : set, x2 ∈ ω → (∀y2 : set, y2 ∈ x2 → ap z y2 < ap z x2))
Hypothesis H7 : (∀x2 : set, x2 ∈ ω → (ap w x2 + - (eps_ x2)) < x)
Hypothesis H8 : (∀x2 : set, x2 ∈ ω → x < ap w x2)
Hypothesis H9 : (∀x2 : set, x2 ∈ ω → (∀y2 : set, y2 ∈ x2 → ap w x2 < ap w y2))
Hypothesis H10 : u ∈ setexp (SNoS_ ω) ω
Hypothesis H11 : v ∈ setexp (SNoS_ ω) ω
Hypothesis H12 : (∀x2 : set, x2 ∈ ω → ap u x2 < y)
Hypothesis H13 : (∀x2 : set, x2 ∈ ω → y < ap u x2 + eps_ x2)
Hypothesis H14 : (∀x2 : set, x2 ∈ ω → (∀y2 : set, y2 ∈ x2 → ap u y2 < ap u x2))
Hypothesis H15 : (∀x2 : set, x2 ∈ ω → (ap v x2 + - (eps_ x2)) < y)
Hypothesis H17 : (∀x2 : set, x2 ∈ ω → (∀y2 : set, y2 ∈ x2 → ap v x2 < ap v y2))
Hypothesis H18 : SNo x
Hypothesis H19 : SNo y
Hypothesis H20 : SNo (x + y)
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
Variable x : set
Variable y : set
Variable z : set
Variable w : set
Variable u : set
Variable v : set
Hypothesis H0 : x ∈ real
Hypothesis H1 : y ∈ real
Hypothesis H2 : z ∈ setexp (SNoS_ ω) ω
Hypothesis H3 : w ∈ setexp (SNoS_ ω) ω
Hypothesis H4 : (∀x2 : set, x2 ∈ ω → ap z x2 < x)
Hypothesis H5 : (∀x2 : set, x2 ∈ ω → x < ap z x2 + eps_ x2)
Hypothesis H6 : (∀x2 : set, x2 ∈ ω → (∀y2 : set, y2 ∈ x2 → ap z y2 < ap z x2))
Hypothesis H7 : (∀x2 : set, x2 ∈ ω → (ap w x2 + - (eps_ x2)) < x)
Hypothesis H8 : (∀x2 : set, x2 ∈ ω → x < ap w x2)
Hypothesis H9 : (∀x2 : set, x2 ∈ ω → (∀y2 : set, y2 ∈ x2 → ap w x2 < ap w y2))
Hypothesis H10 : u ∈ setexp (SNoS_ ω) ω
Hypothesis H11 : v ∈ setexp (SNoS_ ω) ω
Hypothesis H12 : (∀x2 : set, x2 ∈ ω → ap u x2 < y)
Hypothesis H13 : (∀x2 : set, x2 ∈ ω → y < ap u x2 + eps_ x2)
Hypothesis H14 : (∀x2 : set, x2 ∈ ω → (∀y2 : set, y2 ∈ x2 → ap u y2 < ap u x2))
Hypothesis H15 : (∀x2 : set, x2 ∈ ω → (ap v x2 + - (eps_ x2)) < y)
Hypothesis H16 : (∀x2 : set, x2 ∈ ω → y < ap v x2)
Hypothesis H17 : (∀x2 : set, x2 ∈ ω → (∀y2 : set, y2 ∈ x2 → ap v x2 < ap v y2))
Hypothesis H18 : SNo x
Hypothesis H19 : SNo y
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
Variable x : set
Variable y : set
Variable z : set
Variable w : set
Variable u : set
Variable v : set
Hypothesis H1 : y ∈ real
Hypothesis H2 : z ∈ setexp (SNoS_ ω) ω
Hypothesis H3 : w ∈ setexp (SNoS_ ω) ω
Hypothesis H4 : (∀x2 : set, x2 ∈ ω → ap z x2 < x)
Hypothesis H5 : (∀x2 : set, x2 ∈ ω → x < ap z x2 + eps_ x2)
Hypothesis H6 : (∀x2 : set, x2 ∈ ω → (∀y2 : set, y2 ∈ x2 → ap z y2 < ap z x2))
Hypothesis H7 : (∀x2 : set, x2 ∈ ω → (ap w x2 + - (eps_ x2)) < x)
Hypothesis H8 : (∀x2 : set, x2 ∈ ω → x < ap w x2)
Hypothesis H9 : (∀x2 : set, x2 ∈ ω → (∀y2 : set, y2 ∈ x2 → ap w x2 < ap w y2))
Hypothesis H10 : u ∈ setexp (SNoS_ ω) ω
Hypothesis H11 : v ∈ setexp (SNoS_ ω) ω
Hypothesis H12 : (∀x2 : set, x2 ∈ ω → ap u x2 < y)
Hypothesis H13 : (∀x2 : set, x2 ∈ ω → y < ap u x2 + eps_ x2)
Hypothesis H14 : (∀x2 : set, x2 ∈ ω → (∀y2 : set, y2 ∈ x2 → ap u y2 < ap u x2))
Hypothesis H15 : (∀x2 : set, x2 ∈ ω → (ap v x2 + - (eps_ x2)) < y)
Hypothesis H16 : (∀x2 : set, x2 ∈ ω → y < ap v x2)
Hypothesis H17 : (∀x2 : set, x2 ∈ ω → (∀y2 : set, y2 ∈ x2 → ap v x2 < ap v y2))
Hypothesis H18 : SNo x
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
Variable x : set
Variable y : set
Variable z : set
Variable w : set
Variable u : set
Variable v : set
Variable x2 : set
Hypothesis H0 : SNo x
Hypothesis H1 : SNo y
Hypothesis H3 : SNo (- (x * y))
Hypothesis H4 : (∀y2 : set, y2 ∈ ω → abs_SNo (z + - (x * y)) < eps_ y2)
Hypothesis H5 : SNo z
Hypothesis H6 : x * y < z
Hypothesis H7 : SNo w
Hypothesis H8 : SNo u
Hypothesis H9 : (w * y + x * u) ≤ z + w * u
Hypothesis H10 : SNo (w * y)
Hypothesis H11 : SNo (x * u)
Hypothesis H12 : SNo (w * u)
Hypothesis H13 : SNo (- (w * u))
Hypothesis H14 : SNo (w + - x)
Hypothesis H15 : SNo (y + - u)
Hypothesis H16 : v ∈ ω
Hypothesis H17 : eps_ v ≤ w + - x
Hypothesis H18 : x2 ∈ ω
Hypothesis H19 : eps_ x2 ≤ y + - u
Hypothesis H20 : SNo (eps_ v)
Hypothesis H21 : SNo (eps_ x2)
Hypothesis H22 : SNo (eps_ (v + x2))
Hypothesis H23 : SNo (eps_ v * eps_ x2)
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
Variable x : set
Variable y : set
Variable z : set
Variable w : set
Variable u : set
Variable v : set
Variable x2 : set
Hypothesis H0 : SNo x
Hypothesis H1 : SNo y
Hypothesis H2 : SNo (x * y)
Hypothesis H3 : SNo (- (x * y))
Hypothesis H4 : (∀y2 : set, y2 ∈ ω → abs_SNo (z + - (x * y)) < eps_ y2)
Hypothesis H5 : SNo z
Hypothesis H6 : x * y < z
Hypothesis H7 : SNo w
Hypothesis H8 : SNo u
Hypothesis H9 : (w * y + x * u) ≤ z + w * u
Hypothesis H10 : SNo (w * y)
Hypothesis H12 : SNo (w * u)
Hypothesis H13 : SNo (- (w * u))
Hypothesis H14 : SNo (w + - x)
Hypothesis H15 : SNo (y + - u)
Hypothesis H16 : v ∈ ω
Hypothesis H17 : eps_ v ≤ w + - x
Hypothesis H18 : x2 ∈ ω
Hypothesis H19 : eps_ x2 ≤ y + - u
Hypothesis H20 : SNo (eps_ v)
Hypothesis H21 : SNo (eps_ x2)
Hypothesis H22 : SNo (eps_ (v + x2))
Hypothesis H23 : SNo (eps_ v * eps_ x2)
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
Variable x : set
Variable y : set
Variable z : set
Variable w : set
Variable u : set
Variable v : set
Variable x2 : set
Hypothesis H0 : SNo x
Hypothesis H1 : SNo y
Hypothesis H2 : SNo (x * y)
Hypothesis H3 : SNo (- (x * y))
Hypothesis H4 : (∀y2 : set, y2 ∈ ω → abs_SNo (z + - (x * y)) < eps_ y2)
Hypothesis H5 : SNo z
Hypothesis H6 : x * y < z
Hypothesis H7 : SNo w
Hypothesis H8 : SNo u
Hypothesis H9 : (w * y + x * u) ≤ z + w * u
Hypothesis H10 : SNo (w * y)
Hypothesis H11 : SNo (x * u)
Hypothesis H12 : SNo (w * u)
Hypothesis H13 : SNo (- (w * u))
Hypothesis H14 : SNo (w + - x)
Hypothesis H15 : SNo (y + - u)
Hypothesis H16 : v ∈ ω
Hypothesis H17 : eps_ v ≤ w + - x
Hypothesis H18 : x2 ∈ ω
Hypothesis H19 : eps_ x2 ≤ y + - u
Hypothesis H20 : SNo (eps_ v)
Hypothesis H22 : SNo (eps_ (v + x2))
Hypothesis H23 : SNo (eps_ v * eps_ x2)
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
Variable x : set
Variable y : set
Variable z : set
Variable w : set
Variable u : set
Variable v : set
Variable x2 : set
Hypothesis H0 : SNo x
Hypothesis H1 : SNo y
Hypothesis H2 : SNo (x * y)
Hypothesis H3 : SNo (- (x * y))
Hypothesis H4 : (∀y2 : set, y2 ∈ ω → abs_SNo (z + - (x * y)) < eps_ y2)
Hypothesis H5 : SNo z
Hypothesis H6 : x * y < z
Hypothesis H7 : SNo w
Hypothesis H8 : SNo u
Hypothesis H9 : (w * y + x * u) ≤ z + w * u
Hypothesis H10 : SNo (w * y)
Hypothesis H11 : SNo (x * u)
Hypothesis H12 : SNo (w * u)
Hypothesis H13 : SNo (- (w * u))
Hypothesis H14 : SNo (w + - x)
Hypothesis H15 : SNo (y + - u)
Hypothesis H16 : v ∈ ω
Hypothesis H17 : eps_ v ≤ w + - x
Hypothesis H18 : x2 ∈ ω
Hypothesis H19 : eps_ x2 ≤ y + - u
Hypothesis H20 : SNo (eps_ v)
Hypothesis H21 : SNo (eps_ x2)
Hypothesis H23 : SNo (eps_ v * eps_ x2)
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
Variable x : set
Variable y : set
Variable z : set
Variable w : set
Variable u : set
Variable v : set
Variable x2 : set
Hypothesis H0 : SNo x
Hypothesis H1 : SNo y
Hypothesis H2 : SNo (x * y)
Hypothesis H3 : SNo (- (x * y))
Hypothesis H4 : (∀y2 : set, y2 ∈ ω → abs_SNo (z + - (x * y)) < eps_ y2)
Hypothesis H5 : SNo z
Hypothesis H6 : x * y < z
Hypothesis H7 : SNo w
Hypothesis H8 : SNo u
Hypothesis H9 : (w * y + x * u) ≤ z + w * u
Hypothesis H11 : SNo (x * u)
Hypothesis H12 : SNo (w * u)
Hypothesis H13 : SNo (- (w * u))
Hypothesis H14 : SNo (w + - x)
Hypothesis H15 : SNo (y + - u)
Hypothesis H16 : v ∈ ω
Hypothesis H17 : eps_ v ≤ w + - x
Hypothesis H18 : x2 ∈ ω
Hypothesis H19 : eps_ x2 ≤ y + - u
Hypothesis H20 : SNo (eps_ v)
Hypothesis H21 : SNo (eps_ x2)
Hypothesis H22 : v + x2 ∈ ω
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
Variable x : set
Variable y : set
Variable z : set
Variable w : set
Variable u : set
Variable v : set
Variable x2 : set
Hypothesis H0 : SNo x
Hypothesis H2 : SNo (x * y)
Hypothesis H3 : SNo (- (x * y))
Hypothesis H4 : (∀y2 : set, y2 ∈ ω → abs_SNo (z + - (x * y)) < eps_ y2)
Hypothesis H5 : SNo z
Hypothesis H6 : x * y < z
Hypothesis H7 : SNo w
Hypothesis H8 : SNo u
Hypothesis H9 : (w * y + x * u) ≤ z + w * u
Hypothesis H10 : SNo (w * y)
Hypothesis H11 : SNo (x * u)
Hypothesis H12 : SNo (w * u)
Hypothesis H13 : SNo (- (w * u))
Hypothesis H14 : SNo (w + - x)
Hypothesis H15 : SNo (y + - u)
Hypothesis H16 : v ∈ ω
Hypothesis H17 : eps_ v ≤ w + - x
Hypothesis H18 : x2 ∈ ω
Hypothesis H19 : eps_ x2 ≤ y + - u
Hypothesis H20 : SNo (eps_ v)
Hypothesis H21 : SNo (eps_ x2)
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
Variable x : set
Variable y : set
Variable z : set
Variable w : set
Variable u : set
Variable v : set
Variable x2 : set
Hypothesis H0 : SNo x
Hypothesis H1 : SNo y
Hypothesis H2 : SNo (x * y)
Hypothesis H3 : SNo (- (x * y))
Hypothesis H4 : (∀y2 : set, y2 ∈ ω → abs_SNo (z + - (x * y)) < eps_ y2)
Hypothesis H6 : x * y < z
Hypothesis H7 : SNo w
Hypothesis H8 : SNo u
Hypothesis H9 : (w * y + x * u) ≤ z + w * u
Hypothesis H10 : SNo (w * y)
Hypothesis H11 : SNo (x * u)
Hypothesis H12 : SNo (w * u)
Hypothesis H13 : SNo (- (w * u))
Hypothesis H14 : SNo (w + - x)
Hypothesis H15 : SNo (y + - u)
Hypothesis H16 : v ∈ ω
Hypothesis H17 : eps_ v ≤ w + - x
Hypothesis H18 : x2 ∈ ω
Hypothesis H19 : eps_ x2 ≤ y + - u
Hypothesis H20 : SNo (eps_ v)
Hypothesis H21 : SNo (eps_ x2)
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
Variable x : set
Variable y : set
Variable z : set
Variable w : set
Variable u : set
Variable v : set
Variable x2 : set
Hypothesis H0 : SNo x
Hypothesis H1 : SNo y
Hypothesis H2 : SNo (x * y)
Hypothesis H3 : SNo (- (x * y))
Hypothesis H4 : (∀y2 : set, y2 ∈ ω → abs_SNo (z + - (x * y)) < eps_ y2)
Hypothesis H5 : SNo z
Hypothesis H6 : x * y < z
Hypothesis H7 : SNo w
Hypothesis H8 : SNo u
Hypothesis H9 : (w * y + x * u) ≤ z + w * u
Hypothesis H10 : SNo (w * y)
Hypothesis H11 : SNo (x * u)
Hypothesis H12 : SNo (w * u)
Hypothesis H13 : SNo (- (w * u))
Hypothesis H14 : SNo (w + - x)
Hypothesis H15 : SNo (y + - u)
Hypothesis H16 : v ∈ ω
Hypothesis H17 : eps_ v ≤ w + - x
Hypothesis H18 : x2 ∈ ω
Hypothesis H19 : eps_ x2 ≤ y + - u
Hypothesis H20 : SNo (eps_ v)
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
Variable x : set
Variable y : set
Variable z : set
Variable w : set
Variable u : set
Variable v : set
Variable x2 : set
Hypothesis H0 : SNo x
Hypothesis H1 : SNo y
Hypothesis H2 : SNo (x * y)
Hypothesis H3 : SNo (- (x * y))
Hypothesis H4 : (∀y2 : set, y2 ∈ ω → abs_SNo (z + - (x * y)) < eps_ y2)
Hypothesis H5 : SNo z
Hypothesis H6 : x * y < z
Hypothesis H7 : SNo w
Hypothesis H8 : SNo u
Hypothesis H9 : (w * y + x * u) ≤ z + w * u
Hypothesis H10 : SNo (w * y)
Hypothesis H11 : SNo (x * u)
Hypothesis H12 : SNo (w * u)
Hypothesis H14 : SNo (w + - x)
Hypothesis H15 : SNo (y + - u)
Hypothesis H16 : v ∈ ω
Hypothesis H17 : eps_ v ≤ w + - x
Hypothesis H18 : x2 ∈ ω
Hypothesis H19 : eps_ x2 ≤ y + - u
Hypothesis H20 : SNo (eps_ v)
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
Variable x : set
Variable y : set
Variable z : set
Variable w : set
Variable u : set
Variable v : set
Variable x2 : set
Hypothesis H1 : SNo y
Hypothesis H2 : SNo (x * y)
Hypothesis H3 : SNo (- (x * y))
Hypothesis H4 : (∀y2 : set, y2 ∈ ω → abs_SNo (z + - (x * y)) < eps_ y2)
Hypothesis H5 : SNo z
Hypothesis H6 : x * y < z
Hypothesis H7 : SNo w
Hypothesis H8 : SNo u
Hypothesis H9 : (w * y + x * u) ≤ z + w * u
Hypothesis H10 : SNo (w * y)
Hypothesis H11 : SNo (x * u)
Hypothesis H12 : SNo (w * u)
Hypothesis H13 : SNo (- (w * u))
Hypothesis H14 : SNo (w + - x)
Hypothesis H15 : SNo (y + - u)
Hypothesis H16 : v ∈ ω
Hypothesis H17 : eps_ v ≤ w + - x
Hypothesis H18 : x2 ∈ ω
Hypothesis H19 : eps_ x2 ≤ y + - u
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
Variable x : set
Variable y : set
Variable z : set
Variable w : set
Variable u : set
Variable v : set
Variable x2 : set
Hypothesis H0 : SNo x
Hypothesis H1 : SNo y
Hypothesis H2 : SNo (x * y)
Hypothesis H3 : SNo (- (x * y))
Hypothesis H4 : (∀y2 : set, y2 ∈ ω → abs_SNo (z + - (x * y)) < eps_ y2)
Hypothesis H6 : x * y < z
Hypothesis H7 : SNo w
Hypothesis H8 : SNo u
Hypothesis H9 : (w * y + x * u) ≤ z + w * u
Hypothesis H10 : SNo (w * y)
Hypothesis H11 : SNo (x * u)
Hypothesis H12 : SNo (w * u)
Hypothesis H13 : SNo (- (w * u))
Hypothesis H14 : SNo (w + - x)
Hypothesis H15 : SNo (y + - u)
Hypothesis H16 : v ∈ ω
Hypothesis H17 : eps_ v ≤ w + - x
Hypothesis H18 : x2 ∈ ω
Hypothesis H19 : eps_ x2 ≤ y + - u
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
Variable x : set
Variable y : set
Variable z : set
Variable w : set
Variable u : set
Hypothesis H0 : SNo x
Hypothesis H1 : SNo y
Hypothesis H2 : SNo (x * y)
Hypothesis H3 : SNo (- (x * y))
Hypothesis H4 : (∀v : set, v ∈ SNoR x → (∀P : prop, (∀x2 : set, x2 ∈ ω → eps_ x2 ≤ v + - x → P) → P))
Hypothesis H5 : (∀v : set, v ∈ SNoL y → (∀P : prop, (∀x2 : set, x2 ∈ ω → eps_ x2 ≤ y + - v → P) → P))
Hypothesis H6 : (∀v : set, v ∈ ω → abs_SNo (z + - (x * y)) < eps_ v)
Hypothesis H7 : SNo z
Hypothesis H8 : x * y < z
Hypothesis H9 : w ∈ SNoR x
Hypothesis H10 : u ∈ SNoL y
Hypothesis H11 : SNo w
Hypothesis H12 : SNo u
Hypothesis H13 : (w * y + x * u) ≤ z + w * u
Hypothesis H15 : SNo (x * u)
Hypothesis H16 : SNo (w * u)
Hypothesis H17 : SNo (- (w * u))
Hypothesis H18 : SNo (w + - x)
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
Variable x : set
Variable y : set
Variable z : set
Variable w : set
Variable u : set
Hypothesis H0 : SNo x
Hypothesis H1 : SNo y
Hypothesis H2 : SNo (x * y)
Hypothesis H4 : (∀v : set, v ∈ SNoR x → (∀P : prop, (∀x2 : set, x2 ∈ ω → eps_ x2 ≤ v + - x → P) → P))
Hypothesis H5 : (∀v : set, v ∈ SNoL y → (∀P : prop, (∀x2 : set, x2 ∈ ω → eps_ x2 ≤ y + - v → P) → P))
Hypothesis H6 : (∀v : set, v ∈ ω → abs_SNo (z + - (x * y)) < eps_ v)
Hypothesis H7 : SNo z
Hypothesis H8 : x * y < z
Hypothesis H9 : w ∈ SNoR x
Hypothesis H10 : u ∈ SNoL y
Hypothesis H11 : SNo w
Hypothesis H12 : SNo u
Hypothesis H13 : (w * y + x * u) ≤ z + w * u
Hypothesis H14 : SNo (w * y)
Hypothesis H15 : SNo (x * u)
Hypothesis H16 : SNo (w * u)
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
Variable x : set
Variable y : set
Variable z : set
Variable w : set
Variable u : set
Hypothesis H0 : SNo x
Hypothesis H1 : SNo y
Hypothesis H2 : SNo (x * y)
Hypothesis H3 : SNo (- (x * y))
Hypothesis H4 : (∀v : set, v ∈ SNoR x → (∀P : prop, (∀x2 : set, x2 ∈ ω → eps_ x2 ≤ v + - x → P) → P))
Hypothesis H5 : (∀v : set, v ∈ SNoL y → (∀P : prop, (∀x2 : set, x2 ∈ ω → eps_ x2 ≤ y + - v → P) → P))
Hypothesis H6 : (∀v : set, v ∈ ω → abs_SNo (z + - (x * y)) < eps_ v)
Hypothesis H7 : SNo z
Hypothesis H8 : x * y < z
Hypothesis H9 : w ∈ SNoR x
Hypothesis H10 : u ∈ SNoL y
Hypothesis H11 : SNo w
Hypothesis H13 : (w * y + x * u) ≤ z + w * u
Hypothesis H14 : SNo (w * y)
Hypothesis H15 : SNo (x * u)
Hypothesis H16 : SNo (w * u)
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
Variable x : set
Variable y : set
Variable z : set
Variable w : set
Variable u : set
Variable v : set
Variable x2 : set
Hypothesis H0 : SNo x
Hypothesis H1 : SNo y
Hypothesis H2 : SNo (x * y)
Hypothesis H3 : SNo (- (x * y))
Hypothesis H4 : SNo z
Hypothesis H5 : x * y < z
Hypothesis H6 : SNo w
Hypothesis H7 : SNo u
Hypothesis H8 : (w * y + x * u) ≤ z + w * u
Hypothesis H10 : SNo (x * u)
Hypothesis H11 : SNo (w * u)
Hypothesis H12 : SNo (- (w * u))
Hypothesis H13 : SNo (x + - w)
Hypothesis H14 : SNo (u + - y)
Hypothesis H15 : v ∈ ω
Hypothesis H16 : eps_ v ≤ x + - w
Hypothesis H17 : x2 ∈ ω
Hypothesis H18 : eps_ x2 ≤ u + - y
Hypothesis H19 : SNo (eps_ v)
Hypothesis H20 : SNo (eps_ x2)
Hypothesis H21 : SNo (eps_ (v + x2))
Hypothesis H22 : SNo (eps_ v * eps_ x2)
Hypothesis H23 : abs_SNo (z + - (x * y)) < eps_ (v + x2)
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
Variable x : set
Variable y : set
Variable z : set
Variable w : set
Variable u : set
Variable v : set
Variable x2 : set
Hypothesis H0 : SNo x
Hypothesis H1 : SNo y
Hypothesis H2 : SNo (x * y)
Hypothesis H3 : SNo (- (x * y))
Hypothesis H4 : (∀y2 : set, y2 ∈ ω → abs_SNo (z + - (x * y)) < eps_ y2)
Hypothesis H6 : x * y < z
Hypothesis H7 : SNo w
Hypothesis H8 : SNo u
Hypothesis H9 : (w * y + x * u) ≤ z + w * u
Hypothesis H10 : SNo (w * y)
Hypothesis H11 : SNo (x * u)
Hypothesis H12 : SNo (w * u)
Hypothesis H13 : SNo (- (w * u))
Hypothesis H14 : SNo (x + - w)
Hypothesis H15 : SNo (u + - y)
Hypothesis H16 : v ∈ ω
Hypothesis H17 : eps_ v ≤ x + - w
Hypothesis H18 : x2 ∈ ω
Hypothesis H19 : eps_ x2 ≤ u + - y
Hypothesis H20 : SNo (eps_ v)
Hypothesis H21 : SNo (eps_ x2)
Hypothesis H22 : SNo (eps_ (v + x2))
Hypothesis H23 : SNo (eps_ v * eps_ x2)
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
Variable x : set
Variable y : set
Variable z : set
Variable w : set
Variable u : set
Variable v : set
Variable x2 : set
Hypothesis H0 : SNo x
Hypothesis H1 : SNo y
Hypothesis H2 : SNo (x * y)
Hypothesis H3 : SNo (- (x * y))
Hypothesis H4 : (∀y2 : set, y2 ∈ ω → abs_SNo (z + - (x * y)) < eps_ y2)
Hypothesis H5 : SNo z
Hypothesis H6 : x * y < z
Hypothesis H7 : SNo w
Hypothesis H8 : SNo u
Hypothesis H9 : (w * y + x * u) ≤ z + w * u
Hypothesis H10 : SNo (w * y)
Hypothesis H12 : SNo (w * u)
Hypothesis H13 : SNo (- (w * u))
Hypothesis H14 : SNo (x + - w)
Hypothesis H15 : SNo (u + - y)
Hypothesis H16 : v ∈ ω
Hypothesis H17 : eps_ v ≤ x + - w
Hypothesis H18 : x2 ∈ ω
Hypothesis H19 : eps_ x2 ≤ u + - y
Hypothesis H20 : SNo (eps_ v)
Hypothesis H21 : SNo (eps_ x2)
Hypothesis H22 : SNo (eps_ (v + x2))
Hypothesis H23 : SNo (eps_ v * eps_ x2)
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
Variable x : set
Variable y : set
Variable z : set
Variable w : set
Variable u : set
Variable v : set
Variable x2 : set
Hypothesis H0 : SNo x
Hypothesis H1 : SNo y
Hypothesis H2 : SNo (x * y)
Hypothesis H3 : SNo (- (x * y))
Hypothesis H4 : (∀y2 : set, y2 ∈ ω → abs_SNo (z + - (x * y)) < eps_ y2)
Hypothesis H5 : SNo z
Hypothesis H6 : x * y < z
Hypothesis H7 : SNo w
Hypothesis H8 : SNo u
Hypothesis H9 : (w * y + x * u) ≤ z + w * u
Hypothesis H10 : SNo (w * y)
Hypothesis H11 : SNo (x * u)
Hypothesis H13 : SNo (- (w * u))
Hypothesis H14 : SNo (x + - w)
Hypothesis H15 : SNo (u + - y)
Hypothesis H16 : v ∈ ω
Hypothesis H17 : eps_ v ≤ x + - w
Hypothesis H18 : x2 ∈ ω
Hypothesis H19 : eps_ x2 ≤ u + - y
Hypothesis H20 : SNo (eps_ v)
Hypothesis H21 : SNo (eps_ x2)
Hypothesis H22 : SNo (eps_ (v + x2))
Hypothesis H23 : SNo (eps_ v * eps_ x2)
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
Variable x : set
Variable y : set
Variable z : set
Variable w : set
Variable u : set
Variable v : set
Variable x2 : set
Hypothesis H0 : SNo x
Hypothesis H1 : SNo y
Hypothesis H2 : SNo (x * y)
Hypothesis H3 : SNo (- (x * y))
Hypothesis H4 : (∀y2 : set, y2 ∈ ω → abs_SNo (z + - (x * y)) < eps_ y2)
Hypothesis H5 : SNo z
Hypothesis H6 : x * y < z
Hypothesis H7 : SNo w
Hypothesis H8 : SNo u
Hypothesis H9 : (w * y + x * u) ≤ z + w * u
Hypothesis H11 : SNo (x * u)
Hypothesis H12 : SNo (w * u)
Hypothesis H13 : SNo (- (w * u))
Hypothesis H14 : SNo (x + - w)
Hypothesis H15 : SNo (u + - y)
Hypothesis H16 : v ∈ ω
Hypothesis H17 : eps_ v ≤ x + - w
Hypothesis H18 : x2 ∈ ω
Hypothesis H19 : eps_ x2 ≤ u + - y
Hypothesis H20 : SNo (eps_ v)
Hypothesis H21 : SNo (eps_ x2)
Hypothesis H22 : v + x2 ∈ ω
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
Variable x : set
Variable y : set
Variable z : set
Variable w : set
Variable u : set
Variable v : set
Variable x2 : set
Hypothesis H0 : SNo x
Hypothesis H1 : SNo y
Hypothesis H2 : SNo (x * y)
Hypothesis H3 : SNo (- (x * y))
Hypothesis H4 : (∀y2 : set, y2 ∈ ω → abs_SNo (z + - (x * y)) < eps_ y2)
Hypothesis H6 : x * y < z
Hypothesis H7 : SNo w
Hypothesis H8 : SNo u
Hypothesis H9 : (w * y + x * u) ≤ z + w * u
Hypothesis H10 : SNo (w * y)
Hypothesis H11 : SNo (x * u)
Hypothesis H12 : SNo (w * u)
Hypothesis H13 : SNo (- (w * u))
Hypothesis H14 : SNo (x + - w)
Hypothesis H15 : SNo (u + - y)
Hypothesis H16 : v ∈ ω
Hypothesis H17 : eps_ v ≤ x + - w
Hypothesis H18 : x2 ∈ ω
Hypothesis H19 : eps_ x2 ≤ u + - y
Hypothesis H20 : SNo (eps_ v)
Hypothesis H21 : SNo (eps_ x2)
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
Variable x : set
Variable y : set
Variable z : set
Variable w : set
Variable u : set
Variable v : set
Variable x2 : set
Hypothesis H0 : SNo x
Hypothesis H1 : SNo y
Hypothesis H2 : SNo (x * y)
Hypothesis H3 : SNo (- (x * y))
Hypothesis H4 : (∀y2 : set, y2 ∈ ω → abs_SNo (z + - (x * y)) < eps_ y2)
Hypothesis H5 : SNo z
Hypothesis H6 : x * y < z
Hypothesis H7 : SNo w
Hypothesis H8 : SNo u
Hypothesis H9 : (w * y + x * u) ≤ z + w * u
Hypothesis H10 : SNo (w * y)
Hypothesis H11 : SNo (x * u)
Hypothesis H12 : SNo (w * u)
Hypothesis H13 : SNo (- (w * u))
Hypothesis H14 : SNo (x + - w)
Hypothesis H15 : SNo (u + - y)
Hypothesis H16 : v ∈ ω
Hypothesis H17 : eps_ v ≤ x + - w
Hypothesis H18 : x2 ∈ ω
Hypothesis H19 : eps_ x2 ≤ u + - y
Hypothesis H20 : SNo (eps_ v)
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
Variable x : set
Variable y : set
Variable z : set
Variable w : set
Variable u : set
Hypothesis H0 : SNo x
Hypothesis H2 : SNo (x * y)
Hypothesis H3 : SNo (- (x * y))
Hypothesis H4 : (∀v : set, v ∈ SNoL x → (∀P : prop, (∀x2 : set, x2 ∈ ω → eps_ x2 ≤ x + - v → P) → P))
Hypothesis H5 : (∀v : set, v ∈ SNoR y → (∀P : prop, (∀x2 : set, x2 ∈ ω → eps_ x2 ≤ v + - y → P) → P))
Hypothesis H6 : (∀v : set, v ∈ ω → abs_SNo (z + - (x * y)) < eps_ v)
Hypothesis H7 : SNo z
Hypothesis H8 : x * y < z
Hypothesis H9 : w ∈ SNoL x
Hypothesis H10 : u ∈ SNoR y
Hypothesis H11 : SNo w
Hypothesis H12 : SNo u
Hypothesis H13 : (w * y + x * u) ≤ z + w * u
Hypothesis H14 : SNo (w * y)
Hypothesis H15 : SNo (x * u)
Hypothesis H16 : SNo (w * u)
Hypothesis H17 : SNo (- (w * u))
Hypothesis H18 : SNo (x + - w)
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
Variable x : set
Variable y : set
Variable z : set
Variable w : set
Variable u : set
Hypothesis H0 : SNo x
Hypothesis H1 : SNo y
Hypothesis H3 : SNo (- (x * y))
Hypothesis H4 : (∀v : set, v ∈ SNoL x → (∀P : prop, (∀x2 : set, x2 ∈ ω → eps_ x2 ≤ x + - v → P) → P))
Hypothesis H5 : (∀v : set, v ∈ SNoR y → (∀P : prop, (∀x2 : set, x2 ∈ ω → eps_ x2 ≤ v + - y → P) → P))
Hypothesis H6 : (∀v : set, v ∈ ω → abs_SNo (z + - (x * y)) < eps_ v)
Hypothesis H7 : SNo z
Hypothesis H8 : x * y < z
Hypothesis H9 : w ∈ SNoL x
Hypothesis H10 : u ∈ SNoR y
Hypothesis H11 : SNo w
Hypothesis H12 : SNo u
Hypothesis H13 : (w * y + x * u) ≤ z + w * u
Hypothesis H14 : SNo (w * y)
Hypothesis H15 : SNo (x * u)
Hypothesis H16 : SNo (w * u)
Hypothesis H17 : SNo (- (w * u))
Hypothesis H18 : SNo (x + - w)
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
Variable x : set
Variable y : set
Variable z : set
Variable w : set
Variable u : set
Hypothesis H0 : SNo x
Hypothesis H2 : SNo (x * y)
Hypothesis H3 : SNo (- (x * y))
Hypothesis H4 : (∀v : set, v ∈ SNoL x → (∀P : prop, (∀x2 : set, x2 ∈ ω → eps_ x2 ≤ x + - v → P) → P))
Hypothesis H5 : (∀v : set, v ∈ SNoR y → (∀P : prop, (∀x2 : set, x2 ∈ ω → eps_ x2 ≤ v + - y → P) → P))
Hypothesis H6 : (∀v : set, v ∈ ω → abs_SNo (z + - (x * y)) < eps_ v)
Hypothesis H7 : SNo z
Hypothesis H8 : x * y < z
Hypothesis H9 : w ∈ SNoL x
Hypothesis H10 : u ∈ SNoR y
Hypothesis H11 : SNo w
Hypothesis H12 : SNo u
Hypothesis H13 : (w * y + x * u) ≤ z + w * u
Hypothesis H14 : SNo (w * y)
Hypothesis H15 : SNo (x * u)
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
Variable x : set
Variable y : set
Variable z : set
Variable w : set
Variable u : set
Variable v : set
Variable x2 : set
Hypothesis H0 : SNo x
Hypothesis H1 : SNo y
Hypothesis H2 : SNo (x * y)
Hypothesis H3 : SNo z
Hypothesis H4 : z < x * y
Hypothesis H5 : SNo w
Hypothesis H6 : SNo u
Hypothesis H7 : (z + w * u) ≤ w * y + x * u
Hypothesis H8 : SNo (w * u)
Hypothesis H10 : SNo (- (x * u))
Hypothesis H11 : SNo (w * y)
Hypothesis H12 : SNo (- (w * y))
Hypothesis H13 : SNo (w + - x)
Hypothesis H14 : SNo (u + - y)
Hypothesis H15 : v ∈ ω
Hypothesis H16 : eps_ v ≤ w + - x
Hypothesis H17 : x2 ∈ ω
Hypothesis H18 : eps_ x2 ≤ u + - y
Hypothesis H19 : SNo (eps_ v)
Hypothesis H20 : SNo (eps_ x2)
Hypothesis H21 : SNo (eps_ (v + x2))
Hypothesis H22 : SNo (eps_ v * eps_ x2)
Hypothesis H23 : abs_SNo (z + - (x * y)) < eps_ (v + x2)
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
Variable x : set
Variable y : set
Variable z : set
Variable w : set
Variable u : set
Variable v : set
Variable x2 : set
Hypothesis H0 : SNo x
Hypothesis H1 : SNo y
Hypothesis H2 : SNo (x * y)
Hypothesis H3 : (∀y2 : set, y2 ∈ ω → abs_SNo (z + - (x * y)) < eps_ y2)
Hypothesis H4 : SNo z
Hypothesis H5 : z < x * y
Hypothesis H6 : SNo w
Hypothesis H7 : SNo u
Hypothesis H8 : (z + w * u) ≤ w * y + x * u
Hypothesis H9 : SNo (w * u)
Hypothesis H10 : SNo (x * u)
Hypothesis H11 : SNo (- (x * u))
Hypothesis H13 : SNo (- (w * y))
Hypothesis H14 : SNo (w + - x)
Hypothesis H15 : SNo (u + - y)
Hypothesis H16 : v ∈ ω
Hypothesis H17 : eps_ v ≤ w + - x
Hypothesis H18 : x2 ∈ ω
Hypothesis H19 : eps_ x2 ≤ u + - y
Hypothesis H20 : SNo (eps_ v)
Hypothesis H21 : SNo (eps_ x2)
Hypothesis H22 : SNo (eps_ (v + x2))
Hypothesis H23 : SNo (eps_ v * eps_ x2)
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
Variable x : set
Variable y : set
Variable z : set
Variable w : set
Variable u : set
Variable v : set
Variable x2 : set
Hypothesis H0 : SNo x
Hypothesis H1 : SNo y
Hypothesis H2 : SNo (x * y)
Hypothesis H3 : (∀y2 : set, y2 ∈ ω → abs_SNo (z + - (x * y)) < eps_ y2)
Hypothesis H4 : SNo z
Hypothesis H5 : z < x * y
Hypothesis H6 : SNo w
Hypothesis H7 : SNo u
Hypothesis H8 : (z + w * u) ≤ w * y + x * u
Hypothesis H9 : SNo (w * u)
Hypothesis H10 : SNo (x * u)
Hypothesis H11 : SNo (- (x * u))
Hypothesis H12 : SNo (w * y)
Hypothesis H13 : SNo (- (w * y))
Hypothesis H14 : SNo (w + - x)
Hypothesis H16 : v ∈ ω
Hypothesis H17 : eps_ v ≤ w + - x
Hypothesis H18 : x2 ∈ ω
Hypothesis H19 : eps_ x2 ≤ u + - y
Hypothesis H20 : SNo (eps_ v)
Hypothesis H21 : SNo (eps_ x2)
Hypothesis H22 : SNo (eps_ (v + x2))
Hypothesis H23 : SNo (eps_ v * eps_ x2)
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
Variable x : set
Variable y : set
Variable z : set
Variable w : set
Variable u : set
Variable v : set
Variable x2 : set
Hypothesis H0 : SNo x
Hypothesis H2 : SNo (x * y)
Hypothesis H3 : (∀y2 : set, y2 ∈ ω → abs_SNo (z + - (x * y)) < eps_ y2)
Hypothesis H4 : SNo z
Hypothesis H5 : z < x * y
Hypothesis H6 : SNo w
Hypothesis H7 : SNo u
Hypothesis H8 : (z + w * u) ≤ w * y + x * u
Hypothesis H9 : SNo (w * u)
Hypothesis H10 : SNo (x * u)
Hypothesis H11 : SNo (- (x * u))
Hypothesis H12 : SNo (w * y)
Hypothesis H13 : SNo (- (w * y))
Hypothesis H14 : SNo (w + - x)
Hypothesis H15 : SNo (u + - y)
Hypothesis H16 : v ∈ ω
Hypothesis H17 : eps_ v ≤ w + - x
Hypothesis H18 : x2 ∈ ω
Hypothesis H19 : eps_ x2 ≤ u + - y
Hypothesis H20 : SNo (eps_ v)
Hypothesis H21 : SNo (eps_ x2)
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
Variable x : set
Variable y : set
Variable z : set
Variable w : set
Variable u : set
Variable v : set
Variable x2 : set
Hypothesis H0 : SNo x
Hypothesis H1 : SNo y
Hypothesis H2 : SNo (x * y)
Hypothesis H3 : (∀y2 : set, y2 ∈ ω → abs_SNo (z + - (x * y)) < eps_ y2)
Hypothesis H4 : SNo z
Hypothesis H5 : z < x * y
Hypothesis H6 : SNo w
Hypothesis H7 : SNo u
Hypothesis H8 : (z + w * u) ≤ w * y + x * u
Hypothesis H9 : SNo (w * u)
Hypothesis H10 : SNo (x * u)
Hypothesis H11 : SNo (- (x * u))
Hypothesis H12 : SNo (w * y)
Hypothesis H13 : SNo (- (w * y))
Hypothesis H14 : SNo (w + - x)
Hypothesis H15 : SNo (u + - y)
Hypothesis H16 : v ∈ ω
Hypothesis H17 : eps_ v ≤ w + - x
Hypothesis H19 : eps_ x2 ≤ u + - y
Hypothesis H20 : SNo (eps_ v)
Hypothesis H21 : SNo (eps_ x2)
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
Variable x : set
Variable y : set
Variable z : set
Variable w : set
Variable u : set
Variable v : set
Variable x2 : set
Hypothesis H0 : SNo x
Hypothesis H1 : SNo y
Hypothesis H2 : SNo (x * y)
Hypothesis H3 : (∀y2 : set, y2 ∈ ω → abs_SNo (z + - (x * y)) < eps_ y2)
Hypothesis H4 : SNo z
Hypothesis H5 : z < x * y
Hypothesis H6 : SNo w
Hypothesis H7 : SNo u
Hypothesis H8 : (z + w * u) ≤ w * y + x * u
Hypothesis H10 : SNo (x * u)
Hypothesis H11 : SNo (- (x * u))
Hypothesis H12 : SNo (w * y)
Hypothesis H13 : SNo (- (w * y))
Hypothesis H14 : SNo (w + - x)
Hypothesis H15 : SNo (u + - y)
Hypothesis H16 : v ∈ ω
Hypothesis H17 : eps_ v ≤ w + - x
Hypothesis H18 : x2 ∈ ω
Hypothesis H19 : eps_ x2 ≤ u + - y
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
Variable x : set
Variable y : set
Variable z : set
Variable w : set
Variable u : set
Hypothesis H1 : SNo y
Hypothesis H2 : SNo (x * y)
Hypothesis H3 : (∀v : set, v ∈ SNoR x → (∀P : prop, (∀x2 : set, x2 ∈ ω → eps_ x2 ≤ v + - x → P) → P))
Hypothesis H4 : (∀v : set, v ∈ SNoR y → (∀P : prop, (∀x2 : set, x2 ∈ ω → eps_ x2 ≤ v + - y → P) → P))
Hypothesis H5 : (∀v : set, v ∈ ω → abs_SNo (z + - (x * y)) < eps_ v)
Hypothesis H6 : SNo z
Hypothesis H7 : z < x * y
Hypothesis H8 : w ∈ SNoR x
Hypothesis H9 : u ∈ SNoR y
Hypothesis H10 : SNo w
Hypothesis H11 : SNo u
Hypothesis H12 : (z + w * u) ≤ w * y + x * u
Hypothesis H13 : SNo (w * u)
Hypothesis H14 : SNo (x * u)
Hypothesis H15 : SNo (- (x * u))
Hypothesis H16 : SNo (w * y)
Hypothesis H17 : SNo (- (w * y))
Hypothesis H18 : SNo (w + - x)
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
Variable x : set
Variable y : set
Variable z : set
Variable w : set
Variable u : set
Hypothesis H0 : SNo x
Hypothesis H1 : SNo y
Hypothesis H2 : SNo (x * y)
Hypothesis H3 : (∀v : set, v ∈ SNoR x → (∀P : prop, (∀x2 : set, x2 ∈ ω → eps_ x2 ≤ v + - x → P) → P))
Hypothesis H4 : (∀v : set, v ∈ SNoR y → (∀P : prop, (∀x2 : set, x2 ∈ ω → eps_ x2 ≤ v + - y → P) → P))
Hypothesis H5 : (∀v : set, v ∈ ω → abs_SNo (z + - (x * y)) < eps_ v)
Hypothesis H6 : SNo z
Hypothesis H7 : z < x * y
Hypothesis H8 : w ∈ SNoR x
Hypothesis H10 : SNo w
Hypothesis H11 : SNo u
Hypothesis H12 : (z + w * u) ≤ w * y + x * u
Hypothesis H13 : SNo (w * u)
Hypothesis H14 : SNo (x * u)
Hypothesis H15 : SNo (- (x * u))
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
Variable x : set
Variable y : set
Variable z : set
Variable w : set
Variable u : set
Hypothesis H0 : SNo x
Hypothesis H1 : SNo y
Hypothesis H2 : SNo (x * y)
Hypothesis H3 : (∀v : set, v ∈ SNoR x → (∀P : prop, (∀x2 : set, x2 ∈ ω → eps_ x2 ≤ v + - x → P) → P))
Hypothesis H4 : (∀v : set, v ∈ SNoR y → (∀P : prop, (∀x2 : set, x2 ∈ ω → eps_ x2 ≤ v + - y → P) → P))
Hypothesis H5 : (∀v : set, v ∈ ω → abs_SNo (z + - (x * y)) < eps_ v)
Hypothesis H6 : SNo z
Hypothesis H7 : z < x * y
Hypothesis H8 : w ∈ SNoR x
Hypothesis H9 : u ∈ SNoR y
Hypothesis H10 : SNo w
Hypothesis H11 : SNo u
Hypothesis H13 : SNo (w * u)
Hypothesis H14 : SNo (x * u)
Hypothesis H15 : SNo (- (x * u))
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
Variable x : set
Variable y : set
Variable z : set
Variable w : set
Variable u : set
Hypothesis H0 : SNo x
Hypothesis H2 : SNo (x * y)
Hypothesis H3 : (∀v : set, v ∈ SNoR x → (∀P : prop, (∀x2 : set, x2 ∈ ω → eps_ x2 ≤ v + - x → P) → P))
Hypothesis H4 : (∀v : set, v ∈ SNoR y → (∀P : prop, (∀x2 : set, x2 ∈ ω → eps_ x2 ≤ v + - y → P) → P))
Hypothesis H5 : (∀v : set, v ∈ ω → abs_SNo (z + - (x * y)) < eps_ v)
Hypothesis H6 : SNo z
Hypothesis H7 : z < x * y
Hypothesis H8 : w ∈ SNoR x
Hypothesis H9 : u ∈ SNoR y
Hypothesis H10 : SNo w
Hypothesis H11 : SNo u
Hypothesis H12 : (z + w * u) ≤ w * y + x * u
Hypothesis H13 : SNo (w * u)
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
Variable x : set
Variable y : set
Variable z : set
Variable w : set
Variable u : set
Hypothesis H0 : SNo x
Hypothesis H1 : SNo y
Hypothesis H2 : SNo (x * y)
Hypothesis H4 : (∀v : set, v ∈ SNoR y → (∀P : prop, (∀x2 : set, x2 ∈ ω → eps_ x2 ≤ v + - y → P) → P))
Hypothesis H5 : (∀v : set, v ∈ ω → abs_SNo (z + - (x * y)) < eps_ v)
Hypothesis H6 : SNo z
Hypothesis H7 : z < x * y
Hypothesis H8 : w ∈ SNoR x
Hypothesis H9 : u ∈ SNoR y
Hypothesis H10 : SNo w
Hypothesis H11 : SNo u
Hypothesis H12 : (z + w * u) ≤ w * y + x * u
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
Variable x : set
Variable y : set
Variable z : set
Variable w : set
Variable u : set
Variable v : set
Variable x2 : set
Hypothesis H0 : SNo x
Hypothesis H1 : SNo y
Hypothesis H2 : SNo (x * y)
Hypothesis H3 : SNo (- (x * y))
Hypothesis H4 : (∀y2 : set, y2 ∈ ω → abs_SNo (z + - (x * y)) < eps_ y2)
Hypothesis H6 : z < x * y
Hypothesis H7 : SNo w
Hypothesis H8 : SNo u
Hypothesis H9 : (z + w * u) ≤ w * y + x * u
Hypothesis H10 : SNo (w * u)
Hypothesis H11 : SNo (x * u)
Hypothesis H12 : SNo (- (x * u))
Hypothesis H13 : SNo (w * y)
Hypothesis H14 : SNo (- (w * y))
Hypothesis H15 : SNo (x + - w)
Hypothesis H16 : SNo (y + - u)
Hypothesis H17 : v ∈ ω
Hypothesis H18 : eps_ v ≤ x + - w
Hypothesis H19 : x2 ∈ ω
Hypothesis H20 : eps_ x2 ≤ y + - u
Hypothesis H21 : SNo (eps_ v)
Hypothesis H22 : SNo (eps_ x2)
Hypothesis H23 : SNo (eps_ (v + x2))
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
Variable x : set
Variable y : set
Variable z : set
Variable w : set
Variable u : set
Variable v : set
Variable x2 : set
Hypothesis H0 : SNo x
Hypothesis H1 : SNo y
Hypothesis H2 : SNo (x * y)
Hypothesis H3 : SNo (- (x * y))
Hypothesis H4 : (∀y2 : set, y2 ∈ ω → abs_SNo (z + - (x * y)) < eps_ y2)
Hypothesis H5 : SNo z
Hypothesis H6 : z < x * y
Hypothesis H7 : SNo w
Hypothesis H8 : SNo u
Hypothesis H9 : (z + w * u) ≤ w * y + x * u
Hypothesis H10 : SNo (w * u)
Hypothesis H11 : SNo (x * u)
Hypothesis H12 : SNo (- (x * u))
Hypothesis H13 : SNo (w * y)
Hypothesis H15 : SNo (x + - w)
Hypothesis H16 : SNo (y + - u)
Hypothesis H17 : v ∈ ω
Hypothesis H18 : eps_ v ≤ x + - w
Hypothesis H19 : x2 ∈ ω
Hypothesis H20 : eps_ x2 ≤ y + - u
Hypothesis H21 : SNo (eps_ v)
Hypothesis H22 : SNo (eps_ x2)
Hypothesis H23 : SNo (eps_ (v + x2))
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
Variable x : set
Variable y : set
Variable z : set
Variable w : set
Variable u : set
Variable v : set
Variable x2 : set
Hypothesis H0 : SNo x
Hypothesis H1 : SNo y
Hypothesis H2 : SNo (x * y)
Hypothesis H3 : SNo (- (x * y))
Hypothesis H4 : (∀y2 : set, y2 ∈ ω → abs_SNo (z + - (x * y)) < eps_ y2)
Hypothesis H5 : SNo z
Hypothesis H6 : z < x * y
Hypothesis H7 : SNo w
Hypothesis H8 : SNo u
Hypothesis H9 : (z + w * u) ≤ w * y + x * u
Hypothesis H10 : SNo (w * u)
Hypothesis H11 : SNo (x * u)
Hypothesis H13 : SNo (w * y)
Hypothesis H14 : SNo (- (w * y))
Hypothesis H15 : SNo (x + - w)
Hypothesis H16 : SNo (y + - u)
Hypothesis H17 : v ∈ ω
Hypothesis H18 : eps_ v ≤ x + - w
Hypothesis H19 : x2 ∈ ω
Hypothesis H20 : eps_ x2 ≤ y + - u
Hypothesis H21 : SNo (eps_ v)
Hypothesis H22 : SNo (eps_ x2)
Hypothesis H23 : v + x2 ∈ ω
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
Variable x : set
Variable y : set
Variable z : set
Variable w : set
Variable u : set
Variable v : set
Variable x2 : set
Hypothesis H0 : SNo x
Hypothesis H1 : SNo y
Hypothesis H2 : SNo (x * y)
Hypothesis H3 : SNo (- (x * y))
Hypothesis H4 : (∀y2 : set, y2 ∈ ω → abs_SNo (z + - (x * y)) < eps_ y2)
Hypothesis H5 : SNo z
Hypothesis H6 : z < x * y
Hypothesis H7 : SNo w
Hypothesis H8 : SNo u
Hypothesis H9 : (z + w * u) ≤ w * y + x * u
Hypothesis H10 : SNo (w * u)
Hypothesis H11 : SNo (x * u)
Hypothesis H12 : SNo (- (x * u))
Hypothesis H13 : SNo (w * y)
Hypothesis H14 : SNo (- (w * y))
Hypothesis H15 : SNo (x + - w)
Hypothesis H16 : SNo (y + - u)
Hypothesis H17 : v ∈ ω
Hypothesis H19 : x2 ∈ ω
Hypothesis H20 : eps_ x2 ≤ y + - u
Hypothesis H21 : SNo (eps_ v)
Hypothesis H22 : SNo (eps_ x2)
Hypothesis H23 : v + x2 ∈ ω
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
Variable x : set
Variable y : set
Variable z : set
Variable w : set
Variable u : set
Variable v : set
Variable x2 : set
Hypothesis H0 : SNo x
Hypothesis H1 : SNo y
Hypothesis H2 : SNo (x * y)
Hypothesis H3 : SNo (- (x * y))
Hypothesis H4 : (∀y2 : set, y2 ∈ ω → abs_SNo (z + - (x * y)) < eps_ y2)
Hypothesis H5 : SNo z
Hypothesis H6 : z < x * y
Hypothesis H7 : SNo w
Hypothesis H8 : SNo u
Hypothesis H9 : (z + w * u) ≤ w * y + x * u
Hypothesis H10 : SNo (w * u)
Hypothesis H11 : SNo (x * u)
Hypothesis H12 : SNo (- (x * u))
Hypothesis H13 : SNo (w * y)
Hypothesis H14 : SNo (- (w * y))
Hypothesis H15 : SNo (x + - w)
Hypothesis H16 : SNo (y + - u)
Hypothesis H17 : v ∈ ω
Hypothesis H18 : eps_ v ≤ x + - w
Hypothesis H19 : x2 ∈ ω
Hypothesis H20 : eps_ x2 ≤ y + - u
Hypothesis H22 : SNo (eps_ x2)
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
Variable x : set
Variable y : set
Variable z : set
Variable w : set
Variable u : set
Variable v : set
Variable x2 : set
Hypothesis H0 : SNo x
Hypothesis H1 : SNo y
Hypothesis H2 : SNo (x * y)
Hypothesis H3 : SNo (- (x * y))
Hypothesis H4 : (∀y2 : set, y2 ∈ ω → abs_SNo (z + - (x * y)) < eps_ y2)
Hypothesis H5 : SNo z
Hypothesis H6 : z < x * y
Hypothesis H7 : SNo w
Hypothesis H8 : SNo u
Hypothesis H9 : (z + w * u) ≤ w * y + x * u
Hypothesis H10 : SNo (w * u)
Hypothesis H11 : SNo (x * u)
Hypothesis H12 : SNo (- (x * u))
Hypothesis H13 : SNo (w * y)
Hypothesis H14 : SNo (- (w * y))
Hypothesis H15 : SNo (x + - w)
Hypothesis H16 : SNo (y + - u)
Hypothesis H17 : v ∈ ω
Hypothesis H18 : eps_ v ≤ x + - w
Hypothesis H20 : eps_ x2 ≤ y + - u
Hypothesis H21 : SNo (eps_ v)
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
Variable x : set
Variable y : set
Variable z : set
Variable w : set
Variable u : set
Variable v : set
Variable x2 : set
Hypothesis H0 : SNo x
Hypothesis H1 : SNo y
Hypothesis H2 : SNo (x * y)
Hypothesis H3 : SNo (- (x * y))
Hypothesis H5 : SNo z
Hypothesis H6 : z < x * y
Hypothesis H7 : SNo w
Hypothesis H8 : SNo u
Hypothesis H9 : (z + w * u) ≤ w * y + x * u
Hypothesis H10 : SNo (w * u)
Hypothesis H11 : SNo (x * u)
Hypothesis H12 : SNo (- (x * u))
Hypothesis H13 : SNo (w * y)
Hypothesis H14 : SNo (- (w * y))
Hypothesis H15 : SNo (x + - w)
Hypothesis H16 : SNo (y + - u)
Hypothesis H17 : v ∈ ω
Hypothesis H18 : eps_ v ≤ x + - w
Hypothesis H19 : x2 ∈ ω
Hypothesis H20 : eps_ x2 ≤ y + - u
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
Variable x : set
Variable y : set
Variable z : set
Variable w : set
Variable u : set
Hypothesis H0 : SNo x
Hypothesis H1 : SNo y
Hypothesis H3 : SNo (- (x * y))
Hypothesis H4 : (∀v : set, v ∈ SNoL x → (∀P : prop, (∀x2 : set, x2 ∈ ω → eps_ x2 ≤ x + - v → P) → P))
Hypothesis H5 : (∀v : set, v ∈ SNoL y → (∀P : prop, (∀x2 : set, x2 ∈ ω → eps_ x2 ≤ y + - v → P) → P))
Hypothesis H6 : (∀v : set, v ∈ ω → abs_SNo (z + - (x * y)) < eps_ v)
Hypothesis H7 : SNo z
Hypothesis H8 : z < x * y
Hypothesis H9 : w ∈ SNoL x
Hypothesis H10 : u ∈ SNoL y
Hypothesis H11 : SNo w
Hypothesis H12 : SNo u
Hypothesis H13 : (z + w * u) ≤ w * y + x * u
Hypothesis H14 : SNo (w * u)
Hypothesis H15 : SNo (x * u)
Hypothesis H16 : SNo (- (x * u))
Hypothesis H17 : SNo (w * y)
Hypothesis H18 : SNo (- (w * y))
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
Variable x : set
Variable y : set
Variable z : set
Variable w : set
Variable u : set
Hypothesis H1 : SNo y
Hypothesis H2 : SNo (x * y)
Hypothesis H3 : SNo (- (x * y))
Hypothesis H4 : (∀v : set, v ∈ SNoL x → (∀P : prop, (∀x2 : set, x2 ∈ ω → eps_ x2 ≤ x + - v → P) → P))
Hypothesis H5 : (∀v : set, v ∈ SNoL y → (∀P : prop, (∀x2 : set, x2 ∈ ω → eps_ x2 ≤ y + - v → P) → P))
Hypothesis H6 : (∀v : set, v ∈ ω → abs_SNo (z + - (x * y)) < eps_ v)
Hypothesis H7 : SNo z
Hypothesis H8 : z < x * y
Hypothesis H9 : w ∈ SNoL x
Hypothesis H10 : u ∈ SNoL y
Hypothesis H11 : SNo w
Hypothesis H12 : SNo u
Hypothesis H13 : (z + w * u) ≤ w * y + x * u
Hypothesis H14 : SNo (w * u)
Hypothesis H15 : SNo (x * u)
Hypothesis H16 : SNo (- (x * u))
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
Variable x : set
Variable y : set
Variable z : set
Variable w : set
Variable u : set
Hypothesis H0 : SNo x
Hypothesis H2 : SNo (x * y)
Hypothesis H3 : SNo (- (x * y))
Hypothesis H4 : (∀v : set, v ∈ SNoL x → (∀P : prop, (∀x2 : set, x2 ∈ ω → eps_ x2 ≤ x + - v → P) → P))
Hypothesis H5 : (∀v : set, v ∈ SNoL y → (∀P : prop, (∀x2 : set, x2 ∈ ω → eps_ x2 ≤ y + - v → P) → P))
Hypothesis H6 : (∀v : set, v ∈ ω → abs_SNo (z + - (x * y)) < eps_ v)
Hypothesis H7 : SNo z
Hypothesis H8 : z < x * y
Hypothesis H9 : w ∈ SNoL x
Hypothesis H10 : u ∈ SNoL y
Hypothesis H11 : SNo w
Hypothesis H12 : SNo u
Hypothesis H13 : (z + w * u) ≤ w * y + x * u
Hypothesis H14 : SNo (w * u)
Hypothesis H15 : SNo (x * u)
Hypothesis H16 : SNo (- (x * u))
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
Variable x : set
Variable y : set
Variable z : set
Variable w : set
Variable u : set
Hypothesis H0 : SNo x
Hypothesis H1 : SNo y
Hypothesis H2 : SNo (x * y)
Hypothesis H3 : SNo (- (x * y))
Hypothesis H4 : (∀v : set, v ∈ SNoL x → (∀P : prop, (∀x2 : set, x2 ∈ ω → eps_ x2 ≤ x + - v → P) → P))
Hypothesis H5 : (∀v : set, v ∈ SNoL y → (∀P : prop, (∀x2 : set, x2 ∈ ω → eps_ x2 ≤ y + - v → P) → P))
Hypothesis H7 : SNo z
Hypothesis H8 : z < x * y
Hypothesis H9 : w ∈ SNoL x
Hypothesis H10 : u ∈ SNoL y
Hypothesis H11 : SNo w
Hypothesis H12 : SNo u
Hypothesis H13 : (z + w * u) ≤ w * y + x * u
Hypothesis H14 : SNo (w * u)
Hypothesis H15 : SNo (x * u)
Hypothesis H16 : SNo (- (x * u))
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
Variable x : set
Variable y : set
Variable z : set
Variable w : set
Variable u : set
Hypothesis H0 : SNo x
Hypothesis H1 : SNo y
Hypothesis H2 : SNo (x * y)
Hypothesis H3 : SNo (- (x * y))
Hypothesis H4 : (∀v : set, v ∈ SNoL x → (∀P : prop, (∀x2 : set, x2 ∈ ω → eps_ x2 ≤ x + - v → P) → P))
Hypothesis H5 : (∀v : set, v ∈ SNoL y → (∀P : prop, (∀x2 : set, x2 ∈ ω → eps_ x2 ≤ y + - v → P) → P))
Hypothesis H6 : (∀v : set, v ∈ ω → abs_SNo (z + - (x * y)) < eps_ v)
Hypothesis H7 : SNo z
Hypothesis H8 : z < x * y
Hypothesis H9 : w ∈ SNoL x
Hypothesis H11 : SNo w
Hypothesis H12 : SNo u
Hypothesis H13 : (z + w * u) ≤ w * y + x * u
Hypothesis H14 : SNo (w * u)
Hypothesis H15 : SNo (x * u)
Hypothesis H16 : SNo (- (x * u))
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
Variable x : set
Variable y : set
Variable z : set
Variable w : set
Variable u : set
Hypothesis H0 : SNo x
Hypothesis H1 : SNo y
Hypothesis H2 : SNo (x * y)
Hypothesis H3 : SNo (- (x * y))
Hypothesis H4 : (∀v : set, v ∈ SNoL x → (∀P : prop, (∀x2 : set, x2 ∈ ω → eps_ x2 ≤ x + - v → P) → P))
Hypothesis H5 : (∀v : set, v ∈ SNoL y → (∀P : prop, (∀x2 : set, x2 ∈ ω → eps_ x2 ≤ y + - v → P) → P))
Hypothesis H6 : (∀v : set, v ∈ ω → abs_SNo (z + - (x * y)) < eps_ v)
Hypothesis H7 : SNo z
Hypothesis H8 : z < x * y
Hypothesis H9 : w ∈ SNoL x
Hypothesis H10 : u ∈ SNoL y
Hypothesis H11 : SNo w
Hypothesis H12 : SNo u
Hypothesis H13 : (z + w * u) ≤ w * y + x * u
Hypothesis H15 : SNo (x * u)
Hypothesis H16 : SNo (- (x * u))
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
Variable x : set
Variable y : set
Variable z : set
Variable w : set
Variable u : set
Hypothesis H0 : SNo x
Hypothesis H1 : SNo y
Hypothesis H2 : SNo (x * y)
Hypothesis H3 : SNo (- (x * y))
Hypothesis H4 : (∀v : set, v ∈ SNoL x → (∀P : prop, (∀x2 : set, x2 ∈ ω → eps_ x2 ≤ x + - v → P) → P))
Hypothesis H5 : (∀v : set, v ∈ SNoL y → (∀P : prop, (∀x2 : set, x2 ∈ ω → eps_ x2 ≤ y + - v → P) → P))
Hypothesis H6 : (∀v : set, v ∈ ω → abs_SNo (z + - (x * y)) < eps_ v)
Hypothesis H7 : SNo z
Hypothesis H9 : w ∈ SNoL x
Hypothesis H10 : u ∈ SNoL y
Hypothesis H11 : SNo w
Hypothesis H12 : SNo u
Hypothesis H13 : (z + w * u) ≤ w * y + x * u
Hypothesis H14 : SNo (w * u)
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
Variable x : set
Variable y : set
Variable z : set
Variable w : set
Variable u : set
Variable v : set
Hypothesis H0 : SNo x
Hypothesis H1 : SNo y
Hypothesis H2 : z ∈ ω
Hypothesis H3 : eps_ z * x < ordsucc Empty
Hypothesis H4 : eps_ z * y < ordsucc Empty
Hypothesis H6 : w + ordsucc Empty ∈ ω
Hypothesis H7 : w + ordsucc (ordsucc Empty) ∈ ω
Hypothesis H8 : u < x
Hypothesis H9 : SNo u
Hypothesis H10 : v < y
Hypothesis H11 : SNo v
Hypothesis H12 : SNo (eps_ z)
Hypothesis H13 : SNo (eps_ (w + ordsucc Empty))
Hypothesis H14 : SNo (eps_ (w + ordsucc (ordsucc Empty)))
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
Variable x : set
Variable y : set
Variable z : set
Variable w : set
Variable u : set
Variable v : set
Variable x2 : set
Variable y2 : set
Variable z2 : set
Hypothesis H0 : SNo x
Hypothesis H1 : SNo y
Hypothesis H2 : SNo (x * y)
Hypothesis H3 : SNo (- (x * y))
Hypothesis H4 : SNo (SNoCut (Repl ω (ap z)) (Repl ω (ap w)))
Hypothesis H5 : (∀w2 : set, w2 ∈ ω → SNoCut (Repl ω (ap z)) (Repl ω (ap w)) < ap w w2)
Hypothesis H6 : (∀w2 : set, w2 ∈ ω → (ap w w2 + - (eps_ w2)) < x * y)
Hypothesis H7 : u = v * y + x * x2 + - (v * x2)
Hypothesis H8 : y2 ∈ ω
Hypothesis H9 : eps_ y2 ≤ v + - x
Hypothesis H10 : z2 ∈ ω
Hypothesis H11 : eps_ z2 ≤ y + - x2
Hypothesis H12 : SNo v
Hypothesis H13 : SNo x2
Hypothesis H14 : SNo (eps_ y2)
Hypothesis H15 : SNo (eps_ z2)
Hypothesis H16 : y2 + z2 ∈ ω
Hypothesis H17 : SNo (eps_ (y2 + z2))
Hypothesis H18 : SNo (- (eps_ (y2 + z2)))
Hypothesis H19 : SNo (ap w (y2 + z2))
Hypothesis H20 : SNo (v * y)
Hypothesis H22 : SNo (- (v * x2))
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
Variable x : set
Variable y : set
Variable z : set
Variable w : set
Variable u : set
Variable v : set
Variable x2 : set
Variable y2 : set
Variable z2 : set
Hypothesis H0 : SNo x
Hypothesis H2 : SNo (x * y)
Hypothesis H3 : SNo (- (x * y))
Hypothesis H4 : SNo (SNoCut (Repl ω (ap z)) (Repl ω (ap w)))
Hypothesis H5 : (∀w2 : set, w2 ∈ ω → SNoCut (Repl ω (ap z)) (Repl ω (ap w)) < ap w w2)
Hypothesis H6 : (∀w2 : set, w2 ∈ ω → (ap w w2 + - (eps_ w2)) < x * y)
Hypothesis H7 : u = v * y + x * x2 + - (v * x2)
Hypothesis H8 : y2 ∈ ω
Hypothesis H9 : eps_ y2 ≤ v + - x
Hypothesis H10 : z2 ∈ ω
Hypothesis H11 : eps_ z2 ≤ y + - x2
Hypothesis H12 : SNo v
Hypothesis H13 : SNo x2
Hypothesis H14 : SNo (eps_ y2)
Hypothesis H15 : SNo (eps_ z2)
Hypothesis H16 : y2 + z2 ∈ ω
Hypothesis H17 : SNo (eps_ (y2 + z2))
Hypothesis H18 : SNo (- (eps_ (y2 + z2)))
Hypothesis H19 : SNo (ap w (y2 + z2))
Hypothesis H20 : SNo (v * y)
Hypothesis H21 : SNo (x * x2)
Hypothesis H22 : SNo (v * x2)
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
Variable x : set
Variable y : set
Variable z : set
Variable w : set
Variable u : set
Variable v : set
Variable x2 : set
Variable y2 : set
Variable z2 : set
Hypothesis H0 : SNo x
Hypothesis H1 : SNo y
Hypothesis H2 : SNo (x * y)
Hypothesis H4 : SNo (SNoCut (Repl ω (ap z)) (Repl ω (ap w)))
Hypothesis H5 : (∀w2 : set, w2 ∈ ω → SNoCut (Repl ω (ap z)) (Repl ω (ap w)) < ap w w2)
Hypothesis H6 : (∀w2 : set, w2 ∈ ω → (ap w w2 + - (eps_ w2)) < x * y)
Hypothesis H7 : u = v * y + x * x2 + - (v * x2)
Hypothesis H8 : y2 ∈ ω
Hypothesis H9 : eps_ y2 ≤ v + - x
Hypothesis H10 : z2 ∈ ω
Hypothesis H11 : eps_ z2 ≤ y + - x2
Hypothesis H12 : SNo v
Hypothesis H13 : SNo x2
Hypothesis H14 : SNo (eps_ y2)
Hypothesis H15 : SNo (eps_ z2)
Hypothesis H16 : y2 + z2 ∈ ω
Hypothesis H17 : SNo (eps_ (y2 + z2))
Hypothesis H18 : SNo (- (eps_ (y2 + z2)))
Hypothesis H19 : SNo (ap w (y2 + z2))
Hypothesis H20 : SNo (v * y)
Hypothesis H21 : SNo (x * x2)
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
Variable x : set
Variable y : set
Variable z : set
Variable w : set
Variable u : set
Variable v : set
Variable x2 : set
Variable y2 : set
Variable z2 : set
Hypothesis H0 : SNo x
Hypothesis H1 : SNo y
Hypothesis H2 : SNo (x * y)
Hypothesis H3 : SNo (- (x * y))
Hypothesis H4 : SNo (SNoCut (Repl ω (ap z)) (Repl ω (ap w)))
Hypothesis H5 : (∀w2 : set, w2 ∈ ω → SNoCut (Repl ω (ap z)) (Repl ω (ap w)) < ap w w2)
Hypothesis H6 : (∀w2 : set, w2 ∈ ω → (ap w w2 + - (eps_ w2)) < x * y)
Hypothesis H7 : u = v * y + x * x2 + - (v * x2)
Hypothesis H8 : y2 ∈ ω
Hypothesis H9 : eps_ y2 ≤ v + - x
Hypothesis H10 : z2 ∈ ω
Hypothesis H11 : eps_ z2 ≤ y + - x2
Hypothesis H12 : SNo v
Hypothesis H13 : SNo x2
Hypothesis H14 : SNo (eps_ y2)
Hypothesis H15 : SNo (eps_ z2)
Hypothesis H16 : y2 + z2 ∈ ω
Hypothesis H17 : SNo (eps_ (y2 + z2))
Hypothesis H19 : SNo (ap w (y2 + z2))
Hypothesis H20 : SNo (v * y)
Hypothesis H21 : SNo (x * x2)
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
Variable x : set
Variable y : set
Variable z : set
Variable w : set
Variable u : set
Variable v : set
Variable x2 : set
Variable y2 : set
Variable z2 : set
Hypothesis H0 : SNo x
Hypothesis H1 : SNo y
Hypothesis H2 : SNo (x * y)
Hypothesis H3 : SNo (- (x * y))
Hypothesis H4 : SNo (SNoCut (Repl ω (ap z)) (Repl ω (ap w)))
Hypothesis H5 : (∀w2 : set, w2 ∈ ω → SNoCut (Repl ω (ap z)) (Repl ω (ap w)) < ap w w2)
Hypothesis H6 : (∀w2 : set, w2 ∈ ω → (ap w w2 + - (eps_ w2)) < x * y)
Hypothesis H7 : u = v * y + x * x2 + - (v * x2)
Hypothesis H8 : y2 ∈ ω
Hypothesis H10 : z2 ∈ ω
Hypothesis H11 : eps_ z2 ≤ y + - x2
Hypothesis H12 : SNo v
Hypothesis H13 : SNo x2
Hypothesis H14 : SNo (eps_ y2)
Hypothesis H15 : SNo (eps_ z2)
Hypothesis H16 : y2 + z2 ∈ ω
Hypothesis H17 : SNo (eps_ (y2 + z2))
Hypothesis H18 : SNo (- (eps_ (y2 + z2)))
Hypothesis H19 : SNo (ap w (y2 + z2))
Hypothesis H20 : SNo (v * y)
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
Variable x : set
Variable y : set
Variable z : set
Variable w : set
Variable u : set
Variable v : set
Variable x2 : set
Variable y2 : set
Variable z2 : set
Hypothesis H0 : SNo x
Hypothesis H1 : SNo y
Hypothesis H2 : SNo (x * y)
Hypothesis H3 : SNo (- (x * y))
Hypothesis H4 : SNo (SNoCut (Repl ω (ap z)) (Repl ω (ap w)))
Hypothesis H5 : (∀w2 : set, w2 ∈ ω → SNoCut (Repl ω (ap z)) (Repl ω (ap w)) < ap w w2)
Hypothesis H6 : (∀w2 : set, w2 ∈ ω → (ap w w2 + - (eps_ w2)) < x * y)
Hypothesis H7 : u = v * y + x * x2 + - (v * x2)
Hypothesis H8 : y2 ∈ ω
Hypothesis H9 : eps_ y2 ≤ v + - x
Hypothesis H10 : z2 ∈ ω
Hypothesis H11 : eps_ z2 ≤ y + - x2
Hypothesis H12 : SNo v
Hypothesis H14 : SNo (eps_ y2)
Hypothesis H15 : SNo (eps_ z2)
Hypothesis H16 : y2 + z2 ∈ ω
Hypothesis H17 : SNo (eps_ (y2 + z2))
Hypothesis H18 : SNo (- (eps_ (y2 + z2)))
Hypothesis H19 : SNo (ap w (y2 + z2))
Hypothesis H20 : SNo (v * y)
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
Variable x : set
Variable y : set
Variable z : set
Variable w : set
Variable u : set
Variable v : set
Variable x2 : set
Variable y2 : set
Variable z2 : set
Hypothesis H0 : SNo x
Hypothesis H1 : SNo y
Hypothesis H2 : SNo (x * y)
Hypothesis H3 : SNo (- (x * y))
Hypothesis H4 : SNo (SNoCut (Repl ω (ap z)) (Repl ω (ap w)))
Hypothesis H5 : (∀w2 : set, w2 ∈ ω → SNoCut (Repl ω (ap z)) (Repl ω (ap w)) < ap w w2)
Hypothesis H6 : (∀w2 : set, w2 ∈ ω → (ap w w2 + - (eps_ w2)) < x * y)
Hypothesis H7 : u = v * y + x * x2 + - (v * x2)
Hypothesis H8 : y2 ∈ ω
Hypothesis H9 : eps_ y2 ≤ v + - x
Hypothesis H10 : z2 ∈ ω
Hypothesis H11 : eps_ z2 ≤ y + - x2
Hypothesis H12 : SNo v
Hypothesis H13 : SNo x2
Hypothesis H14 : SNo (eps_ y2)
Hypothesis H15 : SNo (eps_ z2)
Hypothesis H17 : SNo (eps_ (y2 + z2))
Hypothesis H18 : SNo (- (eps_ (y2 + z2)))
Hypothesis H19 : SNo (ap w (y2 + z2))
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
Variable x : set
Variable y : set
Variable z : set
Variable w : set
Variable u : set
Variable v : set
Variable x2 : set
Variable y2 : set
Variable z2 : set
Hypothesis H0 : SNo x
Hypothesis H1 : SNo y
Hypothesis H2 : SNo (x * y)
Hypothesis H3 : SNo (- (x * y))
Hypothesis H4 : (∀w2 : set, w2 ∈ ω → SNo (ap w w2))
Hypothesis H5 : SNo (SNoCut (Repl ω (ap z)) (Repl ω (ap w)))
Hypothesis H6 : (∀w2 : set, w2 ∈ ω → SNoCut (Repl ω (ap z)) (Repl ω (ap w)) < ap w w2)
Hypothesis H7 : (∀w2 : set, w2 ∈ ω → (ap w w2 + - (eps_ w2)) < x * y)
Hypothesis H8 : u = v * y + x * x2 + - (v * x2)
Hypothesis H9 : y2 ∈ ω
Hypothesis H10 : eps_ y2 ≤ v + - x
Hypothesis H11 : z2 ∈ ω
Hypothesis H12 : eps_ z2 ≤ y + - x2
Hypothesis H13 : SNo v
Hypothesis H14 : SNo x2
Hypothesis H15 : SNo (eps_ y2)
Hypothesis H17 : y2 + z2 ∈ ω
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
Variable x : set
Variable y : set
Variable z : set
Variable w : set
Variable u : set
Variable v : set
Variable x2 : set
Variable y2 : set
Variable z2 : set
Hypothesis H0 : SNo x
Hypothesis H1 : SNo y
Hypothesis H2 : SNo (x * y)
Hypothesis H3 : SNo (- (x * y))
Hypothesis H4 : (∀w2 : set, w2 ∈ ω → SNo (ap w w2))
Hypothesis H5 : SNo (SNoCut (Repl ω (ap z)) (Repl ω (ap w)))
Hypothesis H6 : (∀w2 : set, w2 ∈ ω → SNoCut (Repl ω (ap z)) (Repl ω (ap w)) < ap w w2)
Hypothesis H7 : (∀w2 : set, w2 ∈ ω → (ap w w2 + - (eps_ w2)) < x * y)
Hypothesis H8 : u = v * y + x * x2 + - (v * x2)
Hypothesis H9 : y2 ∈ ω
Hypothesis H11 : z2 ∈ ω
Hypothesis H12 : eps_ z2 ≤ y + - x2
Hypothesis H13 : SNo v
Hypothesis H14 : SNo x2
Hypothesis H15 : SNo (eps_ y2)
Hypothesis H16 : SNo (eps_ z2)
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
Variable x : set
Variable y : set
Variable z : set
Variable w : set
Variable u : set
Variable v : set
Variable x2 : set
Variable y2 : set
Variable z2 : set
Hypothesis H0 : SNo x
Hypothesis H1 : SNo y
Hypothesis H3 : SNo (- (x * y))
Hypothesis H4 : SNo (SNoCut (Repl ω (ap z)) (Repl ω (ap w)))
Hypothesis H5 : (∀w2 : set, w2 ∈ ω → SNoCut (Repl ω (ap z)) (Repl ω (ap w)) < ap w w2)
Hypothesis H6 : (∀w2 : set, w2 ∈ ω → (ap w w2 + - (eps_ w2)) < x * y)
Hypothesis H7 : u = v * y + x * x2 + - (v * x2)
Hypothesis H8 : y2 ∈ ω
Hypothesis H9 : eps_ y2 ≤ x + - v
Hypothesis H10 : z2 ∈ ω
Hypothesis H11 : eps_ z2 ≤ x2 + - y
Hypothesis H12 : SNo v
Hypothesis H13 : SNo x2
Hypothesis H14 : SNo (eps_ y2)
Hypothesis H15 : SNo (eps_ z2)
Hypothesis H16 : y2 + z2 ∈ ω
Hypothesis H17 : SNo (eps_ (y2 + z2))
Hypothesis H18 : SNo (- (eps_ (y2 + z2)))
Hypothesis H19 : SNo (ap w (y2 + z2))
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
Variable x : set
Variable y : set
Variable z : set
Variable w : set
Variable u : set
Variable v : set
Variable x2 : set
Variable y2 : set
Variable z2 : set
Hypothesis H0 : SNo x
Hypothesis H1 : SNo y
Hypothesis H2 : SNo (x * y)
Hypothesis H3 : SNo (- (x * y))
Hypothesis H4 : (∀w2 : set, w2 ∈ ω → SNo (ap w w2))
Hypothesis H5 : SNo (SNoCut (Repl ω (ap z)) (Repl ω (ap w)))
Hypothesis H7 : (∀w2 : set, w2 ∈ ω → (ap w w2 + - (eps_ w2)) < x * y)
Hypothesis H8 : u = v * y + x * x2 + - (v * x2)
Hypothesis H9 : y2 ∈ ω
Hypothesis H10 : eps_ y2 ≤ x + - v
Hypothesis H11 : z2 ∈ ω
Hypothesis H12 : eps_ z2 ≤ x2 + - y
Hypothesis H13 : SNo v
Hypothesis H14 : SNo x2
Hypothesis H15 : SNo (eps_ y2)
Hypothesis H16 : SNo (eps_ z2)
Hypothesis H17 : y2 + z2 ∈ ω
Hypothesis H18 : SNo (eps_ (y2 + z2))
Hypothesis H19 : SNo (- (eps_ (y2 + z2)))
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
Variable x : set
Variable y : set
Variable z : set
Variable w : set
Variable u : set
Variable v : set
Variable x2 : set
Variable y2 : set
Variable z2 : set
Hypothesis H0 : SNo x
Hypothesis H1 : SNo y
Hypothesis H2 : SNo (x * y)
Hypothesis H3 : SNo (- (x * y))
Hypothesis H4 : (∀w2 : set, w2 ∈ ω → SNo (ap w w2))
Hypothesis H5 : SNo (SNoCut (Repl ω (ap z)) (Repl ω (ap w)))
Hypothesis H6 : (∀w2 : set, w2 ∈ ω → SNoCut (Repl ω (ap z)) (Repl ω (ap w)) < ap w w2)
Hypothesis H7 : (∀w2 : set, w2 ∈ ω → (ap w w2 + - (eps_ w2)) < x * y)
Hypothesis H8 : u = v * y + x * x2 + - (v * x2)
Hypothesis H9 : y2 ∈ ω
Hypothesis H10 : eps_ y2 ≤ x + - v
Hypothesis H11 : z2 ∈ ω
Hypothesis H12 : eps_ z2 ≤ x2 + - y
Hypothesis H13 : SNo v
Hypothesis H15 : SNo (eps_ y2)
Hypothesis H16 : SNo (eps_ z2)
Hypothesis H17 : y2 + z2 ∈ ω
Hypothesis H18 : SNo (eps_ (y2 + z2))
Hypothesis H19 : SNo (- (eps_ (y2 + z2)))
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
Variable x : set
Variable y : set
Variable z : set
Variable w : set
Variable u : set
Variable v : set
Variable x2 : set
Variable y2 : set
Variable z2 : set
Hypothesis H0 : SNo x
Hypothesis H1 : SNo y
Hypothesis H2 : SNo (x * y)
Hypothesis H3 : SNo (- (x * y))
Hypothesis H4 : (∀w2 : set, w2 ∈ ω → SNo (ap w w2))
Hypothesis H5 : SNo (SNoCut (Repl ω (ap z)) (Repl ω (ap w)))
Hypothesis H6 : (∀w2 : set, w2 ∈ ω → SNoCut (Repl ω (ap z)) (Repl ω (ap w)) < ap w w2)
Hypothesis H7 : (∀w2 : set, w2 ∈ ω → (ap w w2 + - (eps_ w2)) < x * y)
Hypothesis H8 : u = v * y + x * x2 + - (v * x2)
Hypothesis H9 : y2 ∈ ω
Hypothesis H10 : eps_ y2 ≤ x + - v
Hypothesis H11 : z2 ∈ ω
Hypothesis H12 : eps_ z2 ≤ x2 + - y
Hypothesis H13 : SNo v
Hypothesis H14 : SNo x2
Hypothesis H15 : SNo (eps_ y2)
Hypothesis H16 : SNo (eps_ z2)
Hypothesis H17 : y2 + z2 ∈ ω
Hypothesis H18 : SNo (eps_ (y2 + z2))
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
Variable x : set
Variable y : set
Variable z : set
Variable w : set
Variable u : set
Variable v : set
Variable x2 : set
Variable y2 : set
Variable z2 : set
Hypothesis H0 : SNo x
Hypothesis H1 : SNo y
Hypothesis H2 : SNo (x * y)
Hypothesis H3 : SNo (- (x * y))
Hypothesis H5 : SNo (SNoCut (Repl ω (ap z)) (Repl ω (ap w)))
Hypothesis H6 : (∀w2 : set, w2 ∈ ω → SNoCut (Repl ω (ap z)) (Repl ω (ap w)) < ap w w2)
Hypothesis H7 : (∀w2 : set, w2 ∈ ω → (ap w w2 + - (eps_ w2)) < x * y)
Hypothesis H8 : u = v * y + x * x2 + - (v * x2)
Hypothesis H9 : y2 ∈ ω
Hypothesis H10 : eps_ y2 ≤ x + - v
Hypothesis H11 : z2 ∈ ω
Hypothesis H12 : eps_ z2 ≤ x2 + - y
Hypothesis H13 : SNo v
Hypothesis H14 : SNo x2
Hypothesis H15 : SNo (eps_ y2)
Hypothesis H16 : SNo (eps_ z2)
Hypothesis H17 : y2 + z2 ∈ ω
Hypothesis H18 : SNo (eps_ (y2 + z2))
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
Variable x : set
Variable y : set
Variable z : set
Variable w : set
Variable u : set
Variable v : set
Variable x2 : set
Variable y2 : set
Variable z2 : set
Hypothesis H0 : SNo x
Hypothesis H1 : SNo y
Hypothesis H2 : SNo (x * y)
Hypothesis H3 : SNo (- (x * y))
Hypothesis H4 : (∀w2 : set, w2 ∈ ω → SNo (ap w w2))
Hypothesis H5 : SNo (SNoCut (Repl ω (ap z)) (Repl ω (ap w)))
Hypothesis H6 : (∀w2 : set, w2 ∈ ω → SNoCut (Repl ω (ap z)) (Repl ω (ap w)) < ap w w2)
Hypothesis H7 : (∀w2 : set, w2 ∈ ω → (ap w w2 + - (eps_ w2)) < x * y)
Hypothesis H8 : u = v * y + x * x2 + - (v * x2)
Hypothesis H9 : y2 ∈ ω
Hypothesis H10 : eps_ y2 ≤ x + - v
Hypothesis H11 : z2 ∈ ω
Hypothesis H12 : eps_ z2 ≤ x2 + - y
Hypothesis H13 : SNo v
Hypothesis H14 : SNo x2
Hypothesis H15 : SNo (eps_ y2)
Hypothesis H16 : SNo (eps_ z2)
Hypothesis H17 : y2 + z2 ∈ ω
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
Variable x : set
Variable y : set
Variable z : set
Variable w : set
Variable u : set
Variable v : set
Variable x2 : set
Variable y2 : set
Variable z2 : set
Hypothesis H0 : SNo x
Hypothesis H1 : SNo y
Hypothesis H2 : SNo (x * y)
Hypothesis H3 : SNo (- (x * y))
Hypothesis H4 : (∀w2 : set, w2 ∈ ω → SNo (ap w w2))
Hypothesis H5 : SNo (SNoCut (Repl ω (ap z)) (Repl ω (ap w)))
Hypothesis H6 : (∀w2 : set, w2 ∈ ω → SNoCut (Repl ω (ap z)) (Repl ω (ap w)) < ap w w2)
Hypothesis H7 : (∀w2 : set, w2 ∈ ω → (ap w w2 + - (eps_ w2)) < x * y)
Hypothesis H8 : u = v * y + x * x2 + - (v * x2)
Hypothesis H9 : y2 ∈ ω
Hypothesis H10 : eps_ y2 ≤ x + - v
Hypothesis H11 : z2 ∈ ω
Hypothesis H12 : eps_ z2 ≤ x2 + - y
Hypothesis H13 : SNo v
Hypothesis H15 : SNo (eps_ y2)
Hypothesis H16 : SNo (eps_ z2)
Hypothesis H17 : y2 + z2 ∈ ω
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
Variable x : set
Variable y : set
Variable z : set
Variable w : set
Variable u : set
Variable v : set
Variable x2 : set
Variable y2 : set
Variable z2 : set
Hypothesis H0 : SNo x
Hypothesis H1 : SNo y
Hypothesis H2 : SNo (x * y)
Hypothesis H3 : SNo (- (x * y))
Hypothesis H4 : (∀w2 : set, w2 ∈ ω → SNo (ap w w2))
Hypothesis H5 : SNo (SNoCut (Repl ω (ap z)) (Repl ω (ap w)))
Hypothesis H6 : (∀w2 : set, w2 ∈ ω → SNoCut (Repl ω (ap z)) (Repl ω (ap w)) < ap w w2)
Hypothesis H7 : (∀w2 : set, w2 ∈ ω → (ap w w2 + - (eps_ w2)) < x * y)
Hypothesis H8 : u = v * y + x * x2 + - (v * x2)
Hypothesis H9 : y2 ∈ ω
Hypothesis H11 : z2 ∈ ω
Hypothesis H12 : eps_ z2 ≤ x2 + - y
Hypothesis H13 : SNo v
Hypothesis H14 : SNo x2
Hypothesis H15 : SNo (eps_ y2)
Hypothesis H16 : SNo (eps_ z2)
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
Variable x : set
Variable y : set
Variable z : set
Variable w : set
Variable u : set
Variable v : set
Variable x2 : set
Variable y2 : set
Variable z2 : set
Hypothesis H0 : SNo x
Hypothesis H1 : SNo y
Hypothesis H2 : SNo (x * y)
Hypothesis H3 : SNo (- (x * y))
Hypothesis H4 : (∀w2 : set, w2 ∈ ω → SNo (ap w w2))
Hypothesis H5 : SNo (SNoCut (Repl ω (ap z)) (Repl ω (ap w)))
Hypothesis H6 : (∀w2 : set, w2 ∈ ω → SNoCut (Repl ω (ap z)) (Repl ω (ap w)) < ap w w2)
Hypothesis H7 : (∀w2 : set, w2 ∈ ω → (ap w w2 + - (eps_ w2)) < x * y)
Hypothesis H8 : u = v * y + x * x2 + - (v * x2)
Hypothesis H9 : y2 ∈ ω
Hypothesis H10 : eps_ y2 ≤ x + - v
Hypothesis H11 : z2 ∈ ω
Hypothesis H12 : eps_ z2 ≤ x2 + - y
Hypothesis H13 : SNo v
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
Variable x : set
Variable y : set
Variable z : set
Variable w : set
Variable u : set
Variable v : set
Variable x2 : set
Variable y2 : set
Variable z2 : set
Hypothesis H0 : SNo x
Hypothesis H2 : SNo (x * y)
Hypothesis H3 : SNo (SNoCut (Repl ω (ap z)) (Repl ω (ap w)))
Hypothesis H4 : (∀w2 : set, w2 ∈ ω → ap z w2 < SNoCut (Repl ω (ap z)) (Repl ω (ap w)))
Hypothesis H5 : (∀w2 : set, w2 ∈ ω → x * y < ap z w2 + eps_ w2)
Hypothesis H6 : u = v * y + x * x2 + - (v * x2)
Hypothesis H7 : y2 ∈ ω
Hypothesis H8 : eps_ y2 ≤ v + - x
Hypothesis H9 : z2 ∈ ω
Hypothesis H10 : eps_ z2 ≤ x2 + - y
Hypothesis H11 : SNo v
Hypothesis H12 : SNo x2
Hypothesis H13 : SNo (eps_ y2)
Hypothesis H14 : SNo (eps_ z2)
Hypothesis H15 : y2 + z2 ∈ ω
Hypothesis H16 : SNo (eps_ (y2 + z2))
Hypothesis H17 : SNo (ap z (y2 + z2))
Hypothesis H18 : SNo (v * y)
Hypothesis H19 : SNo (x * x2)
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
Variable x : set
Variable y : set
Variable z : set
Variable w : set
Variable u : set
Variable v : set
Variable x2 : set
Variable y2 : set
Variable z2 : set
Hypothesis H0 : SNo x
Hypothesis H1 : SNo y
Hypothesis H2 : SNo (x * y)
Hypothesis H3 : SNo (SNoCut (Repl ω (ap z)) (Repl ω (ap w)))
Hypothesis H4 : (∀w2 : set, w2 ∈ ω → ap z w2 < SNoCut (Repl ω (ap z)) (Repl ω (ap w)))
Hypothesis H5 : (∀w2 : set, w2 ∈ ω → x * y < ap z w2 + eps_ w2)
Hypothesis H6 : u = v * y + x * x2 + - (v * x2)
Hypothesis H7 : y2 ∈ ω
Hypothesis H8 : eps_ y2 ≤ v + - x
Hypothesis H9 : z2 ∈ ω
Hypothesis H10 : eps_ z2 ≤ x2 + - y
Hypothesis H11 : SNo v
Hypothesis H12 : SNo x2
Hypothesis H13 : SNo (eps_ y2)
Hypothesis H14 : SNo (eps_ z2)
Hypothesis H15 : y2 + z2 ∈ ω
Hypothesis H17 : SNo (ap z (y2 + z2))
Hypothesis H18 : SNo (v * y)
Hypothesis H19 : SNo (x * x2)
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
Variable x : set
Variable y : set
Variable z : set
Variable w : set
Variable u : set
Variable v : set
Variable x2 : set
Variable y2 : set
Variable z2 : set
Hypothesis H0 : SNo x
Hypothesis H1 : SNo y
Hypothesis H2 : SNo (x * y)
Hypothesis H3 : SNo (SNoCut (Repl ω (ap z)) (Repl ω (ap w)))
Hypothesis H4 : (∀w2 : set, w2 ∈ ω → ap z w2 < SNoCut (Repl ω (ap z)) (Repl ω (ap w)))
Hypothesis H5 : (∀w2 : set, w2 ∈ ω → x * y < ap z w2 + eps_ w2)
Hypothesis H7 : y2 ∈ ω
Hypothesis H8 : eps_ y2 ≤ v + - x
Hypothesis H9 : z2 ∈ ω
Hypothesis H10 : eps_ z2 ≤ x2 + - y
Hypothesis H11 : SNo v
Hypothesis H12 : SNo x2
Hypothesis H13 : SNo (eps_ y2)
Hypothesis H14 : SNo (eps_ z2)
Hypothesis H15 : y2 + z2 ∈ ω
Hypothesis H16 : SNo (eps_ (y2 + z2))
Hypothesis H17 : SNo (ap z (y2 + z2))
Hypothesis H18 : SNo (v * y)
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
Variable x : set
Variable y : set
Variable z : set
Variable w : set
Variable u : set
Variable v : set
Variable x2 : set
Variable y2 : set
Variable z2 : set
Hypothesis H0 : SNo x
Hypothesis H1 : SNo y
Hypothesis H2 : SNo (x * y)
Hypothesis H3 : SNo (SNoCut (Repl ω (ap z)) (Repl ω (ap w)))
Hypothesis H4 : (∀w2 : set, w2 ∈ ω → ap z w2 < SNoCut (Repl ω (ap z)) (Repl ω (ap w)))
Hypothesis H5 : (∀w2 : set, w2 ∈ ω → x * y < ap z w2 + eps_ w2)
Hypothesis H6 : u = v * y + x * x2 + - (v * x2)
Hypothesis H8 : eps_ y2 ≤ v + - x
Hypothesis H9 : z2 ∈ ω
Hypothesis H10 : eps_ z2 ≤ x2 + - y
Hypothesis H11 : SNo v
Hypothesis H12 : SNo x2
Hypothesis H13 : SNo (eps_ y2)
Hypothesis H14 : SNo (eps_ z2)
Hypothesis H15 : y2 + z2 ∈ ω
Hypothesis H16 : SNo (eps_ (y2 + z2))
Hypothesis H17 : SNo (ap z (y2 + z2))
Hypothesis H18 : SNo (v * y)
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
Variable x : set
Variable y : set
Variable z : set
Variable w : set
Variable u : set
Variable v : set
Variable x2 : set
Variable y2 : set
Variable z2 : set
Hypothesis H0 : SNo x
Hypothesis H1 : SNo y
Hypothesis H2 : SNo (x * y)
Hypothesis H3 : SNo (SNoCut (Repl ω (ap z)) (Repl ω (ap w)))
Hypothesis H4 : (∀w2 : set, w2 ∈ ω → ap z w2 < SNoCut (Repl ω (ap z)) (Repl ω (ap w)))
Hypothesis H5 : (∀w2 : set, w2 ∈ ω → x * y < ap z w2 + eps_ w2)
Hypothesis H6 : u = v * y + x * x2 + - (v * x2)
Hypothesis H7 : y2 ∈ ω
Hypothesis H8 : eps_ y2 ≤ v + - x
Hypothesis H9 : z2 ∈ ω
Hypothesis H10 : eps_ z2 ≤ x2 + - y
Hypothesis H11 : SNo v
Hypothesis H12 : SNo x2
Hypothesis H13 : SNo (eps_ y2)
Hypothesis H15 : y2 + z2 ∈ ω
Hypothesis H16 : SNo (eps_ (y2 + z2))
Hypothesis H17 : SNo (ap z (y2 + z2))
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
Variable x : set
Variable y : set
Variable z : set
Variable w : set
Variable u : set
Variable v : set
Variable x2 : set
Variable y2 : set
Variable z2 : set
Hypothesis H0 : SNo x
Hypothesis H1 : SNo y
Hypothesis H2 : SNo (x * y)
Hypothesis H3 : (∀w2 : set, w2 ∈ ω → SNo (ap z w2))
Hypothesis H4 : SNo (SNoCut (Repl ω (ap z)) (Repl ω (ap w)))
Hypothesis H5 : (∀w2 : set, w2 ∈ ω → ap z w2 < SNoCut (Repl ω (ap z)) (Repl ω (ap w)))
Hypothesis H6 : (∀w2 : set, w2 ∈ ω → x * y < ap z w2 + eps_ w2)
Hypothesis H7 : u = v * y + x * x2 + - (v * x2)
Hypothesis H8 : y2 ∈ ω
Hypothesis H10 : z2 ∈ ω
Hypothesis H11 : eps_ z2 ≤ x2 + - y
Hypothesis H12 : SNo v
Hypothesis H13 : SNo x2
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
Variable x : set
Variable y : set
Variable z : set
Variable w : set
Variable u : set
Variable v : set
Variable x2 : set
Variable y2 : set
Variable z2 : set
Hypothesis H0 : SNo x
Hypothesis H1 : SNo y
Hypothesis H2 : SNo (x * y)
Hypothesis H3 : (∀w2 : set, w2 ∈ ω → SNo (ap z w2))
Hypothesis H4 : SNo (SNoCut (Repl ω (ap z)) (Repl ω (ap w)))
Hypothesis H5 : (∀w2 : set, w2 ∈ ω → ap z w2 < SNoCut (Repl ω (ap z)) (Repl ω (ap w)))
Hypothesis H6 : (∀w2 : set, w2 ∈ ω → x * y < ap z w2 + eps_ w2)
Hypothesis H7 : u = v * y + x * x2 + - (v * x2)
Hypothesis H8 : y2 ∈ ω
Hypothesis H9 : eps_ y2 ≤ v + - x
Hypothesis H10 : z2 ∈ ω
Hypothesis H11 : eps_ z2 ≤ x2 + - y
Hypothesis H12 : SNo v
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
Variable x : set
Variable y : set
Variable z : set
Variable w : set
Variable u : set
Variable v : set
Variable x2 : set
Variable y2 : set
Variable z2 : set
Hypothesis H0 : SNo x
Hypothesis H2 : SNo (x * y)
Hypothesis H3 : SNo (SNoCut (Repl ω (ap z)) (Repl ω (ap w)))
Hypothesis H4 : (∀w2 : set, w2 ∈ ω → ap z w2 < SNoCut (Repl ω (ap z)) (Repl ω (ap w)))
Hypothesis H5 : (∀w2 : set, w2 ∈ ω → x * y < ap z w2 + eps_ w2)
Hypothesis H6 : u = v * y + x * x2 + - (v * x2)
Hypothesis H7 : y2 ∈ ω
Hypothesis H8 : eps_ y2 ≤ x + - v
Hypothesis H9 : z2 ∈ ω
Hypothesis H10 : eps_ z2 ≤ y + - x2
Hypothesis H11 : SNo v
Hypothesis H12 : SNo x2
Hypothesis H13 : SNo (eps_ y2)
Hypothesis H14 : SNo (eps_ z2)
Hypothesis H15 : y2 + z2 ∈ ω
Hypothesis H16 : SNo (eps_ (y2 + z2))
Hypothesis H17 : SNo (ap z (y2 + z2))
Hypothesis H18 : SNo (v * y)
Hypothesis H19 : SNo (x * x2)
Hypothesis H20 : SNo (- (v * x2))
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
Variable x : set
Variable y : set
Variable z : set
Variable w : set
Variable u : set
Variable v : set
Variable x2 : set
Variable y2 : set
Variable z2 : set
Hypothesis H0 : SNo x
Hypothesis H1 : SNo y
Hypothesis H2 : SNo (x * y)
Hypothesis H3 : SNo (SNoCut (Repl ω (ap z)) (Repl ω (ap w)))
Hypothesis H4 : (∀w2 : set, w2 ∈ ω → ap z w2 < SNoCut (Repl ω (ap z)) (Repl ω (ap w)))
Hypothesis H5 : (∀w2 : set, w2 ∈ ω → x * y < ap z w2 + eps_ w2)
Hypothesis H6 : u = v * y + x * x2 + - (v * x2)
Hypothesis H7 : y2 ∈ ω
Hypothesis H8 : eps_ y2 ≤ x + - v
Hypothesis H9 : z2 ∈ ω
Hypothesis H10 : eps_ z2 ≤ y + - x2
Hypothesis H11 : SNo v
Hypothesis H12 : SNo x2
Hypothesis H13 : SNo (eps_ y2)
Hypothesis H14 : SNo (eps_ z2)
Hypothesis H15 : y2 + z2 ∈ ω
Hypothesis H16 : SNo (eps_ (y2 + z2))
Hypothesis H17 : SNo (ap z (y2 + z2))
Hypothesis H18 : SNo (v * y)
Hypothesis H19 : SNo (x * x2)
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
Variable x : set
Variable y : set
Variable z : set
Variable w : set
Variable u : set
Variable v : set
Variable x2 : set
Variable y2 : set
Variable z2 : set
Hypothesis H0 : SNo x
Hypothesis H1 : SNo y
Hypothesis H2 : SNo (x * y)
Hypothesis H3 : SNo (SNoCut (Repl ω (ap z)) (Repl ω (ap w)))
Hypothesis H5 : (∀w2 : set, w2 ∈ ω → x * y < ap z w2 + eps_ w2)
Hypothesis H6 : u = v * y + x * x2 + - (v * x2)
Hypothesis H7 : y2 ∈ ω
Hypothesis H8 : eps_ y2 ≤ x + - v
Hypothesis H9 : z2 ∈ ω
Hypothesis H10 : eps_ z2 ≤ y + - x2
Hypothesis H11 : SNo v
Hypothesis H12 : SNo x2
Hypothesis H13 : SNo (eps_ y2)
Hypothesis H14 : SNo (eps_ z2)
Hypothesis H15 : y2 + z2 ∈ ω
Hypothesis H16 : SNo (eps_ (y2 + z2))
Hypothesis H17 : SNo (ap z (y2 + z2))
Hypothesis H18 : SNo (v * y)
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
Variable x : set
Variable y : set
Variable z : set
Variable w : set
Variable u : set
Variable v : set
Variable x2 : set
Variable y2 : set
Variable z2 : set
Hypothesis H0 : SNo x
Hypothesis H1 : SNo y
Hypothesis H2 : SNo (x * y)
Hypothesis H3 : SNo (SNoCut (Repl ω (ap z)) (Repl ω (ap w)))
Hypothesis H4 : (∀w2 : set, w2 ∈ ω → ap z w2 < SNoCut (Repl ω (ap z)) (Repl ω (ap w)))
Hypothesis H5 : (∀w2 : set, w2 ∈ ω → x * y < ap z w2 + eps_ w2)
Hypothesis H7 : y2 ∈ ω
Hypothesis H8 : eps_ y2 ≤ x + - v
Hypothesis H9 : z2 ∈ ω
Hypothesis H10 : eps_ z2 ≤ y + - x2
Hypothesis H11 : SNo v
Hypothesis H12 : SNo x2
Hypothesis H13 : SNo (eps_ y2)
Hypothesis H14 : SNo (eps_ z2)
Hypothesis H15 : y2 + z2 ∈ ω
Hypothesis H16 : SNo (eps_ (y2 + z2))
Hypothesis H17 : SNo (ap z (y2 + z2))
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
Variable x : set
Variable y : set
Variable z : set
Variable w : set
Variable u : set
Variable v : set
Variable x2 : set
Variable y2 : set
Variable z2 : set
Hypothesis H0 : SNo x
Hypothesis H1 : SNo y
Hypothesis H2 : SNo (x * y)
Hypothesis H3 : (∀w2 : set, w2 ∈ ω → SNo (ap z w2))
Hypothesis H4 : SNo (SNoCut (Repl ω (ap z)) (Repl ω (ap w)))
Hypothesis H5 : (∀w2 : set, w2 ∈ ω → ap z w2 < SNoCut (Repl ω (ap z)) (Repl ω (ap w)))
Hypothesis H6 : (∀w2 : set, w2 ∈ ω → x * y < ap z w2 + eps_ w2)
Hypothesis H7 : u = v * y + x * x2 + - (v * x2)
Hypothesis H8 : y2 ∈ ω
Hypothesis H9 : eps_ y2 ≤ x + - v
Hypothesis H10 : z2 ∈ ω
Hypothesis H11 : eps_ z2 ≤ y + - x2
Hypothesis H12 : SNo v
Hypothesis H13 : SNo x2
Hypothesis H14 : SNo (eps_ y2)
Hypothesis H15 : SNo (eps_ z2)
Hypothesis H16 : y2 + z2 ∈ ω
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
Variable x : set
Variable y : set
Variable z : set
Variable w : set
Variable u : set
Variable v : set
Variable x2 : set
Variable y2 : set
Variable z2 : set
Hypothesis H0 : SNo x
Hypothesis H1 : SNo y
Hypothesis H2 : SNo (x * y)
Hypothesis H3 : (∀w2 : set, w2 ∈ ω → SNo (ap z w2))
Hypothesis H4 : SNo (SNoCut (Repl ω (ap z)) (Repl ω (ap w)))
Hypothesis H5 : (∀w2 : set, w2 ∈ ω → ap z w2 < SNoCut (Repl ω (ap z)) (Repl ω (ap w)))
Hypothesis H6 : (∀w2 : set, w2 ∈ ω → x * y < ap z w2 + eps_ w2)
Hypothesis H7 : u = v * y + x * x2 + - (v * x2)
Hypothesis H8 : y2 ∈ ω
Hypothesis H9 : eps_ y2 ≤ x + - v
Hypothesis H10 : z2 ∈ ω
Hypothesis H11 : eps_ z2 ≤ y + - x2
Hypothesis H12 : SNo v
Hypothesis H13 : SNo x2
Hypothesis H14 : SNo (eps_ y2)
Hypothesis H15 : SNo (eps_ z2)
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
Variable x : set
Variable y : set
Variable z : set
Variable w : set
Variable u : set
Variable v : set
Variable x2 : set
Variable y2 : set
Variable z2 : set
Hypothesis H0 : SNo x
Hypothesis H1 : SNo y
Hypothesis H2 : SNo (x * y)
Hypothesis H3 : (∀w2 : set, w2 ∈ ω → SNo (ap z w2))
Hypothesis H4 : SNo (SNoCut (Repl ω (ap z)) (Repl ω (ap w)))
Hypothesis H5 : (∀w2 : set, w2 ∈ ω → ap z w2 < SNoCut (Repl ω (ap z)) (Repl ω (ap w)))
Hypothesis H7 : u = v * y + x * x2 + - (v * x2)
Hypothesis H8 : y2 ∈ ω
Hypothesis H9 : eps_ y2 ≤ x + - v
Hypothesis H10 : z2 ∈ ω
Hypothesis H11 : eps_ z2 ≤ y + - x2
Hypothesis H12 : SNo v
Hypothesis H13 : SNo x2
Hypothesis H14 : SNo (eps_ y2)
Hypothesis H15 : SNo (eps_ z2)
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
Variable x : set
Variable y : set
Variable z : set
Variable w : set
Variable u : set
Variable v : set
Variable x2 : set
Variable y2 : set
Variable z2 : set
Hypothesis H0 : SNo x
Hypothesis H1 : SNo y
Hypothesis H2 : SNo (x * y)
Hypothesis H3 : (∀w2 : set, w2 ∈ ω → SNo (ap z w2))
Hypothesis H4 : SNo (SNoCut (Repl ω (ap z)) (Repl ω (ap w)))
Hypothesis H5 : (∀w2 : set, w2 ∈ ω → ap z w2 < SNoCut (Repl ω (ap z)) (Repl ω (ap w)))
Hypothesis H6 : (∀w2 : set, w2 ∈ ω → x * y < ap z w2 + eps_ w2)
Hypothesis H7 : u = v * y + x * x2 + - (v * x2)
Hypothesis H8 : y2 ∈ ω
Hypothesis H9 : eps_ y2 ≤ x + - v
Hypothesis H10 : z2 ∈ ω
Hypothesis H12 : SNo v
Hypothesis H13 : SNo x2
Hypothesis H14 : SNo (eps_ y2)
Hypothesis H15 : SNo (eps_ z2)
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
Variable x : set
Variable y : set
Variable z : set
Variable w : set
Variable u : set
Variable v : set
Variable x2 : set
Variable y2 : set
Variable z2 : set
Hypothesis H0 : SNo x
Hypothesis H1 : SNo y
Hypothesis H2 : SNo (x * y)
Hypothesis H4 : SNo (SNoCut (Repl ω (ap z)) (Repl ω (ap w)))
Hypothesis H5 : (∀w2 : set, w2 ∈ ω → ap z w2 < SNoCut (Repl ω (ap z)) (Repl ω (ap w)))
Hypothesis H6 : (∀w2 : set, w2 ∈ ω → x * y < ap z w2 + eps_ w2)
Hypothesis H7 : u = v * y + x * x2 + - (v * x2)
Hypothesis H8 : y2 ∈ ω
Hypothesis H9 : eps_ y2 ≤ x + - v
Hypothesis H10 : z2 ∈ ω
Hypothesis H11 : eps_ z2 ≤ y + - x2
Hypothesis H12 : SNo v
Hypothesis H13 : SNo x2
Hypothesis H14 : SNo (eps_ y2)
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
Variable x : set
Variable y : set
Variable z : set
Variable w : set
Variable u : set
Variable v : set
Hypothesis H0 : ¬ x * y ∈ real
Hypothesis H1 : SNo x
Hypothesis H2 : SNo y
Hypothesis H3 : SNo (x * y)
Hypothesis H4 : SNo (- (x * y))
Hypothesis H5 : (∀x2 : set, x2 ∈ SNoL x → (∀P : prop, (∀y2 : set, y2 ∈ ω → eps_ y2 ≤ x + - x2 → P) → P))
Hypothesis H6 : (∀x2 : set, x2 ∈ SNoR x → (∀P : prop, (∀y2 : set, y2 ∈ ω → eps_ y2 ≤ x2 + - x → P) → P))
Hypothesis H7 : (∀x2 : set, x2 ∈ SNoL y → (∀P : prop, (∀y2 : set, y2 ∈ ω → eps_ y2 ≤ y + - x2 → P) → P))
Hypothesis H8 : (∀x2 : set, x2 ∈ SNoR y → (∀P : prop, (∀y2 : set, y2 ∈ ω → eps_ y2 ≤ x2 + - y → P) → P))
Hypothesis H9 : SNoCutP z w
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))
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))
Hypothesis H12 : x * y = SNoCut z w
Hypothesis H13 : u ∈ setexp (SNoS_ ω) ω
Hypothesis H14 : v ∈ setexp (SNoS_ ω) ω
Hypothesis H15 : (∀x2 : set, x2 ∈ ω → SNo (ap u x2))
Hypothesis H16 : (∀x2 : set, x2 ∈ ω → SNo (ap v x2))
Hypothesis H17 : (∀x2 : set, x2 ∈ ω → ap u x2 < x * y)
Hypothesis H18 : (∀x2 : set, x2 ∈ ω → x * y < ap v x2)
Hypothesis H19 : SNoCutP (Repl ω (ap u)) (Repl ω (ap v))
Hypothesis H20 : SNo (SNoCut (Repl ω (ap u)) (Repl ω (ap v)))
Hypothesis H21 : (∀x2 : set, x2 ∈ ω → ap u x2 < SNoCut (Repl ω (ap u)) (Repl ω (ap v)))
Hypothesis H22 : (∀x2 : set, x2 ∈ ω → SNoCut (Repl ω (ap u)) (Repl ω (ap v)) < ap v x2)
Hypothesis H24 : (∀x2 : set, x2 ∈ ω → (∀y2 : set, y2 ∈ x2 → ap u y2 < ap u x2))
Hypothesis H25 : (∀x2 : set, x2 ∈ ω → (ap v x2 + - (eps_ x2)) < x * y)
Hypothesis H26 : (∀x2 : set, x2 ∈ ω → (∀y2 : set, y2 ∈ x2 → ap v x2 < ap v y2))
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
Variable x : set
Variable y : set
Variable z : set
Variable w : set
Variable u : set
Variable v : set
Hypothesis H0 : ¬ x * y ∈ real
Hypothesis H1 : SNo x
Hypothesis H2 : SNo y
Hypothesis H3 : SNo (x * y)
Hypothesis H4 : SNo (- (x * y))
Hypothesis H5 : (∀x2 : set, x2 ∈ SNoL x → (∀P : prop, (∀y2 : set, y2 ∈ ω → eps_ y2 ≤ x + - x2 → P) → P))
Hypothesis H6 : (∀x2 : set, x2 ∈ SNoR x → (∀P : prop, (∀y2 : set, y2 ∈ ω → eps_ y2 ≤ x2 + - x → P) → P))
Hypothesis H7 : (∀x2 : set, x2 ∈ SNoL y → (∀P : prop, (∀y2 : set, y2 ∈ ω → eps_ y2 ≤ y + - x2 → P) → P))
Hypothesis H8 : (∀x2 : set, x2 ∈ SNoR y → (∀P : prop, (∀y2 : set, y2 ∈ ω → eps_ y2 ≤ x2 + - y → P) → P))
Hypothesis H9 : SNoCutP z w
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))
Hypothesis H12 : x * y = SNoCut z w
Hypothesis H13 : u ∈ setexp (SNoS_ ω) ω
Hypothesis H14 : v ∈ setexp (SNoS_ ω) ω
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))
Hypothesis H16 : (∀x2 : set, x2 ∈ ω → SNo (ap u x2))
Hypothesis H17 : (∀x2 : set, x2 ∈ ω → SNo (ap v x2))
Hypothesis H18 : (∀x2 : set, x2 ∈ ω → ap u x2 < x * y)
Hypothesis H19 : (∀x2 : set, x2 ∈ ω → x * y < ap v x2)
Hypothesis H20 : SNoCutP (Repl ω (ap u)) (Repl ω (ap v))
Hypothesis H21 : SNo (SNoCut (Repl ω (ap u)) (Repl ω (ap v)))
Hypothesis H22 : (∀x2 : set, x2 ∈ ω → ap u x2 < SNoCut (Repl ω (ap u)) (Repl ω (ap v)))
Hypothesis H23 : (∀x2 : set, x2 ∈ ω → SNoCut (Repl ω (ap u)) (Repl ω (ap v)) < ap v x2)
Hypothesis H24 : (∀x2 : set, x2 ∈ ω → x * y < ap u x2 + eps_ x2)
Hypothesis H25 : (∀x2 : set, x2 ∈ ω → (∀y2 : set, y2 ∈ x2 → ap u y2 < ap u x2))
Hypothesis H26 : (∀x2 : set, x2 ∈ ω → (ap v x2 + - (eps_ x2)) < x * y)
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
Variable x : set
Variable y : set
Variable z : set
Variable w : set
Variable u : set
Variable v : set
Hypothesis H0 : ¬ x * y ∈ real
Hypothesis H1 : SNo x
Hypothesis H2 : SNo y
Hypothesis H3 : SNo (x * y)
Hypothesis H4 : SNo (- (x * y))
Hypothesis H5 : (∀x2 : set, x2 ∈ SNoL x → (∀P : prop, (∀y2 : set, y2 ∈ ω → eps_ y2 ≤ x + - x2 → P) → P))
Hypothesis H6 : (∀x2 : set, x2 ∈ SNoR x → (∀P : prop, (∀y2 : set, y2 ∈ ω → eps_ y2 ≤ x2 + - x → P) → P))
Hypothesis H7 : (∀x2 : set, x2 ∈ SNoL y → (∀P : prop, (∀y2 : set, y2 ∈ ω → eps_ y2 ≤ y + - x2 → P) → P))
Hypothesis H8 : (∀x2 : set, x2 ∈ SNoR y → (∀P : prop, (∀y2 : set, y2 ∈ ω → eps_ y2 ≤ x2 + - y → P) → P))
Hypothesis H9 : SNoCutP z w
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))
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))
Hypothesis H12 : x * y = SNoCut z w
Hypothesis H13 : u ∈ setexp (SNoS_ ω) ω
Hypothesis H14 : v ∈ setexp (SNoS_ ω) ω
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))
Hypothesis H16 : (∀x2 : set, x2 ∈ ω → SNo (ap u x2))
Hypothesis H17 : (∀x2 : set, x2 ∈ ω → SNo (ap v x2))
Hypothesis H18 : (∀x2 : set, x2 ∈ ω → ap u x2 < x * y)
Hypothesis H19 : (∀x2 : set, x2 ∈ ω → x * y < ap v x2)
Hypothesis H20 : SNoCutP (Repl ω (ap u)) (Repl ω (ap v))
Hypothesis H21 : SNo (SNoCut (Repl ω (ap u)) (Repl ω (ap v)))
Hypothesis H23 : (∀x2 : set, x2 ∈ ω → SNoCut (Repl ω (ap u)) (Repl ω (ap v)) < ap v x2)
Hypothesis H24 : (∀x2 : set, x2 ∈ ω → x * y < ap u x2 + eps_ x2)
Hypothesis H25 : (∀x2 : set, x2 ∈ ω → (∀y2 : set, y2 ∈ x2 → ap u y2 < ap u x2))
Hypothesis H26 : (∀x2 : set, x2 ∈ ω → (ap v x2 + - (eps_ x2)) < x * y)
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
Variable x : set
Variable y : set
Variable z : set
Variable w : set
Variable u : set
Variable v : set
Hypothesis H0 : ¬ x * y ∈ real
Hypothesis H1 : SNo x
Hypothesis H2 : SNo y
Hypothesis H3 : SNo (x * y)
Hypothesis H4 : SNo (- (x * y))
Hypothesis H5 : (∀x2 : set, x2 ∈ SNoL x → (∀P : prop, (∀y2 : set, y2 ∈ ω → eps_ y2 ≤ x + - x2 → P) → P))
Hypothesis H7 : (∀x2 : set, x2 ∈ SNoL y → (∀P : prop, (∀y2 : set, y2 ∈ ω → eps_ y2 ≤ y + - x2 → P) → P))
Hypothesis H8 : (∀x2 : set, x2 ∈ SNoR y → (∀P : prop, (∀y2 : set, y2 ∈ ω → eps_ y2 ≤ x2 + - y → P) → P))
Hypothesis H9 : SNoCutP z w
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))
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))
Hypothesis H12 : x * y = SNoCut z w
Hypothesis H13 : u ∈ setexp (SNoS_ ω) ω
Hypothesis H14 : v ∈ setexp (SNoS_ ω) ω
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))
Hypothesis H16 : (∀x2 : set, x2 ∈ ω → SNo (ap u x2))
Hypothesis H17 : (∀x2 : set, x2 ∈ ω → SNo (ap v x2))
Hypothesis H18 : (∀x2 : set, x2 ∈ ω → ap u x2 < x * y)
Hypothesis H19 : (∀x2 : set, x2 ∈ ω → x * y < ap v x2)
Hypothesis H20 : SNoCutP (Repl ω (ap u)) (Repl ω (ap v))
Hypothesis H21 : SNo (SNoCut (Repl ω (ap u)) (Repl ω (ap v)))
Hypothesis H22 : (∀x2 : set, x2 ∈ ω → ap u x2 < SNoCut (Repl ω (ap u)) (Repl ω (ap v)))
Hypothesis H23 : (∀x2 : set, x2 ∈ ω → SNoCut (Repl ω (ap u)) (Repl ω (ap v)) < ap v x2)
Hypothesis H24 : (∀x2 : set, x2 ∈ ω → x * y < ap u x2 + eps_ x2)
Hypothesis H25 : (∀x2 : set, x2 ∈ ω → (∀y2 : set, y2 ∈ x2 → ap u y2 < ap u x2))
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
Variable x : set
Variable y : set
Variable z : set
Variable w : set
Variable u : set
Variable v : set
Hypothesis H0 : ¬ x * y ∈ real
Hypothesis H1 : SNo x
Hypothesis H2 : SNo y
Hypothesis H3 : SNo (x * y)
Hypothesis H4 : SNo (- (x * y))
Hypothesis H5 : (∀x2 : set, x2 ∈ SNoL x → (∀P : prop, (∀y2 : set, y2 ∈ ω → eps_ y2 ≤ x + - x2 → P) → P))
Hypothesis H7 : (∀x2 : set, x2 ∈ SNoL y → (∀P : prop, (∀y2 : set, y2 ∈ ω → eps_ y2 ≤ y + - x2 → P) → P))
Hypothesis H8 : (∀x2 : set, x2 ∈ SNoR y → (∀P : prop, (∀y2 : set, y2 ∈ ω → eps_ y2 ≤ x2 + - y → P) → P))
Hypothesis H9 : SNoCutP z w
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))
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))
Hypothesis H12 : x * y = SNoCut z w
Hypothesis H13 : u ∈ setexp (SNoS_ ω) ω
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))
Hypothesis H15 : v ∈ setexp (SNoS_ ω) ω
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))
Hypothesis H17 : (∀x2 : set, x2 ∈ ω → SNo (ap u x2))
Hypothesis H18 : (∀x2 : set, x2 ∈ ω → SNo (ap v x2))
Hypothesis H19 : (∀x2 : set, x2 ∈ ω → ap u x2 < x * y)
Hypothesis H20 : (∀x2 : set, x2 ∈ ω → x * y < ap v x2)
Hypothesis H21 : SNoCutP (Repl ω (ap u)) (Repl ω (ap v))
Hypothesis H22 : SNo (SNoCut (Repl ω (ap u)) (Repl ω (ap v)))
Hypothesis H23 : (∀x2 : set, x2 ∈ ω → ap u x2 < SNoCut (Repl ω (ap u)) (Repl ω (ap v)))
Hypothesis H24 : (∀x2 : set, x2 ∈ ω → SNoCut (Repl ω (ap u)) (Repl ω (ap v)) < ap v x2)
Hypothesis H25 : (∀x2 : set, x2 ∈ ω → x * y < ap u x2 + eps_ x2)
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
Variable x : set
Variable y : set
Variable z : set
Variable w : set
Variable u : set
Variable v : set
Hypothesis H0 : ¬ x * y ∈ real
Hypothesis H1 : SNo x
Hypothesis H2 : SNo y
Hypothesis H3 : SNo (x * y)
Hypothesis H4 : SNo (- (x * y))
Hypothesis H5 : (∀x2 : set, x2 ∈ SNoL x → (∀P : prop, (∀y2 : set, y2 ∈ ω → eps_ y2 ≤ x + - x2 → P) → P))
Hypothesis H6 : (∀x2 : set, x2 ∈ SNoR x → (∀P : prop, (∀y2 : set, y2 ∈ ω → eps_ y2 ≤ x2 + - x → P) → P))
Hypothesis H7 : (∀x2 : set, x2 ∈ SNoL y → (∀P : prop, (∀y2 : set, y2 ∈ ω → eps_ y2 ≤ y + - x2 → P) → P))
Hypothesis H8 : (∀x2 : set, x2 ∈ SNoR y → (∀P : prop, (∀y2 : set, y2 ∈ ω → eps_ y2 ≤ x2 + - y → P) → P))
Hypothesis H9 : SNoCutP z w
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))
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))
Hypothesis H12 : x * y = SNoCut z w
Hypothesis H13 : u ∈ setexp (SNoS_ ω) ω
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))
Hypothesis H15 : v ∈ setexp (SNoS_ ω) ω
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))
Hypothesis H17 : (∀x2 : set, x2 ∈ ω → SNo (ap u x2))
Hypothesis H18 : (∀x2 : set, x2 ∈ ω → SNo (ap v x2))
Hypothesis H19 : (∀x2 : set, x2 ∈ ω → ap u x2 < x * y)
Hypothesis H20 : (∀x2 : set, x2 ∈ ω → x * y < ap v x2)
Hypothesis H22 : SNo (SNoCut (Repl ω (ap u)) (Repl ω (ap v)))
Hypothesis H23 : (∀x2 : set, x2 ∈ ω → ap u x2 < SNoCut (Repl ω (ap u)) (Repl ω (ap v)))
Hypothesis H24 : (∀x2 : set, x2 ∈ ω → SNoCut (Repl ω (ap u)) (Repl ω (ap v)) < ap v x2)
Hypothesis H25 : (∀x2 : set, x2 ∈ ω → x * y < ap u x2 + eps_ x2)
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
Variable x : set
Variable y : set
Variable z : set
Variable w : set
Variable u : set
Variable v : set
Hypothesis H0 : ¬ x * y ∈ real
Hypothesis H1 : SNo x
Hypothesis H2 : SNo y
Hypothesis H3 : SNo (x * y)
Hypothesis H4 : SNo (- (x * y))
Hypothesis H5 : (∀x2 : set, x2 ∈ SNoL x → (∀P : prop, (∀y2 : set, y2 ∈ ω → eps_ y2 ≤ x + - x2 → P) → P))
Hypothesis H6 : (∀x2 : set, x2 ∈ SNoR x → (∀P : prop, (∀y2 : set, y2 ∈ ω → eps_ y2 ≤ x2 + - x → P) → P))
Hypothesis H7 : (∀x2 : set, x2 ∈ SNoL y → (∀P : prop, (∀y2 : set, y2 ∈ ω → eps_ y2 ≤ y + - x2 → P) → P))
Hypothesis H8 : (∀x2 : set, x2 ∈ SNoR y → (∀P : prop, (∀y2 : set, y2 ∈ ω → eps_ y2 ≤ x2 + - y → P) → P))
Hypothesis H9 : SNoCutP z w
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))
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))
Hypothesis H12 : x * y = SNoCut z w
Hypothesis H13 : u ∈ setexp (SNoS_ ω) ω
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))
Hypothesis H15 : v ∈ setexp (SNoS_ ω) ω
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))
Hypothesis H17 : (∀x2 : set, x2 ∈ ω → SNo (ap u x2))
Hypothesis H18 : (∀x2 : set, x2 ∈ ω → SNo (ap v x2))
Hypothesis H19 : (∀x2 : set, x2 ∈ ω → ap u x2 < x * y)
Hypothesis H20 : (∀x2 : set, x2 ∈ ω → x * y < ap v x2)
Hypothesis H21 : SNoCutP (Repl ω (ap u)) (Repl ω (ap v))
Hypothesis H22 : SNo (SNoCut (Repl ω (ap u)) (Repl ω (ap v)))
Hypothesis H23 : (∀x2 : set, x2 ∈ ω → ap u x2 < SNoCut (Repl ω (ap u)) (Repl ω (ap v)))
Hypothesis H24 : (∀x2 : set, x2 ∈ ω → SNoCut (Repl ω (ap u)) (Repl ω (ap v)) < ap v x2)
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
Variable x : set
Variable y : set
Variable z : set
Variable w : set
Variable u : set
Variable v : set
Hypothesis H0 : ¬ x * y ∈ real
Hypothesis H1 : SNo x
Hypothesis H2 : SNo y
Hypothesis H3 : SNo (x * y)
Hypothesis H4 : SNo (- (x * y))
Hypothesis H5 : (∀x2 : set, x2 ∈ SNoL x → (∀P : prop, (∀y2 : set, y2 ∈ ω → eps_ y2 ≤ x + - x2 → P) → P))
Hypothesis H6 : (∀x2 : set, x2 ∈ SNoR x → (∀P : prop, (∀y2 : set, y2 ∈ ω → eps_ y2 ≤ x2 + - x → P) → P))
Hypothesis H7 : (∀x2 : set, x2 ∈ SNoL y → (∀P : prop, (∀y2 : set, y2 ∈ ω → eps_ y2 ≤ y + - x2 → P) → P))
Hypothesis H8 : (∀x2 : set, x2 ∈ SNoR y → (∀P : prop, (∀y2 : set, y2 ∈ ω → eps_ y2 ≤ x2 + - y → P) → P))
Hypothesis H9 : SNoCutP z w
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))
Hypothesis H12 : x * y = SNoCut z w
Hypothesis H13 : u ∈ setexp (SNoS_ ω) ω
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))
Hypothesis H15 : v ∈ setexp (SNoS_ ω) ω
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))
Hypothesis H17 : (∀x2 : set, x2 ∈ ω → SNo (ap u x2))
Hypothesis H18 : (∀x2 : set, x2 ∈ ω → SNo (ap v x2))
Hypothesis H19 : (∀x2 : set, x2 ∈ ω → ap u x2 < x * y)
Hypothesis H20 : (∀x2 : set, x2 ∈ ω → x * y < ap v x2)
Hypothesis H21 : SNoCutP (Repl ω (ap u)) (Repl ω (ap v))
Hypothesis H22 : SNo (SNoCut (Repl ω (ap u)) (Repl ω (ap v)))
Hypothesis H23 : (∀x2 : set, x2 ∈ ω → ap u x2 < SNoCut (Repl ω (ap u)) (Repl ω (ap v)))
Hypothesis H24 : (∀x2 : set, x2 ∈ ω → SNoCut (Repl ω (ap u)) (Repl ω (ap v)) < ap v x2)
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
Variable x : set
Variable y : set
Variable z : set
Variable w : set
Variable u : set
Variable v : set
Hypothesis H0 : ¬ x * y ∈ real
Hypothesis H1 : SNo x
Hypothesis H2 : SNo y
Hypothesis H3 : SNo (x * y)
Hypothesis H4 : SNo (- (x * y))
Hypothesis H5 : (∀x2 : set, x2 ∈ SNoL x → (∀P : prop, (∀y2 : set, y2 ∈ ω → eps_ y2 ≤ x + - x2 → P) → P))
Hypothesis H6 : (∀x2 : set, x2 ∈ SNoR x → (∀P : prop, (∀y2 : set, y2 ∈ ω → eps_ y2 ≤ x2 + - x → P) → P))
Hypothesis H7 : (∀x2 : set, x2 ∈ SNoL y → (∀P : prop, (∀y2 : set, y2 ∈ ω → eps_ y2 ≤ y + - x2 → P) → P))
Hypothesis H8 : (∀x2 : set, x2 ∈ SNoR y → (∀P : prop, (∀y2 : set, y2 ∈ ω → eps_ y2 ≤ x2 + - y → P) → P))
Hypothesis H9 : SNoCutP z w
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))
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))
Hypothesis H12 : x * y = SNoCut z w
Hypothesis H13 : u ∈ setexp (SNoS_ ω) ω
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))
Hypothesis H15 : v ∈ setexp (SNoS_ ω) ω
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))
Hypothesis H17 : (∀x2 : set, x2 ∈ ω → SNo (ap u x2))
Hypothesis H18 : (∀x2 : set, x2 ∈ ω → SNo (ap v x2))
Hypothesis H19 : (∀x2 : set, x2 ∈ ω → ap u x2 < x * y)
Hypothesis H20 : (∀x2 : set, x2 ∈ ω → x * y < ap v x2)
Hypothesis H22 : SNo (SNoCut (Repl ω (ap u)) (Repl ω (ap v)))
Hypothesis H23 : (∀x2 : set, x2 ∈ ω → ap u x2 < SNoCut (Repl ω (ap u)) (Repl ω (ap v)))
Hypothesis H24 : (∀x2 : set, x2 ∈ ω → SNoCut (Repl ω (ap u)) (Repl ω (ap v)) < ap v x2)
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
Variable x : set
Variable y : set
Variable z : set
Variable w : set
Variable u : set
Variable v : set
Hypothesis H1 : SNo x
Hypothesis H2 : SNo y
Hypothesis H3 : SNo (x * y)
Hypothesis H4 : SNo (- (x * y))
Hypothesis H5 : (∀x2 : set, x2 ∈ SNoL x → (∀P : prop, (∀y2 : set, y2 ∈ ω → eps_ y2 ≤ x + - x2 → P) → P))
Hypothesis H6 : (∀x2 : set, x2 ∈ SNoR x → (∀P : prop, (∀y2 : set, y2 ∈ ω → eps_ y2 ≤ x2 + - x → P) → P))
Hypothesis H7 : (∀x2 : set, x2 ∈ SNoL y → (∀P : prop, (∀y2 : set, y2 ∈ ω → eps_ y2 ≤ y + - x2 → P) → P))
Hypothesis H8 : (∀x2 : set, x2 ∈ SNoR y → (∀P : prop, (∀y2 : set, y2 ∈ ω → eps_ y2 ≤ x2 + - y → P) → P))
Hypothesis H9 : SNoCutP z w
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))
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))
Hypothesis H12 : x * y = SNoCut z w
Hypothesis H13 : u ∈ setexp (SNoS_ ω) ω
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))
Hypothesis H15 : v ∈ setexp (SNoS_ ω) ω
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))
Hypothesis H17 : (∀x2 : set, x2 ∈ ω → SNo (ap u x2))
Hypothesis H18 : (∀x2 : set, x2 ∈ ω → SNo (ap v x2))
Hypothesis H19 : (∀x2 : set, x2 ∈ ω → ap u x2 < x * y)
Hypothesis H20 : (∀x2 : set, x2 ∈ ω → x * y < ap v x2)
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
Variable x : set
Variable y : set
Variable z : set
Variable w : set
Variable u : set
Variable v : set
Hypothesis H0 : ¬ x * y ∈ real
Hypothesis H1 : SNo x
Hypothesis H2 : SNo y
Hypothesis H3 : SNo (x * y)
Hypothesis H4 : SNo (- (x * y))
Hypothesis H6 : (∀x2 : set, x2 ∈ SNoR x → (∀P : prop, (∀y2 : set, y2 ∈ ω → eps_ y2 ≤ x2 + - x → P) → P))
Hypothesis H7 : (∀x2 : set, x2 ∈ SNoL y → (∀P : prop, (∀y2 : set, y2 ∈ ω → eps_ y2 ≤ y + - x2 → P) → P))
Hypothesis H8 : (∀x2 : set, x2 ∈ SNoR y → (∀P : prop, (∀y2 : set, y2 ∈ ω → eps_ y2 ≤ x2 + - y → P) → P))
Hypothesis H9 : SNoCutP z w
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))
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))
Hypothesis H12 : x * y = SNoCut z w
Hypothesis H13 : u ∈ setexp (SNoS_ ω) ω
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))
Hypothesis H15 : v ∈ setexp (SNoS_ ω) ω
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))
Hypothesis H17 : (∀x2 : set, x2 ∈ ω → SNo (ap u x2))
Hypothesis H18 : (∀x2 : set, x2 ∈ ω → SNo (ap v x2))
Hypothesis H19 : (∀x2 : set, x2 ∈ ω → ap u x2 < x * y)
Hypothesis H20 : (∀x2 : set, x2 ∈ ω → x * y < ap v x2)
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
Variable x : set
Variable y : set
Variable z : set
Variable w : set
Variable u : set
Variable v : set
Hypothesis H1 : SNo x
Hypothesis H2 : SNo y
Hypothesis H3 : SNo (x * y)
Hypothesis H4 : SNo (- (x * y))
Hypothesis H5 : (∀x2 : set, x2 ∈ SNoL x → (∀P : prop, (∀y2 : set, y2 ∈ ω → eps_ y2 ≤ x + - x2 → P) → P))
Hypothesis H6 : (∀x2 : set, x2 ∈ SNoR x → (∀P : prop, (∀y2 : set, y2 ∈ ω → eps_ y2 ≤ x2 + - x → P) → P))
Hypothesis H7 : (∀x2 : set, x2 ∈ SNoL y → (∀P : prop, (∀y2 : set, y2 ∈ ω → eps_ y2 ≤ y + - x2 → P) → P))
Hypothesis H8 : (∀x2 : set, x2 ∈ SNoR y → (∀P : prop, (∀y2 : set, y2 ∈ ω → eps_ y2 ≤ x2 + - y → P) → P))
Hypothesis H9 : SNoCutP z w
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))
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))
Hypothesis H12 : x * y = SNoCut z w
Hypothesis H13 : u ∈ setexp (SNoS_ ω) ω
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))
Hypothesis H15 : v ∈ setexp (SNoS_ ω) ω
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))
Hypothesis H17 : (∀x2 : set, x2 ∈ ω → SNo (ap u x2))
Hypothesis H18 : (∀x2 : set, x2 ∈ ω → SNo (ap v x2))
Hypothesis H19 : (∀x2 : set, x2 ∈ ω → ap u x2 < x * y)
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
Variable x : set
Variable y : set
Variable z : set
Variable w : set
Variable u : set
Variable v : set
Hypothesis H0 : ¬ x * y ∈ real
Hypothesis H1 : SNo x
Hypothesis H2 : SNo y
Hypothesis H3 : SNo (x * y)
Hypothesis H4 : SNo (- (x * y))
Hypothesis H5 : (∀x2 : set, x2 ∈ SNoL x → (∀P : prop, (∀y2 : set, y2 ∈ ω → eps_ y2 ≤ x + - x2 → P) → P))
Hypothesis H6 : (∀x2 : set, x2 ∈ SNoR x → (∀P : prop, (∀y2 : set, y2 ∈ ω → eps_ y2 ≤ x2 + - x → P) → P))
Hypothesis H7 : (∀x2 : set, x2 ∈ SNoL y → (∀P : prop, (∀y2 : set, y2 ∈ ω → eps_ y2 ≤ y + - x2 → P) → P))
Hypothesis H8 : (∀x2 : set, x2 ∈ SNoR y → (∀P : prop, (∀y2 : set, y2 ∈ ω → eps_ y2 ≤ x2 + - y → P) → P))
Hypothesis H9 : SNoCutP z w
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))
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))
Hypothesis H12 : x * y = SNoCut z w
Hypothesis H13 : u ∈ setexp (SNoS_ ω) ω
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))
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))
Hypothesis H17 : (∀x2 : set, x2 ∈ ω → SNo (ap u x2))
Hypothesis H18 : (∀x2 : set, x2 ∈ ω → SNo (ap v x2))
Hypothesis H19 : (∀x2 : set, x2 ∈ ω → ap u x2 < x * y)
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
Variable x : set
Variable y : set
Variable z : set
Variable w : set
Variable u : set
Variable v : set
Hypothesis H0 : ¬ x * y ∈ real
Hypothesis H1 : SNo x
Hypothesis H2 : SNo y
Hypothesis H3 : SNo (x * y)
Hypothesis H4 : SNo (- (x * y))
Hypothesis H5 : (∀x2 : set, x2 ∈ SNoL x → (∀P : prop, (∀y2 : set, y2 ∈ ω → eps_ y2 ≤ x + - x2 → P) → P))
Hypothesis H6 : (∀x2 : set, x2 ∈ SNoR x → (∀P : prop, (∀y2 : set, y2 ∈ ω → eps_ y2 ≤ x2 + - x → P) → P))
Hypothesis H7 : (∀x2 : set, x2 ∈ SNoL y → (∀P : prop, (∀y2 : set, y2 ∈ ω → eps_ y2 ≤ y + - x2 → P) → P))
Hypothesis H8 : (∀x2 : set, x2 ∈ SNoR y → (∀P : prop, (∀y2 : set, y2 ∈ ω → eps_ y2 ≤ x2 + - y → P) → P))
Hypothesis H9 : SNoCutP z w
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))
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))
Hypothesis H12 : x * y = SNoCut z w
Hypothesis H13 : u ∈ setexp (SNoS_ ω) ω
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))
Hypothesis H15 : v ∈ setexp (SNoS_ ω) ω
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))
Hypothesis H17 : (∀x2 : set, x2 ∈ ω → SNo (ap u x2))
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
Variable x : set
Variable y : set
Variable z : set
Variable w : set
Hypothesis H0 : Empty < x
Hypothesis H1 : Empty < y
Hypothesis H2 : ¬ x * y ∈ real
Hypothesis H3 : SNo x
Hypothesis H4 : x ∈ SNoS_ (ordsucc ω)
Hypothesis H5 : x < ω
Hypothesis H6 : SNo y
Hypothesis H7 : y ∈ SNoS_ (ordsucc ω)
Hypothesis H8 : y < ω
Hypothesis H9 : (∀u : set, u ∈ ω → (∀P : prop, (∀v : set, v ∈ SNoS_ ω → Empty < v → v < x → x < v + eps_ u → P) → P))
Hypothesis H10 : (∀u : set, u ∈ ω → (∀P : prop, (∀v : set, v ∈ SNoS_ ω → Empty < v → v < y → y < v + eps_ u → P) → P))
Hypothesis H11 : SNo (x * y)
Hypothesis H12 : SNo (- (x * y))
Hypothesis H13 : (∀u : set, SNo u → SNoLev u ∈ ω → SNoLev u ∈ SNoLev (x * y))
Hypothesis H14 : (∀u : set, u ∈ SNoL x → (∀P : prop, (∀v : set, v ∈ ω → eps_ v ≤ x + - u → P) → P))
Hypothesis H15 : (∀u : set, u ∈ SNoR x → (∀P : prop, (∀v : set, v ∈ ω → eps_ v ≤ u + - x → P) → P))
Hypothesis H16 : (∀u : set, u ∈ SNoL y → (∀P : prop, (∀v : set, v ∈ ω → eps_ v ≤ y + - u → P) → P))
Hypothesis H17 : (∀u : set, u ∈ SNoR y → (∀P : prop, (∀v : set, v ∈ ω → eps_ v ≤ u + - y → P) → P))
Hypothesis H18 : SNoCutP z w
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))
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))
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
Variable x : set
Variable y : set
Hypothesis H0 : Empty < x
Hypothesis H1 : Empty < y
Hypothesis H2 : ¬ x * y ∈ real
Hypothesis H4 : x ∈ SNoS_ (ordsucc ω)
Hypothesis H5 : x < ω
Hypothesis H6 : (∀z : set, z ∈ SNoS_ ω → (∀w : set, w ∈ ω → abs_SNo (z + - x) < eps_ w) → z = x)
Hypothesis H7 : SNo y
Hypothesis H8 : y ∈ SNoS_ (ordsucc ω)
Hypothesis H9 : y < ω
Hypothesis H10 : (∀z : set, z ∈ SNoS_ ω → (∀w : set, w ∈ ω → abs_SNo (z + - y) < eps_ w) → z = y)
Hypothesis H11 : (∀z : set, z ∈ ω → (∀P : prop, (∀w : set, w ∈ SNoS_ ω → Empty < w → w < x → x < w + eps_ z → P) → P))
Hypothesis H12 : (∀z : set, z ∈ ω → (∀P : prop, (∀w : set, w ∈ SNoS_ ω → Empty < w → w < y → y < w + eps_ z → P) → P))
Hypothesis H13 : SNo (x * y)
Hypothesis H14 : SNo (- (x * y))
Hypothesis H15 : (∀z : set, SNo z → SNoLev z ∈ ω → SNoLev z ∈ SNoLev (x * y))
Hypothesis H16 : Subq (SNoL x) (SNoS_ ω)
Hypothesis H17 : Subq (SNoR x) (SNoS_ ω)
Hypothesis H18 : Subq (SNoL y) (SNoS_ ω)
Hypothesis H19 : Subq (SNoR y) (SNoS_ ω)
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
Variable x : set
Variable y : set
Hypothesis H0 : Empty < x
Hypothesis H1 : Empty < y
Hypothesis H2 : ¬ x * y ∈ real
Hypothesis H3 : SNo x
Hypothesis H5 : x < ω
Hypothesis H6 : (∀z : set, z ∈ SNoS_ ω → (∀w : set, w ∈ ω → abs_SNo (z + - x) < eps_ w) → z = x)
Hypothesis H7 : SNo y
Hypothesis H8 : SNoLev y ∈ ordsucc ω
Hypothesis H9 : y ∈ SNoS_ (ordsucc ω)
Hypothesis H10 : y < ω
Hypothesis H11 : (∀z : set, z ∈ SNoS_ ω → (∀w : set, w ∈ ω → abs_SNo (z + - y) < eps_ w) → z = y)
Hypothesis H12 : (∀z : set, z ∈ ω → (∀P : prop, (∀w : set, w ∈ SNoS_ ω → Empty < w → w < x → x < w + eps_ z → P) → P))
Hypothesis H13 : (∀z : set, z ∈ ω → (∀P : prop, (∀w : set, w ∈ SNoS_ ω → Empty < w → w < y → y < w + eps_ z → P) → P))
Hypothesis H14 : SNo (x * y)
Hypothesis H15 : SNo (- (x * y))
Hypothesis H16 : (∀z : set, SNo z → SNoLev z ∈ ω → SNoLev z ∈ SNoLev (x * y))
Hypothesis H17 : Subq (SNoL x) (SNoS_ ω)
Hypothesis H18 : Subq (SNoR x) (SNoS_ ω)
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
Variable x : set
Variable y : set
Hypothesis H1 : Empty < y
Hypothesis H2 : ¬ x * y ∈ real
Hypothesis H3 : SNo x
Hypothesis H4 : SNoLev x ∈ ordsucc ω
Hypothesis H5 : x ∈ SNoS_ (ordsucc ω)
Hypothesis H6 : x < ω
Hypothesis H7 : (∀z : set, z ∈ SNoS_ ω → (∀w : set, w ∈ ω → abs_SNo (z + - x) < eps_ w) → z = x)
Hypothesis H8 : SNo y
Hypothesis H9 : SNoLev y ∈ ordsucc ω
Hypothesis H10 : y ∈ SNoS_ (ordsucc ω)
Hypothesis H11 : y < ω
Hypothesis H12 : (∀z : set, z ∈ SNoS_ ω → (∀w : set, w ∈ ω → abs_SNo (z + - y) < eps_ w) → z = y)
Hypothesis H13 : (∀z : set, z ∈ ω → (∀P : prop, (∀w : set, w ∈ SNoS_ ω → Empty < w → w < x → x < w + eps_ z → P) → P))
Hypothesis H14 : (∀z : set, z ∈ ω → (∀P : prop, (∀w : set, w ∈ SNoS_ ω → Empty < w → w < y → y < w + eps_ z → P) → P))
Hypothesis H15 : SNo (x * y)
Hypothesis H16 : SNo (- (x * y))
Hypothesis H17 : (∀z : set, SNo z → SNoLev z ∈ ω → SNoLev z ∈ SNoLev (x * y))
Hypothesis H18 : Subq (SNoL x) (SNoS_ ω)
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
Variable x : set
Variable y : set
Hypothesis H0 : Empty < x
Hypothesis H1 : Empty < y
Hypothesis H2 : ¬ x * y ∈ real
Hypothesis H3 : SNo x
Hypothesis H5 : x ∈ SNoS_ (ordsucc ω)
Hypothesis H6 : x < ω
Hypothesis H7 : (∀z : set, z ∈ SNoS_ ω → (∀w : set, w ∈ ω → abs_SNo (z + - x) < eps_ w) → z = x)
Hypothesis H8 : SNo y
Hypothesis H9 : SNoLev y ∈ ordsucc ω
Hypothesis H10 : y ∈ SNoS_ (ordsucc ω)
Hypothesis H11 : y < ω
Hypothesis H12 : (∀z : set, z ∈ SNoS_ ω → (∀w : set, w ∈ ω → abs_SNo (z + - y) < eps_ w) → z = y)
Hypothesis H13 : (∀z : set, z ∈ ω → (∀P : prop, (∀w : set, w ∈ SNoS_ ω → Empty < w → w < x → x < w + eps_ z → P) → P))
Hypothesis H14 : (∀z : set, z ∈ ω → (∀P : prop, (∀w : set, w ∈ SNoS_ ω → Empty < w → w < y → y < w + eps_ z → P) → P))
Hypothesis H15 : SNo (x * y)
Hypothesis H16 : SNo (- (x * y))
Hypothesis H17 : nIn (SNoLev (x * y)) ω
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
Variable x : set
Variable y : set
Hypothesis H0 : Empty < x
Hypothesis H1 : Empty < y
Hypothesis H2 : ¬ x * y ∈ real
Hypothesis H3 : SNo x
Hypothesis H4 : SNoLev x ∈ ordsucc ω
Hypothesis H5 : x ∈ SNoS_ (ordsucc ω)
Hypothesis H6 : x < ω
Hypothesis H7 : (∀z : set, z ∈ SNoS_ ω → (∀w : set, w ∈ ω → abs_SNo (z + - x) < eps_ w) → z = x)
Hypothesis H8 : SNo y
Hypothesis H9 : SNoLev y ∈ ordsucc ω
Hypothesis H10 : y ∈ SNoS_ (ordsucc ω)
Hypothesis H11 : y < ω
Hypothesis H13 : (∀z : set, z ∈ ω → (∀P : prop, (∀w : set, w ∈ SNoS_ ω → Empty < w → w < x → x < w + eps_ z → P) → P))
Hypothesis H14 : (∀z : set, z ∈ ω → (∀P : prop, (∀w : set, w ∈ SNoS_ ω → Empty < w → w < y → y < w + eps_ z → P) → P))
Hypothesis H15 : SNo (x * y)
Hypothesis H16 : SNo (- (x * y))
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
Variable x : set
Variable y : set
Hypothesis H0 : Empty < x
Hypothesis H1 : Empty < y
Hypothesis H2 : ¬ x * y ∈ real
Hypothesis H3 : SNo x
Hypothesis H4 : SNoLev x ∈ ordsucc ω
Hypothesis H5 : x ∈ SNoS_ (ordsucc ω)
Hypothesis H6 : x < ω
Hypothesis H7 : (∀z : set, z ∈ SNoS_ ω → (∀w : set, w ∈ ω → abs_SNo (z + - x) < eps_ w) → z = x)
Hypothesis H8 : SNo y
Hypothesis H9 : SNoLev y ∈ ordsucc ω
Hypothesis H11 : y < ω
Hypothesis H12 : (∀z : set, z ∈ SNoS_ ω → (∀w : set, w ∈ ω → abs_SNo (z + - y) < eps_ w) → z = y)
Hypothesis H13 : (∀z : set, z ∈ ω → (∀P : prop, (∀w : set, w ∈ SNoS_ ω → Empty < w → w < x → x < w + eps_ z → P) → P))
Hypothesis H14 : (∀z : set, z ∈ ω → (∀P : prop, (∀w : set, w ∈ SNoS_ ω → Empty < w → w < y → y < w + eps_ z → P) → P))
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
Variable x : set
Variable y : set
Hypothesis H0 : Empty < x
Hypothesis H1 : Empty < y
Hypothesis H2 : ¬ x * y ∈ real
Hypothesis H3 : SNo x
Hypothesis H4 : SNoLev x ∈ ordsucc ω
Hypothesis H5 : x ∈ SNoS_ (ordsucc ω)
Hypothesis H6 : x < ω
Hypothesis H7 : (∀z : set, z ∈ SNoS_ ω → (∀w : set, w ∈ ω → abs_SNo (z + - x) < eps_ w) → z = x)
Hypothesis H8 : SNo y
Hypothesis H9 : SNoLev y ∈ ordsucc ω
Hypothesis H10 : y ∈ SNoS_ (ordsucc ω)
Hypothesis H11 : y < ω
Hypothesis H13 : (∀z : set, z ∈ ω → (∃w : set, w ∈ SNoS_ ω ∧ (w < y ∧ y < w + eps_ z)))
Hypothesis H14 : (∀z : set, z ∈ ω → (∀P : prop, (∀w : set, w ∈ SNoS_ ω → Empty < w → w < x → x < w + eps_ z → P) → P))
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
Variable x : set
Variable y : set
Hypothesis H0 : Empty < x
Hypothesis H1 : Empty < y
Hypothesis H2 : ¬ x * y ∈ real
Hypothesis H3 : SNo x
Hypothesis H5 : x ∈ SNoS_ (ordsucc ω)
Hypothesis H6 : x < ω
Hypothesis H7 : (∀z : set, z ∈ SNoS_ ω → (∀w : set, w ∈ ω → abs_SNo (z + - x) < eps_ w) → z = x)
Hypothesis H8 : (∀z : set, z ∈ ω → (∃w : set, w ∈ SNoS_ ω ∧ (w < x ∧ x < w + eps_ z)))
Hypothesis H9 : SNo y
Hypothesis H10 : SNoLev y ∈ ordsucc ω
Hypothesis H11 : y ∈ SNoS_ (ordsucc ω)
Hypothesis H12 : y < ω
Hypothesis H13 : (∀z : set, z ∈ SNoS_ ω → (∀w : set, w ∈ ω → abs_SNo (z + - y) < eps_ w) → z = y)
Hypothesis H14 : (∀z : set, z ∈ ω → (∃w : set, w ∈ SNoS_ ω ∧ (w < y ∧ y < w + eps_ z)))
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
Variable x : set
Variable y : set
Variable z : set
Hypothesis H0 : SNo x
Hypothesis H2 : SNo z
Hypothesis H3 : y < x + z
Hypothesis H4 : Empty ≤ y + - x
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
Variable x : set
Variable y : set
Variable z : set
Hypothesis H0 : x < ordsucc Empty
Hypothesis H1 : SNo x
Hypothesis H2 : SNo (SNoCut (Sep (SNoS_ ω) (λw : set ⇒ x * w < ordsucc Empty)) (Sep (SNoS_ ω) (λw : set ⇒ ordsucc Empty < x * w)))
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)
Hypothesis H4 : y ∈ SNoS_ ω
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)
Hypothesis H6 : SNo y
Hypothesis H7 : SNo (x * y)
Hypothesis H8 : SNoCut (Sep (SNoS_ ω) (λw : set ⇒ x * w < ordsucc Empty)) (Sep (SNoS_ ω) (λw : set ⇒ ordsucc Empty < x * w)) < y
Hypothesis H10 : ordsucc Empty ≤ x * y + - (eps_ z)
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
Variable x : set
Variable y : set
Hypothesis H1 : ¬ (∃z : set, z ∈ real ∧ x * z = ordsucc Empty)
Hypothesis H2 : SNo x
Hypothesis H3 : SNo (SNoCut (Sep (SNoS_ ω) (λz : set ⇒ x * z < ordsucc Empty)) (Sep (SNoS_ ω) (λz : set ⇒ ordsucc Empty < x * z)))
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)))
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)
Hypothesis H6 : y ∈ SNoS_ ω
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)
Hypothesis H8 : SNo y
Hypothesis H9 : y ∈ real
Hypothesis H10 : SNo (x * y)
Hypothesis H11 : (∀z : set, z ∈ SNoS_ ω → (∀w : set, w ∈ ω → abs_SNo (z + - (x * y)) < eps_ w) → z = x * y)
Hypothesis H12 : x * y < ordsucc Empty → (∃z : set, z ∈ ω ∧ (x * y + eps_ z) ≤ ordsucc Empty)
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
Variable x : set
Variable y : set
Hypothesis H0 : x < ordsucc Empty
Hypothesis H1 : ¬ (∃z : set, z ∈ real ∧ x * z = ordsucc Empty)
Hypothesis H2 : SNo x
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)))
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)
Hypothesis H6 : y ∈ SNoS_ ω
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)
Hypothesis H8 : SNo y
Hypothesis H9 : y ∈ real
Hypothesis H10 : SNo (x * y)
Hypothesis H11 : (∀z : set, z ∈ SNoS_ ω → (∀w : set, w ∈ ω → abs_SNo (z + - (x * y)) < eps_ w) → z = x * y)
Hypothesis H12 : SNo (- (x * y))
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
Variable x : set
Variable y : set
Hypothesis H0 : x ∈ real
Hypothesis H2 : ¬ (∃z : set, z ∈ real ∧ x * z = ordsucc Empty)
Hypothesis H3 : SNo x
Hypothesis H4 : SNo (SNoCut (Sep (SNoS_ ω) (λz : set ⇒ x * z < ordsucc Empty)) (Sep (SNoS_ ω) (λz : set ⇒ ordsucc Empty < x * z)))
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)))
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)
Hypothesis H7 : y ∈ SNoS_ ω
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)
Hypothesis H9 : SNo y
Hypothesis H10 : y ∈ real
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
Variable x : set
Variable y : set
Hypothesis H0 : x ∈ real
Hypothesis H1 : x < ordsucc Empty
Hypothesis H3 : SNo x
Hypothesis H4 : SNo (SNoCut (Sep (SNoS_ ω) (λz : set ⇒ x * z < ordsucc Empty)) (Sep (SNoS_ ω) (λz : set ⇒ ordsucc Empty < x * z)))
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)))
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)
Hypothesis H7 : y ∈ SNoS_ ω
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)
Hypothesis H9 : SNo y
Hypothesis H10 : y ∈ real
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
Variable x : set
Variable y : set
Hypothesis H0 : x ∈ real
Hypothesis H1 : x < ordsucc Empty
Hypothesis H2 : ¬ (∃z : set, z ∈ real ∧ x * z = ordsucc Empty)
Hypothesis H3 : SNo x
Hypothesis H4 : SNo (SNoCut (Sep (SNoS_ ω) (λz : set ⇒ x * z < ordsucc Empty)) (Sep (SNoS_ ω) (λz : set ⇒ ordsucc Empty < x * z)))
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)))
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)
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)
Hypothesis H9 : SNo y
Hypothesis H10 : y ∈ real
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
Variable x : set
Variable y : set
Hypothesis H1 : x < ordsucc Empty
Hypothesis H2 : ¬ (∃z : set, z ∈ real ∧ x * z = ordsucc Empty)
Hypothesis H3 : SNo x
Hypothesis H4 : SNo (SNoCut (Sep (SNoS_ ω) (λz : set ⇒ x * z < ordsucc Empty)) (Sep (SNoS_ ω) (λz : set ⇒ ordsucc Empty < x * z)))
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)))
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)
Hypothesis H7 : y ∈ SNoS_ ω
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)
Hypothesis H9 : SNo y
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
Variable x : set
Variable y : set
Hypothesis H0 : x ∈ real
Hypothesis H1 : x < ordsucc Empty
Hypothesis H2 : ¬ (∃z : set, z ∈ real ∧ x * z = ordsucc Empty)
Hypothesis H3 : SNo x
Hypothesis H4 : SNo (SNoCut (Sep (SNoS_ ω) (λz : set ⇒ x * z < ordsucc Empty)) (Sep (SNoS_ ω) (λz : set ⇒ ordsucc Empty < x * z)))
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)))
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)
Hypothesis H7 : y ∈ SNoS_ ω
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)
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
Variable x : set
Variable y : set
Hypothesis H0 : Empty < x
Hypothesis H1 : SNo x
Hypothesis H2 : SNo (SNoCut (Sep (SNoS_ ω) (λz : set ⇒ x * z < ordsucc Empty)) (Sep (SNoS_ ω) (λz : set ⇒ ordsucc Empty < x * z)))
Hypothesis H4 : y ∈ ω
Hypothesis H5 : eps_ y ≤ x
Hypothesis H6 : nat_p (exp_SNo_nat (ordsucc (ordsucc Empty)) y)
Hypothesis H7 : SNo (exp_SNo_nat (ordsucc (ordsucc Empty)) y)
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
Variable x : set
Variable y : set
Variable z : set
Hypothesis H0 : Empty < x
Hypothesis H2 : SNo x
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)))
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)
Hypothesis H5 : SNo (SNoCut (Sep (SNoS_ ω) (λw : set ⇒ x * w < ordsucc Empty)) (Sep (SNoS_ ω) (λw : set ⇒ ordsucc Empty < x * w)))
Hypothesis H6 : SNo (x * SNoCut (Sep (SNoS_ ω) (λw : set ⇒ x * w < ordsucc Empty)) (Sep (SNoS_ ω) (λw : set ⇒ ordsucc Empty < x * w)))
Hypothesis H7 : SNo (x * SNoCut (Sep (SNoS_ ω) (λw : set ⇒ x * w < ordsucc Empty)) (Sep (SNoS_ ω) (λw : set ⇒ ordsucc Empty < x * w)))
Hypothesis H8 : y ∈ ω
Hypothesis H9 : z ∈ SNoS_ ω
Hypothesis H10 : z < SNoCut (Sep (SNoS_ ω) (λw : set ⇒ x * w < ordsucc Empty)) (Sep (SNoS_ ω) (λw : set ⇒ ordsucc Empty < x * w))
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)
Hypothesis H12 : SNo z
Hypothesis H13 : SNo (x * z)
Hypothesis H14 : SNo (eps_ (ordsucc y))
Hypothesis H15 : SNo (x * eps_ (ordsucc y))
Hypothesis H16 : SNo (z + eps_ (ordsucc y))
Hypothesis H17 : SNo (ordsucc Empty + - (x * z))
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))))
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
Variable x : set
Variable y : set
Variable z : set
Hypothesis H0 : Empty < x
Hypothesis H1 : x < ordsucc Empty
Hypothesis H2 : SNo x
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)))
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)
Hypothesis H5 : SNo (SNoCut (Sep (SNoS_ ω) (λw : set ⇒ x * w < ordsucc Empty)) (Sep (SNoS_ ω) (λw : set ⇒ ordsucc Empty < x * w)))
Hypothesis H7 : SNo (x * SNoCut (Sep (SNoS_ ω) (λw : set ⇒ x * w < ordsucc Empty)) (Sep (SNoS_ ω) (λw : set ⇒ ordsucc Empty < x * w)))
Hypothesis H8 : y ∈ ω
Hypothesis H9 : z ∈ SNoS_ ω
Hypothesis H10 : z < SNoCut (Sep (SNoS_ ω) (λw : set ⇒ x * w < ordsucc Empty)) (Sep (SNoS_ ω) (λw : set ⇒ ordsucc Empty < x * w))
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)
Hypothesis H12 : SNo z
Hypothesis H13 : SNo (x * z)
Hypothesis H14 : SNo (eps_ (ordsucc y))
Hypothesis H15 : SNo (x * eps_ (ordsucc y))
Hypothesis H16 : SNo (z + eps_ (ordsucc y))
Hypothesis H17 : SNo (ordsucc Empty + - (x * z))
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))))
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
Variable x : set
Variable y : set
Variable z : set
Hypothesis H0 : Empty < x
Hypothesis H1 : x < ordsucc Empty
Hypothesis H2 : SNo x
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)))
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)
Hypothesis H5 : SNo (SNoCut (Sep (SNoS_ ω) (λw : set ⇒ x * w < ordsucc Empty)) (Sep (SNoS_ ω) (λw : set ⇒ ordsucc Empty < x * w)))
Hypothesis H6 : SNo (x * SNoCut (Sep (SNoS_ ω) (λw : set ⇒ x * w < ordsucc Empty)) (Sep (SNoS_ ω) (λw : set ⇒ ordsucc Empty < x * w)))
Hypothesis H8 : y ∈ ω
Hypothesis H9 : z ∈ SNoS_ ω
Hypothesis H10 : z < SNoCut (Sep (SNoS_ ω) (λw : set ⇒ x * w < ordsucc Empty)) (Sep (SNoS_ ω) (λw : set ⇒ ordsucc Empty < x * w))
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)
Hypothesis H12 : SNo z
Hypothesis H13 : SNo (x * z)
Hypothesis H14 : SNo (eps_ (ordsucc y))
Hypothesis H15 : SNo (x * eps_ (ordsucc y))
Hypothesis H16 : SNo (z + eps_ (ordsucc y))
Hypothesis H17 : SNo (ordsucc Empty + - (x * z))
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))))
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
Variable x : set
Variable y : set
Variable z : set
Hypothesis H0 : Empty < x
Hypothesis H1 : x < ordsucc Empty
Hypothesis H2 : SNo x
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)))
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)
Hypothesis H5 : SNo (SNoCut (Sep (SNoS_ ω) (λw : set ⇒ x * w < ordsucc Empty)) (Sep (SNoS_ ω) (λw : set ⇒ ordsucc Empty < x * w)))
Hypothesis H6 : SNo (x * SNoCut (Sep (SNoS_ ω) (λw : set ⇒ x * w < ordsucc Empty)) (Sep (SNoS_ ω) (λw : set ⇒ ordsucc Empty < x * w)))
Hypothesis H7 : SNo (x * SNoCut (Sep (SNoS_ ω) (λw : set ⇒ x * w < ordsucc Empty)) (Sep (SNoS_ ω) (λw : set ⇒ ordsucc Empty < x * w)))
Hypothesis H9 : z ∈ SNoS_ ω
Hypothesis H10 : z < SNoCut (Sep (SNoS_ ω) (λw : set ⇒ x * w < ordsucc Empty)) (Sep (SNoS_ ω) (λw : set ⇒ ordsucc Empty < x * w))
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)
Hypothesis H12 : SNo z
Hypothesis H13 : SNo (x * z)
Hypothesis H14 : SNo (eps_ (ordsucc y))
Hypothesis H15 : SNo (x * eps_ (ordsucc y))
Hypothesis H16 : SNo (z + eps_ (ordsucc y))
Hypothesis H17 : SNo (ordsucc Empty + - (x * z))
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
Variable x : set
Variable y : set
Variable z : set
Hypothesis H0 : Empty < x
Hypothesis H1 : x < ordsucc Empty
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)))
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)
Hypothesis H5 : SNo (SNoCut (Sep (SNoS_ ω) (λw : set ⇒ x * w < ordsucc Empty)) (Sep (SNoS_ ω) (λw : set ⇒ ordsucc Empty < x * w)))
Hypothesis H6 : SNo (x * SNoCut (Sep (SNoS_ ω) (λw : set ⇒ x * w < ordsucc Empty)) (Sep (SNoS_ ω) (λw : set ⇒ ordsucc Empty < x * w)))
Hypothesis H7 : SNo (x * SNoCut (Sep (SNoS_ ω) (λw : set ⇒ x * w < ordsucc Empty)) (Sep (SNoS_ ω) (λw : set ⇒ ordsucc Empty < x * w)))
Hypothesis H8 : y ∈ ω
Hypothesis H9 : z ∈ SNoS_ ω
Hypothesis H10 : z < SNoCut (Sep (SNoS_ ω) (λw : set ⇒ x * w < ordsucc Empty)) (Sep (SNoS_ ω) (λw : set ⇒ ordsucc Empty < x * w))
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)
Hypothesis H12 : SNo z
Hypothesis H13 : SNo (x * z)
Hypothesis H14 : SNo (eps_ (ordsucc y))
Hypothesis H15 : SNo (x * eps_ (ordsucc y))
Hypothesis H16 : SNo (z + eps_ (ordsucc y))
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
Variable x : set
Variable y : set
Variable z : set
Hypothesis H1 : x < ordsucc Empty
Hypothesis H2 : SNo x
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)))
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)
Hypothesis H5 : SNo (SNoCut (Sep (SNoS_ ω) (λw : set ⇒ x * w < ordsucc Empty)) (Sep (SNoS_ ω) (λw : set ⇒ ordsucc Empty < x * w)))
Hypothesis H6 : SNo (x * SNoCut (Sep (SNoS_ ω) (λw : set ⇒ x * w < ordsucc Empty)) (Sep (SNoS_ ω) (λw : set ⇒ ordsucc Empty < x * w)))
Hypothesis H7 : SNo (x * SNoCut (Sep (SNoS_ ω) (λw : set ⇒ x * w < ordsucc Empty)) (Sep (SNoS_ ω) (λw : set ⇒ ordsucc Empty < x * w)))
Hypothesis H8 : y ∈ ω
Hypothesis H9 : z ∈ SNoS_ ω
Hypothesis H10 : z < SNoCut (Sep (SNoS_ ω) (λw : set ⇒ x * w < ordsucc Empty)) (Sep (SNoS_ ω) (λw : set ⇒ ordsucc Empty < x * w))
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)
Hypothesis H12 : SNo z
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
Variable x : set
Hypothesis H0 : Empty < x
Hypothesis H1 : x < ordsucc Empty
Hypothesis H2 : ¬ (∃y : set, y ∈ real ∧ x * y = ordsucc Empty)
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)))
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)
Hypothesis H6 : SNoCut (Sep (SNoS_ ω) (λy : set ⇒ x * y < ordsucc Empty)) (Sep (SNoS_ ω) (λy : set ⇒ ordsucc Empty < x * y)) ∈ real
Hypothesis H7 : SNo (SNoCut (Sep (SNoS_ ω) (λy : set ⇒ x * y < ordsucc Empty)) (Sep (SNoS_ ω) (λy : set ⇒ ordsucc Empty < x * y)))
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)))
Hypothesis H9 : x * SNoCut (Sep (SNoS_ ω) (λy : set ⇒ x * y < ordsucc Empty)) (Sep (SNoS_ ω) (λy : set ⇒ ordsucc Empty < x * y)) ∈ real
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
Variable x : set
Hypothesis H0 : Empty < x
Hypothesis H1 : x < ordsucc Empty
Hypothesis H2 : ¬ (∃y : set, y ∈ real ∧ x * y = ordsucc Empty)
Hypothesis H3 : SNo x
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)))
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)
Hypothesis H6 : SNoCut (Sep (SNoS_ ω) (λy : set ⇒ x * y < ordsucc Empty)) (Sep (SNoS_ ω) (λy : set ⇒ ordsucc Empty < x * y)) ∈ real
Hypothesis H7 : SNo (SNoCut (Sep (SNoS_ ω) (λy : set ⇒ x * y < ordsucc Empty)) (Sep (SNoS_ ω) (λy : set ⇒ ordsucc Empty < x * y)))
Hypothesis H9 : x * SNoCut (Sep (SNoS_ ω) (λy : set ⇒ x * y < ordsucc Empty)) (Sep (SNoS_ ω) (λy : set ⇒ ordsucc Empty < x * y)) ∈ real
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
Variable x : set
Hypothesis H0 : Empty < x
Hypothesis H1 : x < ordsucc Empty
Hypothesis H2 : ¬ (∃y : set, y ∈ real ∧ x * y = ordsucc Empty)
Hypothesis H3 : SNo x
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)))
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)
Hypothesis H6 : SNoCut (Sep (SNoS_ ω) (λy : set ⇒ x * y < ordsucc Empty)) (Sep (SNoS_ ω) (λy : set ⇒ ordsucc Empty < x * y)) ∈ real
Hypothesis H7 : SNo (SNoCut (Sep (SNoS_ ω) (λy : set ⇒ x * y < ordsucc Empty)) (Sep (SNoS_ ω) (λy : set ⇒ ordsucc Empty < x * y)))
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)))
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
Variable x : set
Hypothesis H0 : x ∈ real
Hypothesis H1 : Empty < x
Hypothesis H2 : x < ordsucc Empty
Hypothesis H4 : SNo x
Hypothesis H5 : (∀y : set, y ∈ SNoS_ ω → (∀z : set, z ∈ ω → abs_SNo (y + - x) < eps_ z) → y = x)
Hypothesis H6 : SNo (SNoCut (Sep (SNoS_ ω) (λy : set ⇒ x * y < ordsucc Empty)) (Sep (SNoS_ ω) (λy : set ⇒ ordsucc Empty < x * y)))
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)))
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)
Hypothesis H9 : SNoCut (Sep (SNoS_ ω) (λy : set ⇒ x * y < ordsucc Empty)) (Sep (SNoS_ ω) (λy : set ⇒ ordsucc Empty < x * y)) ∈ SNoS_ (ordsucc ω)
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
Variable x : set
Hypothesis H0 : x ∈ real
Hypothesis H1 : Empty < x
Hypothesis H2 : x < ordsucc Empty
Hypothesis H3 : ¬ (∃y : set, y ∈ real ∧ x * y = ordsucc Empty)
Hypothesis H4 : SNo x
Hypothesis H5 : (∀y : set, y ∈ SNoS_ ω → (∀z : set, z ∈ ω → abs_SNo (y + - x) < eps_ z) → y = x)
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))
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
Variable x : set
Variable y : set
Variable z : set
Hypothesis H0 : SNo x
Hypothesis H1 : y ∈ ω
Hypothesis H2 : z ∈ real
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
Variable x : set
Variable y : set
Hypothesis H0 : x ∈ real
Hypothesis H1 : Empty < x
Hypothesis H2 : SNo x
Hypothesis H3 : y ∈ ω
Hypothesis H5 : eps_ y ∈ real
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
Variable x : set
Variable y : set
Variable z : set
Hypothesis H0 : SNo x
Hypothesis H1 : SNo (exp_SNo_nat (ordsucc (ordsucc Empty)) y)
Hypothesis H2 : ordsucc Empty ≤ exp_SNo_nat (ordsucc (ordsucc Empty)) y * x
Hypothesis H4 : SNo z
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
Variable x : set
Variable y : set
Variable z : set
Hypothesis H0 : y ∈ real
Hypothesis H1 : SNo x
Hypothesis H3 : eps_ z ≤ x
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
Variable x : set
Variable y : set
Variable z : set
Variable w : set
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)
Hypothesis H2 : (∀u : set, u ∈ z → ap x u ≤ ap x z)
Hypothesis H3 : w ∈ ordsucc z
Hypothesis H4 : z ∈ ω
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
Variable x : set
Variable y : set
Variable z : set
Variable w : set
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)
Hypothesis H1 : (∀u : set, u ∈ ω → SNo (ap x u))
Hypothesis H2 : (∀u : set, u ∈ z → ap x u ≤ ap x z)
Hypothesis H4 : z ∈ ω
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
Variable x : set
Variable y : set
Variable z : set
Variable w : set
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)
Hypothesis H1 : (∀u : set, u ∈ ω → SNo (ap x u))
Hypothesis H2 : nat_p z
Hypothesis H4 : w ∈ ordsucc z
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
Variable x : set
Variable y : set
Variable z : set
Variable w : set
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)
Hypothesis H2 : (∀u : set, u ∈ ω → SNo (ap y u))
Hypothesis H3 : (∀u : set, nat_p u → (∀v : set, v ∈ u → ap x v ≤ ap x u))
Hypothesis H4 : (∀u : set, nat_p u → (∀v : set, v ∈ u → ap y u ≤ ap y v))
Hypothesis H5 : z ∈ ω
Hypothesis H6 : w ∈ ω
Hypothesis H7 : nat_p z
Hypothesis H8 : nat_p w
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
Variable x : set
Variable y : set
Variable z : set
Hypothesis H1 : (∀w : set, w ∈ ω → SNo (ap y w))
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)))
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)
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)
Hypothesis H5 : z ∈ SNoS_ ω
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))
Hypothesis H7 : SNoLev z ∈ ω
Hypothesis H8 : ordinal (SNoLev z)
Hypothesis H9 : SNo z
Hypothesis H10 : (∀w : set, w ∈ Sep (SNoS_ ω) (λu : set ⇒ ∃v : set, v ∈ ω ∧ u < ap x v) → w < z)
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
Variable x : set
Variable y : set
Variable z : set
Hypothesis H0 : ¬ (∃w : set, w ∈ real ∧ (∀u : set, u ∈ ω → ap x u ≤ w ∧ w ≤ ap y u))
Hypothesis H1 : (∀w : set, w ∈ ω → SNo (ap y w))
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)
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)
Hypothesis H5 : z ∈ SNoS_ ω
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))
Hypothesis H7 : SNoLev z ∈ ω
Hypothesis H8 : ordinal (SNoLev z)
Hypothesis H9 : SNo z
Hypothesis H10 : (∀w : set, w ∈ Sep (SNoS_ ω) (λu : set ⇒ ∃v : set, v ∈ ω ∧ u < ap x v) → w < z)
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
Variable x : set
Variable y : set
Variable z : set
Hypothesis H0 : ¬ (∃w : set, w ∈ real ∧ (∀u : set, u ∈ ω → ap x u ≤ w ∧ w ≤ ap y u))
Hypothesis H1 : (∀w : set, w ∈ ω → SNo (ap x w))
Hypothesis H2 : (∀w : set, w ∈ ω → SNo (ap y w))
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)))
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)
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)
Hypothesis H6 : z ∈ SNoS_ ω
Hypothesis H8 : SNoLev z ∈ ω
Hypothesis H9 : ordinal (SNoLev z)
Hypothesis H10 : SNo z
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
Variable x : set
Variable y : set
Variable z : set
Variable w : set
Variable u : set
Hypothesis H0 : (∀v : set, v ∈ ω → SNo (ap y v))
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)))
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)
Hypothesis H3 : SNo z
Hypothesis H4 : w ∈ ω
Hypothesis H6 : u ∈ ω
Hypothesis H7 : Empty < z + - (ap y w)
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
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
Variable x : set
Variable y : set
Variable z : set
Variable w : set
Variable u : set
Hypothesis H0 : (∀v : set, v ∈ ω → SNo (ap y v))
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)))
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)
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)
Hypothesis H4 : SNo z
Hypothesis H5 : w ∈ ω
Hypothesis H7 : u ∈ ω
Hypothesis H8 : Empty < z + - (ap y w)
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
Variable x : set
Variable y : set
Variable z : set
Variable w : set
Variable u : set
Hypothesis H0 : (∀v : set, v ∈ ω → SNo (ap x v))
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)))
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)))
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)
Hypothesis H4 : SNo z
Hypothesis H5 : w ∈ ω
Hypothesis H6 : z < ap x w
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
Variable x : set
Variable y : set
Hypothesis H0 : x ∈ setexp real ω
Hypothesis H1 : y ∈ setexp real ω
Hypothesis H2 : ¬ (∃z : set, z ∈ real ∧ (∀w : set, w ∈ ω → ap x w ≤ z ∧ z ≤ ap y w))
Hypothesis H3 : (∀z : set, z ∈ ω → SNo (ap x z))
Hypothesis H4 : (∀z : set, z ∈ ω → SNo (ap y z))
Hypothesis H5 : Subq (Sep (SNoS_ ω) (λz : set ⇒ ∃w : set, w ∈ ω ∧ z < ap x w)) (SNoS_ ω)
Hypothesis H6 : Subq (Sep (SNoS_ ω) (λz : set ⇒ ∃w : set, w ∈ ω ∧ ap y w < z)) (SNoS_ ω)
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)))
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)
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)))
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)
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
Variable x : set
Variable y : set
Hypothesis H0 : x ∈ setexp real ω
Hypothesis H1 : y ∈ setexp real ω
Hypothesis H2 : ¬ (∃z : set, z ∈ real ∧ (∀w : set, w ∈ ω → ap x w ≤ z ∧ z ≤ ap y w))
Hypothesis H3 : (∀z : set, z ∈ ω → SNo (ap x z))
Hypothesis H4 : (∀z : set, z ∈ ω → SNo (ap y z))
Hypothesis H5 : (∀z : set, z ∈ ω → (∀w : set, w ∈ ω → ap x z ≤ ap y w))
Hypothesis H6 : Subq (Sep (SNoS_ ω) (λz : set ⇒ ∃w : set, w ∈ ω ∧ z < ap x w)) (SNoS_ ω)
Hypothesis H7 : Subq (Sep (SNoS_ ω) (λz : set ⇒ ∃w : set, w ∈ ω ∧ ap y w < z)) (SNoS_ ω)
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))
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)))
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)
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)))
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
Variable x : set
Variable y : set
Hypothesis H0 : x ∈ setexp real ω
Hypothesis H1 : y ∈ setexp real ω
Hypothesis H2 : ¬ (∃z : set, z ∈ real ∧ (∀w : set, w ∈ ω → ap x w ≤ z ∧ z ≤ ap y w))
Hypothesis H3 : (∀z : set, z ∈ ω → SNo (ap x z))
Hypothesis H4 : (∀z : set, z ∈ ω → SNo (ap y z))
Hypothesis H5 : (∀z : set, z ∈ ω → (∀w : set, w ∈ ω → ap x z ≤ ap y w))
Hypothesis H6 : Subq (Sep (SNoS_ ω) (λz : set ⇒ ∃w : set, w ∈ ω ∧ z < ap x w)) (SNoS_ ω)
Hypothesis H7 : Subq (Sep (SNoS_ ω) (λz : set ⇒ ∃w : set, w ∈ ω ∧ ap y w < z)) (SNoS_ ω)
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))
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)))
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)))
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)
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
Variable x : set
Variable y : set
Hypothesis H0 : x ∈ setexp real ω
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)
Hypothesis H3 : ¬ (∃z : set, z ∈ real ∧ (∀w : set, w ∈ ω → ap x w ≤ z ∧ z ≤ ap y w))
Hypothesis H4 : (∀z : set, z ∈ ω → SNo (ap x z))
Hypothesis H5 : (∀z : set, z ∈ ω → SNo (ap y z))
Hypothesis H6 : (∀z : set, nat_p z → (∀w : set, w ∈ z → ap x w ≤ ap x z))
Hypothesis H7 : (∀z : set, nat_p z → (∀w : set, w ∈ z → ap y z ≤ ap y w))
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
Variable x : set
Variable y : set
Hypothesis H1 : ordinal (SNoLev x)
Hypothesis H2 : Subq x (SNoElts_ (SNoLev x))
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
Variable x : set
Variable y : set
Hypothesis H0 : Sing (ordsucc (ordsucc Empty)) ∈ x
Hypothesis H2 : ordinal y
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
Variable x : set
Variable y : set
Variable z : set
Hypothesis H0 : Sing (ordsucc (ordsucc Empty)) ∈ y
Hypothesis H1 : ordinal (SNoLev x)
Hypothesis H2 : z ∈ SNoLev x
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
Variable x : set
Variable y : set
Variable z : set
Hypothesis H0 : SNo x
Hypothesis H1 : y ∈ x
Hypothesis H3 : z ∈ Sing (Sing (ordsucc (ordsucc Empty)))
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
Variable x : set
Variable y : set
Variable z : set
Variable w : set
Variable u : set
Hypothesis H1 : SNo_pair x y = SNo_pair z w
Hypothesis H2 : u ∈ x
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
Variable x : set
Hypothesis H0 : CSNo x
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
Variable x : set
Variable y : set
Hypothesis H0 : CSNo x
Hypothesis H1 : CSNo y
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
Variable x : set
Variable y : set
Variable z : set
Hypothesis H0 : CSNo x
Hypothesis H1 : CSNo y
Hypothesis H2 : CSNo z
Hypothesis H3 : CSNo (add_CSNo y z)
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
Variable x : set
Variable y : set
Variable z : set
Hypothesis H0 : CSNo x
Hypothesis H1 : CSNo y
Hypothesis H2 : CSNo z
Hypothesis H3 : CSNo (mul_CSNo y z)
Hypothesis H4 : CSNo (mul_CSNo x y)
Hypothesis H5 : CSNo (mul_CSNo x (mul_CSNo y z))
Hypothesis H6 : CSNo (mul_CSNo (mul_CSNo x y) z)
Hypothesis H7 : SNo (CSNo_Re x)
Hypothesis H9 : SNo (CSNo_Re z)
Hypothesis H10 : SNo (CSNo_Im x)
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
Variable x : set
Variable y : set
Variable z : set
Hypothesis H0 : CSNo x
Hypothesis H2 : CSNo z
Hypothesis H3 : CSNo (mul_CSNo y z)
Hypothesis H4 : CSNo (mul_CSNo x y)
Hypothesis H5 : CSNo (mul_CSNo x (mul_CSNo y z))
Hypothesis H6 : CSNo (mul_CSNo (mul_CSNo x y) z)
Hypothesis H7 : SNo (CSNo_Re x)
Hypothesis H8 : SNo (CSNo_Re y)
Hypothesis H9 : SNo (CSNo_Re z)
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
Variable x : set
Variable y : set
Variable z : set
Hypothesis H0 : CSNo x
Hypothesis H1 : CSNo y
Hypothesis H2 : CSNo z
Hypothesis H3 : CSNo (mul_CSNo y z)
Hypothesis H4 : CSNo (mul_CSNo x y)
Hypothesis H5 : CSNo (mul_CSNo x (mul_CSNo y z))
Hypothesis H7 : SNo (CSNo_Re x)
Hypothesis H8 : SNo (CSNo_Re y)
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
Variable x : set
Variable y : set
Variable z : set
Hypothesis H1 : CSNo y
Hypothesis H2 : CSNo z
Hypothesis H3 : CSNo (add_CSNo y z)
Hypothesis H4 : CSNo (mul_CSNo x y)
Hypothesis H5 : CSNo (mul_CSNo x z)
Hypothesis H6 : CSNo (mul_CSNo x (add_CSNo y z))
Hypothesis H7 : CSNo (add_CSNo (mul_CSNo x y) (mul_CSNo x z))
Hypothesis H8 : SNo (CSNo_Re x)
Hypothesis H9 : SNo (CSNo_Re y)
Hypothesis H10 : SNo (CSNo_Re z)
Hypothesis H11 : SNo (CSNo_Im x)
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
Variable x : set
Variable y : set
Hypothesis H0 : CSNo x
Hypothesis H1 : SNo y
Hypothesis H3 : SNo (CSNo_Re x)
Hypothesis H4 : SNo (CSNo_Im x)
Hypothesis H5 : SNo (y * CSNo_Re x)
Hypothesis H6 : CSNo (y * CSNo_Re x)
Hypothesis H7 : SNo (y * CSNo_Im x)
Hypothesis H8 : CSNo (y * CSNo_Im x)
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
Variable x : set
Hypothesis H0 : x ∈ complex
Hypothesis H1 : x ≠ Empty
Hypothesis H2 : CSNo_Re x ∈ real
Hypothesis H3 : CSNo_Im x ∈ real
Hypothesis H5 : SNo (CSNo_Im x * CSNo_Im x)
Hypothesis H6 : CSNo_Re x * CSNo_Re x + CSNo_Im x * CSNo_Im x ∈ real
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