(* ----Denial of simplified third axiom: *)
ntheorem prove_set_difference_3_simplified:
- ∀Univ:Type.∀X:Univ.∀Y:Univ.∀Z:Univ.
+ (∀Univ:Type.∀X:Univ.∀Y:Univ.∀Z:Univ.
∀a:Univ.
∀b:Univ.
∀c:Univ.
∀difference:∀_:Univ.∀_:Univ.Univ.
∀H0:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (difference (difference X Y) Z) (difference (difference X Z) (difference Y Z)).
∀H1:∀X:Univ.∀Y:Univ.eq Univ (difference X (difference X Y)) (difference Y (difference Y X)).
-∀H2:∀X:Univ.∀Y:Univ.eq Univ (difference X (difference Y X)) X.eq Univ (difference (difference a c) b) (difference (difference a b) c)
+∀H2:∀X:Univ.∀Y:Univ.eq Univ (difference X (difference Y X)) X.eq Univ (difference (difference a c) b) (difference (difference a b) c))
.
-#Univ.
-#X.
-#Y.
-#Z.
-#a.
-#b.
-#c.
-#difference.
-#H0.
-#H1.
-#H2.
-nauto by H0,H1,H2;
+#Univ ##.
+#X ##.
+#Y ##.
+#Z ##.
+#a ##.
+#b ##.
+#c ##.
+#difference ##.
+#H0 ##.
+#H1 ##.
+#H2 ##.
+nauto by H0,H1,H2 ##;
nqed.
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