Definition B_sub_V_proof_Out : IN Z V := V_pr1 Z step4.

(* Specifications: *)
Lemma union_th3_proof_Out : False.
Theorem union_th3 : forall (A B X : Ens) (h : ~ IN X A) (k : ~ IN X B), ~ IN X (union A B).
Lemma (*Fact*) step2 : False.
Lemma (*Fact*) step5 : False.
Lemma (*Fact*) step8 : False.
Lemma union_th4_Out : P.
Theorem union_th4 : forall (P : Prop) (A B X : Ens) (i : IN X (union A B)) (a : IN X A -> P) (b : IN X B -> P), P.
Lemma union_refl_proof_Out : EQ A U.
Theorem union_refl : forall A : Ens, EQ A (union A A).
Lemma union_symm_proof_Out : EQ U V.
Theorem union_symm : forall A B : Ens, EQ (union A B) (union B A).