Proof union_th4_Out.
Section union_refl_proof.
Variable A : Ens.
Let U := union A A.
Let step1 : SUB A U.
Proof union_pr1 A A.
Let step2 : SUB A A. Proof SUB_refl A.
Let step3 : SUB U A. Proof union_pr3 A A A step2 step2.

Lemma union_refl_proof_Out : EQ A U.
Proof Build_EQ A U step1 step3.