Proof union_th3_proof_Out.
Section union_th4_proof.
Variable P : Prop.
Variable A B X : Ens.
Hypothesis i : IN X (union A B).
Hypothesis a : IN X A -> P.
Hypothesis b : IN X B -> P.
Section ut4contra.
Hypothesis contra : ~ P.
Section ut4a.
Hypothesis j : IN X A.
Let step1 : P.
Proof a j.
Lemma  (*Fact*) step2 : False.
Proof contra step1.
End ut4a.
Let step3 : ~ IN X A.
Proof step2.
Section ut4b.
Hypothesis k : IN X B.
Let step4 : P.
Proof b k.
Lemma  (*Fact*) step5 : False.
Proof contra step4.
End ut4b.
Let step6 : ~ IN X B.
Proof step5.
Let step7 : ~ IN X (union A B).
Proof union_th3 A B X step3 step6.
Lemma  (*Fact*) step8 : False.
Proof step7 i.
End ut4contra.

Lemma union_th4_Out : P.
Proof excluded_middle P step8.