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.
Proof excluded_middle P step8.
End union_th4_proof.

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.