Section A_sub_V_proof1.
Hypothesis h3 : EQ X Z.
Let step1_1 : IN X A.
Proof Extensionality X Z A h3 h2.
Definition A_sub_V_proof1_Out : False := h step1_1.
End A_sub_V_proof1.
Let step2 : ~ EQ X Z.
Proof A_sub_V_proof1_Out.
Let step3 : IN Z U.
Proof union_th1 A B Z h2.
Let step4 : union_th3_property Z.
Proof Build_union_th3_property Z step3 step2.

Lemma A_sub_V_proof_Out : IN Z V.