Section union_th3_proof.
Variable A : Ens.
Variable B : Ens.
Let U := union A B.
Variable X : Ens.
Hypothesis h : ~ IN X A.
Hypothesis k : ~ IN X B.
Hypothesis m : IN X U.
Record union_th3_property (Y : Ens) : Prop :=
{union_th3_property_in_the_union : IN Y U;
union_th3_property_not_equal_X : ~ EQ X Y}.
Let step1 : Bounded_By U union_th3_property.
Proof
fun (Y : Ens) (h : union_th3_property Y) =>
union_th3_property_in_the_union Y h.
Let b : Bounded union_th3_property.
Proof Bounded_th2 union_th3_property U step1.
Let V := REPLACEMENT union_th3_property.
Let V_pr1 : forall X : Ens, union_th3_property X -> IN X V.
Proof REPLACEMENT_pr1 union_th3_property b.
Let V_pr2 : forall X : Ens, IN X V -> union_th3_property X.
Proof REPLACEMENT_pr2 union_th3_property b.
Section A_sub_V_proof.
Variable Z : Ens.
Hypothesis h2 : IN Z A.
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.
Proof V_pr1 Z step4.