Proof doubleton_th4_Out.
Section doubleton_th4a_proof.
Variable X : Ens.
Variable Y : Ens.
Variable Z : Ens.
Hypothesis k : IN Z (Doubleton X Y).
Hypothesis m : ~ EQ Z X.
Let step1 : ~ ~ EQ Z Y := Doubleton_th4 X Y Z k m.
Definition doubleton_th4a_Out : EQ Z Y := excluded_middle (EQ Z Y) step1.
End doubleton_th4a_proof.

Theorem Doubleton_th4_a :
forall (X Y Z : Ens) (k : IN Z (Doubleton X Y)) (m : ~ EQ Z X), EQ Z Y.