Proof doubleton_th4a_Out.
Section doubleton_th4b_proof.
Variable X : Ens.
Variable Y : Ens.
Variable Z : Ens.
Hypothesis k : IN Z (Doubleton X Y).
Hypothesis n : ~ EQ Z Y.
Let step1 : ~ ~ EQ Z X := fun m : ~ EQ Z X => Doubleton_th4 X Y Z k m n.
Definition doubleton_th4b_Out : EQ Z X := excluded_middle (EQ Z X) step1.
End doubleton_th4b_proof.

Theorem Doubleton_th4_b :
forall (X Y Z : Ens) (k : IN Z (Doubleton X Y)) (n : ~ EQ Z Y), EQ Z X.