Section doubleton_th2_proof.
Variable X : Ens.
Variable Y : Ens.
Let D := Doubleton X Y.
Let iYsY : IN Y (Singleton Y) := Singleton_pr1 Y.
Definition doubleton_th2_Out : IN Y D :=
union_th2 (Singleton X) (Singleton Y) Y iYsY.
End doubleton_th2_proof.

Theorem Doubleton_th2 : forall X Y : Ens, IN Y (Doubleton X Y).
Proof doubleton_th2_Out.