Proof when_given_two_distinct_elements_Out.
(***************************************)
(**********     Doubletons  ************)
(***************************************)
Definition Doubleton (X Y : Ens) := union (Singleton X) (Singleton Y).
Section doubleton_th1_proof.
Variable X : Ens.
Variable Y : Ens.
Let D := Doubleton X Y.
Let iXsX : IN X (Singleton X) := Singleton_pr1 X.
Definition doubleton_th1_Out : IN X D :=
union_th1 (Singleton X) (Singleton Y) X iXsX.
End doubleton_th1_proof.

(* Specifications: *)
Theorem Doubleton_th1 : forall X Y : Ens, IN X (Doubleton X Y).
Proof doubleton_th1_Out.