Section A_in_next_A_proof.
Variable A : Ens.
Let N := Next A.
Let step1 : IN A (Singleton A) := Singleton_pr1 A.
Definition A_in_next_A_Out : IN A N := union_th2 A (Singleton A) A step1.
End A_in_next_A_proof.

Theorem anything_in_next_of_itself : forall A : Ens, IN A (Next A).
Proof A_in_next_A_Out.