Section A_sub_next_A_proof.
Variable A : Ens.
Let N := Next A.
(* = (union A (Singleton A))  *)
Definition A_sub_next_A_Out : SUB A N := union_pr1 A (Singleton A).
End A_sub_next_A_proof.

Theorem anything_sub_next_of_itself : forall A : Ens, SUB A (Next A).
Proof A_sub_next_A_Out.