Instance Included_proper (V: Type): Proper (Same_set ==> Same_set ==> iff) (@Included V).

Lemma Included_refl: forall A P, @Included A P P.
Definition app_same_set {A: Type} {P Q: Ensemble A} (H: Same_set P Q) (x: A): P x <-> Q x := proj1 (Same_set_spec A P Q) H x.