Definition restrict : forall A : part_set E, MAP A F.

(* Specifications: *)
Lemma restrict_prop : forall (A : part_set E) (x : E) (p : in_part x A), Equal (restrict A (Build_subtype (subtype_elt:=x) p)) (f x).
Lemma restrict_prop_in_part : forall (A : part_set E) (x : A), Equal (restrict A x) (f (A x)).