Definition inj_part_included :
  forall A B : part_set E, included A B -> MAP A B.

(* Specifications: *)
Lemma inj_part_included_prop : forall (A B : part_set E) (p : included A B) (x : A), Equal (B (inj_part_included p x)) (A x).
Lemma inj_part_included_injective : forall (A B : part_set E) (p : included A B), injective (inj_part_included p).