red in |- *.
auto with algebra.
Qed.
Definition inj_part_included :
forall A B : part_set E, included A B -> MAP A B.
intros A B H'; try assumption.
red in H'.
apply
(Build_Map (A:=A) (B:=B)
(Ap:=fun x : A => Build_subtype (H' (A x) (subtype_prf (E:=E) (P:=A) x)))).
red in |- *.
simpl in |- *; auto with algebra.
Defined.

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).