Qed.
Hint Resolve empty_included full_included: algebra.
Definition inj_part : forall A : part_set E, MAP A E.
intros A; try assumption.
apply (Build_Map (Ap:=fun x : A => subtype_elt x)).
red in |- *.
auto with algebra.
Defined.

Lemma inj_part_injective : forall A : part_set E, injective (inj_part A).
intros A; try assumption.