simpl in |- *.
unfold subtype_image_equal in |- *.
simpl in |- *.
unfold image_map in |- *.
intros y; try assumption.
elim y.
intros x' subtype_prf; try assumption.
elim subtype_prf.
intros x H'; try assumption.
exists x; try assumption.
elim H'; intros H'0 H'1; try exact H'1; clear H'.
Qed.
Let surj_part_image_fun : forall A : part_set E, A -> image A.
intros A x; try assumption.
elim x.
intros x' H'; try assumption.
cut (in_part (f x') (image A)).
intros H'0; try assumption.
simpl in |- *.
apply (Build_subtype (P:=image A) (subtype_elt:=f x') H'0).
auto with algebra.
Defined.
Definition surj_part_image : forall A : part_set E, MAP A (image A).
intros A; try assumption.
apply (Build_Map (Ap:=surj_part_image_fun (A:=A))).
red in |- *.
simpl in |- *.
unfold subtype_image_equal in |- *.
intros x y; try assumption.
elim x.
elim y.
simpl in |- *.
auto with algebra.
Defined.

(* Specifications: *)
Lemma surj_part_image_surjective :
forall A : part_set E, surjective (surj_part_image A).