simpl in |- *; auto with algebra.
intros A x H'; try assumption.
exists x; split; [ try assumption | idtac ]; auto with algebra.
Qed.
Hint Resolve image_in_image: algebra.
Definition image_map := image (full E).
Let surj_set_image_fun : E -> image_map.
intros x; try assumption.
unfold image_map in |- *.
simpl in |- *.
cut (in_part (f x) (image (full E))).
intros H'; try assumption.
apply (Build_subtype (P:=image (full E)) (subtype_elt:=f x) H').
auto with algebra.
Defined.
Definition surj_set_image : MAP E image_map.
apply (Build_Map (Ap:=surj_set_image_fun)).
red in |- *.
simpl in |- *.
unfold subtype_image_equal in |- *.
simpl in |- *.
auto with algebra.
Defined.

Lemma surj_set_image_surjective : surjective surj_set_image.
red in |- *.