Qed.
Hint Resolve full_prop: algebra.
Definition full_to_set : MAP (full E) E.
apply (Build_Map (Ap:=fun x : full E => full E x)).
red in |- *.
intros x y; try assumption.
elim x.
elim y.
simpl in |- *.
unfold subtype_image_equal in |- *.
simpl in |- *; auto with algebra.
Defined.
Definition set_to_full : MAP E (full E).
apply
(Build_Map (A:=E) (B:=full E)
(Ap:=fun x : E =>
Build_subtype (E:=E) (P:=full E) (subtype_elt:=x) (full_prop x))).
red in |- *.
simpl in |- *; auto with algebra.
Defined.

(* Specifications: *)
Lemma set_full_set : Equal (comp_map_map full_to_set set_to_full) (Id E).
simpl in |- *; auto with algebra.