Definition comp_map_map : MAP E G := Build_Map comp_map_fun_compatible.

(* Specifications: *)
Lemma comp_map_comp : forall (A B C : Setoid) (f f' : MAP A B) (g g' : MAP B C), Equal f f' -> Equal g g' -> Equal (comp_map_map g f) (comp_map_map g' f').
Lemma comp_map_assoc : forall (A B C D : Setoid) (f : MAP A B) (g : MAP B C) (h : MAP C D), Equal (comp_map_map h (comp_map_map g f)) (comp_map_map (comp_map_map h g) f).