Definition curry (f : MAP (cart E F) G) (x : E) (y : F) := f (couple x y).Definition fun2_compatible (f : E -> F -> G) :=
  forall (x x' : E) (y y' : F),
  Equal x x' -> Equal y y' -> Equal (f x y) (f x' y').Definition uncurry :
  forall f : E -> F -> G, fun2_compatible f -> MAP (cart E F) G.Definition map_proj1 : MAP E F := comp_map_map (proj1_map F G) f.Definition map_proj2 : MAP E G := comp_map_map (proj2_map F G) f.Definition map_couple : MAP E F -> MAP E G -> MAP E (cart F G).

(* Specifications: *)
Lemma map_couple_proj_prop : Equal (map_couple map_proj1 map_proj2) f.