Parameter
  surjective_image_full :
    forall (E F : Setoid) (f : MAP E F),
    surjective f -> Equal (image_map f) (full F).

Lemma map_not_injective : forall (A B : Setoid) (f : MAP A B) (n m : nat), cardinal (full A) n -> cardinal (full B) m -> m < n -> ~ injective f.