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

Lemma not_included_exist : forall (E : Setoid) (A B : part_set E), included B A -> ~ Equal B A -> exists x : E, in_part x A /\ ~ in_part x B.