intros E F f n H' H'0 H'1; try assumption.
apply cardinal_image_equal_injective with n; auto with *.
apply cardinal_comp with (full F) n; auto with *.
apply Sym; auto with *.
Qed.

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.