intros E A B; simpl in |- *.
unfold eq_part in |- *; simpl in |- *.
intros H' H'0; try assumption.
cut
(ex
(fun x : E =>
~ ((in_part x B -> in_part x A) /\ (in_part x A -> in_part x B)))).
intros H'1; try assumption.
elim H'1; intros x E0; try exact E0; clear H'1.
exists x; split; [ try assumption | idtac ].
red in E0.
apply NNPP.
red in |- *; intros H'1; try exact H'1.
lapply E0; [ intros H'2; apply H'2; clear E0 | clear E0 ].
split; [ idtac | intros H'2; try assumption ].
intros H'2; try assumption.
apply H'; auto with *.
absurd (in_part x A); auto with *.
red in |- *.
red in E0.
intros H'1; try assumption.
lapply E0; [ intros H'2; apply H'2; clear E0 | clear E0 ].
split; [ try assumption | idtac ].
intros H'2; try assumption.
apply H'; auto with *.
auto with *.
apply
not_all_ex_not
with
(P := fun x : E =>
(in_part x B -> in_part x A) /\ (in_part x A -> in_part x B));
auto with *.
Qed.

Lemma cardinal_included :
forall (E : Setoid) (A : part_set E) (n : nat),
cardinal A n ->
forall B : part_set E,
included B A -> exists m : nat, cardinal B m /\ m <= n.