unfold add_part in |- *.
intros A B0 B x H' H'0 H'1; try assumption.
apply Trans with (inter (union B0 (single x)) B).
auto with *.
apply Trans with (union (inter B0 B) (inter (single x) B)).
apply Trans with (union (inter B0 B) (empty E)).
auto with *.
apply union_comp; auto with *.
apply Sym.
apply in_eq_part.
intros x0 H'2; try assumption.
cut (Equal x x0).
intros H'3; try assumption.
absurd (in_part x0 B).
unfold not in |- *; intro.
unfold not in H'1.
apply H'1.
apply in_part_trans_eq with x0; auto with *.
apply in_part_inter_r with (single x).
auto with *.
cut (in_part x0 (single x)).
auto with *.
apply in_part_inter_l with B.
auto with *.
intros x0 H'2; try assumption.
absurd (in_part x0 (empty E)); auto with *.
auto with *.
auto with *.
Qed.

Lemma cardinal_diff :
forall (a : nat) (A B : part_set E),
cardinal A a ->
exists b : nat,
(exists c : nat,
cardinal (diff A B) b /\ cardinal (inter A B) c /\ a = b + c).