simple induction a; intros.
exists 0; intros.
exists 0; intros.
simpl in |- *.
split.
apply cardinal_empty.
apply Trans with (diff (empty E) B); auto with *.
split.
apply cardinal_empty.
apply Trans with (inter (empty E) B); auto with *.
auto with *.
inversion H0.
elim (H B0 B); intros.
case H6; clear H6; intros.
case H6; clear H6; intros.
case H7; clear H7; intros.
case (classic (in_part x B)); intros.
exists x0.
exists (S x1).
split.
apply cardinal_comp_l with (diff B0 B); auto with *.
apply diff_add_part with x; auto with *.
split.
apply cardinal_add with (inter B0 B) x; auto with *.
rewrite H8.
auto with *.
exists (S x0).
exists x1.
split.
apply cardinal_add with (diff B0 B) x; auto with *.
split.
apply cardinal_comp_l with (inter B0 B); auto with *.
apply inter_add_part_not_in with x; auto with *.
rewrite H8; auto with *.
auto with *.
Qed.

Lemma cardinal_union_inter :
forall (A B : part_set E) (a b c : nat),
cardinal A a ->
cardinal B b -> cardinal (inter A B) c -> cardinal (union A B) (a + b - c).