simple induction a.
intros.
apply cardinal_comp_l with (union (empty E) B); auto with *.
apply union_comp; auto with *.
apply Sym.
auto with *.
apply cardinal_comp_l with B; auto with *.
intros.
inversion H0.
simpl in |- *.
apply cardinal_add with (union B0 B) x; auto with *.
apply H; auto with *.
apply disjoint_inclus with (add_part B0 x); auto with *.
apply disjoint_comp with A B; auto with *.
apply union_not_in; auto with *.
apply disjoint_not_in_r with A; auto with *.
apply in_part_comp_r with (add_part B0 x); auto with *.
apply Trans with (union (add_part B0 x) B); auto with *.
unfold add_part in |- *.
apply Trans with (union B0 (union (single x) B)); auto with *.
apply Trans with (union B0 (union B (single x))); auto with *.
Qed.
Hint Resolve cardinal_union_disjoint: algebra.

Lemma in_eq_part :
forall A B : part_set E,
(forall x : E, in_part x A -> in_part x B) ->
(forall x : E, in_part x B -> in_part x A) -> Equal A B.