simple induction n.
auto with algebra.
intros.
inversion c.
apply (H0 n0 H1 A B x H4 H6 c).
Qed.

(* Specifications: *)
Lemma cardinal_S :
forall (n : nat) (A B : part_set E) (x : E),
~ in_part x B -> Equal A (add_part B x) -> cardinal A (S n) -> cardinal B n.