intros A; case A; simpl in |- *.
intros a pa.
intros x y H' H'0; try assumption.
apply pa with x; auto with *.
Qed.

(* Specifications: *)
Lemma diff_add_part :
forall (A B0 B : part_set E) (x : E),
~ in_part x B0 ->
Equal A (add_part B0 x) -> in_part x B -> Equal (diff B0 B) (diff A B).