intros A B.
case A; case B; simpl in |- *.
intros a pa b pb.
unfold eq_part, add_part, union, single in |- *; simpl in |- *.
intros.
elim (classic (Equal x x0)); intros.
split; intros.
apply pa with x; auto with algebra.
cut (b x).
intro.
absurd (b x); auto with algebra.
apply pb with x0; auto with algebra.
cut (a x).
intro.
absurd (a x); auto with algebra.
apply pa with x0; auto with algebra.
elim (H1 x0); intros.
split; intros.
lapply H3.
intros.
elim H6; intros.
auto with algebra.
absurd (Equal x0 x); auto with algebra.
left; auto with algebra.
lapply H4; intros.
elim H6; intros.
auto with algebra.
absurd (Equal x0 x); auto with algebra.
left.
auto with algebra.
Qed.
(* Minus_part. *)
Definition minus_part (A : part_set E) (x : E) := diff A (single x).

Lemma minus_part_comp :
forall (A A' : part_set E) (x x' : E),
Equal A A' -> Equal x x' -> Equal (minus_part A x) (minus_part A' x').