unfold in_part in |- *.
intros x A B.
case A; case B; simpl in |- *.
intros a pa b pb.
unfold eq_part in |- *; simpl in |- *.
intuition.
elim (H0 x); auto with algebra.
Qed.

(* Specifications: *)
Lemma not_in_part_trans_eq :
forall (x y : E) (A : part_set E),
~ in_part x A -> Equal x y -> ~ in_part y A.