intros A.
case A.
intros a pa.
simpl in |- *.
unfold eq_part, empty in |- *; simpl in |- *.
intuition eauto.
Qed.
Hint Immediate not_in_empty: algebra.

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