intros A B H' x H'0; try assumption.
specialize  H' with (x := x); rename H' into H'1; try exact H'1.
elim H'1; intros H'2 H'3; try exact H'2; clear H'1; auto with algebra.
Qed.
Hint Resolve eq_part_included: algebra.

Lemma empty_included : forall A : part_set E, included (empty E) A.
simpl in |- *; unfold included in |- *; auto with algebra.