simpl in |- *; unfold eq_part, included in |- *.
intros A A' B B' H' H'0 H'1 x H'2; try assumption.
elim (H'0 x); intros H'4 H'5; apply H'4.
lapply (H'1 x); [ intros H'6; apply H'6 | idtac ].
elim (H' x); intros H'6 H'7; apply H'7; auto with algebra.
Qed.

Lemma eq_part_included : forall A B : part_set E, Equal A B -> included A B.
simpl in |- *; unfold eq_part, included in |- *.