unfold union in |- *; simpl in |- *.
unfold eq_part in |- *; simpl in |- *.
intros A A' B B' H' H'0 x; split; [ intros H'1; try assumption | idtac ].
elim H'1; [ intros H'2; try exact H'2; clear H'1 | intros H'2; clear H'1 ].
left; try assumption.
elim (H' x); intros H'3 H'4; lapply H'3;
[ intros H'5; try exact H'5; clear H'3 | clear H'3 ].
auto with algebra.
right; try assumption.
elim (H'0 x); intros H'3 H'4; lapply H'3;
[ intros H'5; try exact H'5; clear H'3 | clear H'3 ].
auto with algebra.
intros H'1; try assumption.
elim H'1; [ intros H'2; try exact H'2; clear H'1 | intros H'2; clear H'1 ].
left; try assumption.
elim (H' x); intros H'3 H'4; lapply H'4;
[ intros H'5; try exact H'5; clear H'4 | clear H'4 ].
auto with algebra.
right; try assumption.
elim (H'0 x); intros H'3 H'4; lapply H'4;
[ intros H'5; try exact H'5; clear H'4 | clear H'4 ].
auto with algebra.
Qed.

Lemma union_assoc :
forall A B C : part_set E, Equal (union A (union B C)) (union (union A B) C).