proposition: Lemma union_comp : forall A A' B B' : part_set E, Equal A A' -> Equal B B' -> Equal (union A B) (union A' B').
proof: Proof. 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.
imports: Parts