unfold included, disjoint in |- *.
intros A B C H' H'0; try assumption.
apply not_in_empty.
unfold not in |- *; intros.
cut (in_part x (inter B C)).
generalize (empty_not_in (A:=inter B C) H'0).
unfold not in |- *; intros.
apply H0 with (x := x).
auto with *.
auto with *.
apply in_part_inter.
apply H'.
apply in_part_inter_l with C.
auto with *.
apply in_part_inter_r with A.
auto with *.
Qed.

Lemma included_add_part :
forall (A : part_set E) (x : E), included A (add_part A x).