Section fparts2_def.
Variable E : Setoid.
Definition disjoint (A B : part_set E) := Equal (inter A B) (empty E).
unfold disjoint in |- *.
intros A A' B B' H' H'0 H'1; try assumption.
apply Trans with (inter A B).
auto with *.
auto with *.
Qed.

Lemma empty_not_in :
forall A : part_set E, Equal A (empty E) -> forall x : E, ~ in_part x A.