Section fparts_in_def.
Variable E : Setoid.
(* Add_part. *)
Definition add_part (A : part_set E) (x : E) := union A (single x).
unfold add_part in |- *; auto with algebra.
Qed.
Hint Resolve add_part_comp: algebra.
simpl in |- *; auto with algebra.
Qed.
Hint Resolve add_part_in: algebra.
unfold add_part in |- *.
intros.
apply Trans with (union A (union (single x) (single y))); auto with algebra.
apply Trans with (union A (union (single y) (single x))); auto with algebra.
Qed.
Hint Immediate add_part_com: algebra.

Lemma add_in :
forall (A : part_set E) (x : E), in_part x A -> Equal (add_part A x) A.