Section fparts_in_def.
Variable E : Setoid.
(* Add_part. *)
Definition add_part (A : part_set E) (x : E) := union A (single x).

Lemma add_part_comp :
forall (A A' : part_set E) (x x' : E),
Equal A A' -> Equal x x' -> Equal (add_part A x) (add_part A' x').