Section Union_of_part.
Variable E : Setoid.
Comments "We define the union of a part of" (part_set E).
Variable P : part_set (part_set E).
Definition union_part : part_set E.
apply
(Build_Predicate
(Pred_fun:=fun x : E => exists A : part_set E, in_part A P /\ in_part x A)).
red in |- *.
intros x y H' H'0; try assumption.
elim H'; intros A E0; elim E0; clear H'.
intros H' H'1; try assumption.
exists A; split; [ try assumption | idtac ].
apply in_part_comp_l with x; auto with algebra.
Defined.

Lemma union_part_prop :
forall x : E,
in_part x union_part -> exists A : part_set E, in_part A P /\ in_part x A.