Definition union_part : part_set E.

(* Specifications: *)
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.
Lemma union_part_prop_rev : forall A : part_set E, in_part A P -> forall x : E, in_part x A -> in_part x union_part.
Lemma union_part_included : forall A : part_set E, in_part A P -> included A union_part.
Lemma union_part_upper_bound : forall Y : part_set E, (forall A : part_set E, in_part A P -> included A Y) -> included union_part Y.
Lemma not_inpart_comp_r : forall (E : Setoid) (x : E) (A B : part_set E), ~ in_part x A -> Equal A B -> ~ in_part x B.
Theorem part_set_is_strictly_greater_than_set1 : False.
Theorem part_set_is_strictly_greater_than_set : forall (E : Setoid) (f : MAP E (part_set E)), ~ surjective f.