Definition image : part_set E -> part_set F.

(* Specifications: *)
Lemma image_in : forall (A : part_set E) (y : F), in_part y (image A) -> exists x : E, in_part x A /\ Equal y (f x).
Lemma in_image : forall (A : part_set E) (x : E) (y : F), in_part x A -> Equal y (f x) -> in_part y (image A).
Lemma image_included : forall A B : part_set E, included A B -> included (image A) (image B).
Lemma image_comp : forall A B : part_set E, Equal A B -> Equal (image A) (image B).
Lemma image_in_image : forall (A : part_set E) (x : E), in_part x A -> in_part (f x) (image A).