Definition invimage : part_set F -> part_set E.

(* Specifications: *)
Lemma invimage_in : forall (A : part_set F) (x : E), in_part x (invimage f A) -> in_part (f x) A.
Lemma in_invimage : forall (A : part_set F) (x : E), in_part (f x) A -> in_part x (invimage f A).
Lemma invimage_included : forall A B : part_set F, included A B -> included (invimage f A) (invimage f B).
Lemma invimage_comp : forall A B : part_set F, Equal A B -> Equal (invimage f A) (invimage f B).
Lemma invimage_image : forall A : part_set F, included (image f (invimage f A)) A.
Lemma image_invimage : forall A : part_set E, included A (invimage f (image f A)).
Lemma invimage_image_invimage : forall A : part_set F, Equal (invimage f (image f (invimage f A))) (invimage f A).
Lemma image_invimage_image : forall A : part_set E, Equal (image f (invimage f (image f A))) (image f A).