Definition COMP (F G : Ens) := REPLACEMENT (Comp_prop F G).
(******* note that this is the full composition of relations, like composition of correspondences in algebraic geometry ************)

(* Specifications: *)
Theorem COMP_pr1 : forall F G X : Ens, Comp_prop F G X -> IN X (COMP F G).
Proof
fun F G X : Ens =>
REPLACEMENT_pr1 (Comp_prop F G) (composition_bounded F G) X.
Theorem COMP_pr2 : forall F G X : Ens, IN X (COMP F G) -> Comp_prop F G X.
Proof
fun F G X : Ens =>
REPLACEMENT_pr2 (Comp_prop F G) (composition_bounded F G) X.
Theorem COMP_th1 : forall F G : Ens, is_a_relation (COMP F G).
Proof
fun (F G X : Ens) (i : IN X (COMP F G)) =>
Comp_pair F G X (COMP_pr2 F G X i).
Theorem COMP_th2 :
forall F G : Ens, SUB (COMP F G) (Cartesian (Domain F) (Range G)).