Section comp_prop_3.
Variable F G X : Ens.
Hypothesis c : Comp_prop F G X.
Let step1 : is_a_pair X.
Proof Comp_pair F G X c.
Let step2 : IN (PR1 X) (Domain F).
Proof Comp_prop_th1 F G X c.
Let step3 : IN (PR2 X) (Range G).
Proof Comp_prop_th2 F G X c.
Let step4 : in_cartesian (Domain F) (Range G) X.
Proof Build_in_cartesian (Domain F) (Range G) X step1 step2 step3.
Proof Cartesian_pr1 (Domain F) (Range G) X step4.
End comp_prop_3.
Proof comp_prop_3_Out.
Proof Comp_prop_th3.
Proof
fun F G : Ens =>
Bounded_th2 (Comp_prop F G) (Cartesian (Domain F) (Range G))
(composition_bounded_by F G).
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 ************)
Proof
fun F G X : Ens =>
REPLACEMENT_pr1 (Comp_prop F G) (composition_bounded F G) X.
Proof
fun F G X : Ens =>
REPLACEMENT_pr2 (Comp_prop F G) (composition_bounded F G) X.
Proof
fun (F G X : Ens) (i : IN X (COMP F G)) =>
Comp_pair F G X (COMP_pr2 F G X i).
Proof
fun (F G X : Ens) (i : IN X (COMP F G)) =>
Comp_prop_th3 F G X (COMP_pr2 F G X i).
Proof
fun (F G X : Ens) (i : IN X (COMP F G)) =>
Comp_INTER_th1 F G X (COMP_pr2 F G X i).

Theorem COMP_th4 :
forall (F G X : Ens) (i : IN X (COMP F G)),
IN (PAIR (Comp_INTER F G X) (PR2 X)) G.