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.

Theorem composition_bounded_by :
forall F G : Ens,
Bounded_By (Cartesian (Domain F) (Range G)) (Comp_prop F G).