Section Range_prop_bounded_proof_1.
Variable Y : Ens.
Hypothesis p : Range_prop R Y.
Let X := inverse_EV R Y.
Let step1_1 : IN (PAIR X Y) R.
Proof inverse_EV_pr R Y p.
Proof Relation_Total_th2 R X Y step1_1.
End Range_prop_bounded_proof_1.
Let step1 : Bounded_By T (Range_prop R).
Proof Range_prop_bounded_proof_1_Out.
Proof Bounded_th2 (Range_prop R) T step1.
End Range_prop_bounded_proof.
Proof Range_prop_bounded_Out.
(****** now we define domain and EV *****)
Definition Domain (R : Ens) := REPLACEMENT (Domain_prop R).
Proof fun R : Ens => REPLACEMENT_pr1 (Domain_prop R) (Domain_prop_bounded R).
Proof fun R : Ens => REPLACEMENT_pr2 (Domain_prop R) (Domain_prop_bounded R).

Theorem EV_th1 :
forall (R X : Ens) (i : IN X (Domain R)), IN (PAIR X (EV R X)) R.