Section Domain_th1_proof.
Variable R : Ens.
Variable X Y : Ens.
Hypothesis i : IN (PAIR X Y) R.
Let step1 : Domain_prop R X.
Proof EXISTS_th1 (fun Y1 : Ens => IN (PAIR X Y1) R) Y i.
Proof Domain_pr1 R X step1.
End Domain_th1_proof.
Proof Domain_th1_Out.
(***** same thing for Range *********)
Definition Range (R : Ens) := REPLACEMENT (Range_prop R).
Proof fun R : Ens => REPLACEMENT_pr1 (Range_prop R) (Range_prop_bounded R).
Proof fun R : Ens => REPLACEMENT_pr2 (Range_prop R) (Range_prop_bounded R).

Theorem inverse_EV_th1 :
forall (R Y : Ens) (i : IN Y (Range R)), IN (PAIR (inverse_EV R Y) Y) R.