Proof eval_in_range_Out.
(******)
Section inv_eval_in_domain_proof.
Variable R Y : Ens.
Hypothesis i : IN Y (Range R).
Let X := inverse_EV R Y.
Let step0 : Range_prop R Y.
Proof Range_pr2 R Y i.
Let step1 : IN (PAIR X Y) R.
Proof inverse_EV_pr R Y step0.

Lemma inv_eval_in_domain_Out : IN X (Domain R).
Proof Domain_th1 R X Y step1.