Proof Range_th1_Out.
(******* now say EV R X is in Range R and inversely ????? *****)
Section eval_in_range_proof.
Variable R X : Ens.
Hypothesis i : IN X (Domain R).
Let Y := EV R X.
Let step1 : IN (PAIR X Y) R.
Proof EV_th1 R X i.

Lemma eval_in_range_Out : IN Y (Range R).
Proof Range_th1 R X Y step1.