Section Range_prop_bounded_proof.
Variable R : Ens.
Let T := Total R.
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.

Lemma Range_prop_bounded_proof_1_Out : IN Y T.
Proof Relation_Total_th2 R X Y step1_1.