Proof total_and_pairs_1_OutB.
(******** ??? also prove that if m: M is a pair and M in R then PAIR_proj1 M m in Total R etc *********)
(************ Domain and Range ***************)
Definition Domain_prop (R X : Ens) := EXISTS (fun Y : Ens => IN (PAIR X Y) R).
Definition Range_prop (R Y : Ens) := EXISTS (fun X : Ens => IN (PAIR X Y) R).
(******** we use CHOICE to define evaluations ******)
Definition EV (R X : Ens) := CHOICE (fun Y : Ens => IN (PAIR X Y) R).

Theorem EV_pr :
forall (R X : Ens) (p : Domain_prop R X), IN (PAIR X (EV R X)) R.