(******** ??? 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).

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