Proof fe_Out.
(************* restricted_to ***********)
Definition restricted_to (A F : Ens) :=
Set_Of F (fun M : Ens => IN (PR1 M) A).

Theorem restricted_to_th1 :
forall (A F U : Ens) (i : IN U F) (j : IN (PR1 U) A),
IN U (restricted_to A F).