Section function_evaluation_proof.
Variable F X Y : Ens.
Hypothesis f : is_a_function F.
Hypothesis i : IN (PAIR X Y) F.
Let Z := EV F X.
Let step1 : IN X (Domain F).
Proof Domain_th1 F X Y i.
Let step2 : IN (PAIR X Z) F.
Proof EV_th1 F X step1.
Proof function_well_definedness F f X Y Z i step2.
End function_evaluation_proof.
Proof fe_Out.
(************* restricted_to ***********)
Definition restricted_to (A F : Ens) :=
Set_Of F (fun M : Ens => IN (PR1 M) A).
Proof
fun (A F U : Ens) (i : IN U F) (j : IN (PR1 U) A) =>
Set_Of_th1 F (fun M : Ens => IN (PR1 M) A) U i j.
Proof
fun (A F U : Ens) (i : IN U (restricted_to A F)) =>
Set_Of_th2 F (fun M : Ens => IN (PR1 M) A) U i.
Proof
fun (A F U : Ens) (i : IN U (restricted_to A F)) =>
Set_Of_th3 F (fun M : Ens => IN (PR1 M) A) U i.

Theorem restricted_to_th4 : forall A F : Ens, SUB (restricted_to A F) F.