Section res_to_3.
Variable A F : Ens.
Hypothesis f : is_a_function F.
Variable X : Ens.
Hypothesis i : IN X (intersection A (Domain F)).
Let Y := EV F X.
Let Z := EV (restricted_to A F) X.
Let step1 : IN X (Domain F).
Proof intersection_th2 A (Domain F) X i.
Let step2 : IN X A.
Proof intersection_th1 A (Domain F) X i.
Let step3 : EQ (intersection A (Domain F)) (Domain (restricted_to A F)).
Proof restricted_to_th5 A F.
Let step4 : IN X (Domain (restricted_to A F)).
Proof
EQ_forwards (intersection A (Domain F)) (Domain (restricted_to A F)) step3
X i.
Let step5 : IN (PAIR X Y) F.
Proof EV_th1 F X step1.
Let step6 : IN (PAIR X Z) (restricted_to A F).
Proof EV_th1 (restricted_to A F) X step4.
Let step7 : IN (PAIR X Z) F.
Proof restricted_to_th4 A F (PAIR X Z) step6.
Proof function_well_definedness F f X Y Z step5 step7.
End res_to_3.

Theorem restricted_to_th8 :
forall (A F : Ens) (f : is_a_function F) (X : Ens)
(i : IN X (intersection A (Domain F))),
EQ (EV F X) (EV (restricted_to A F) X).