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.

Theorem function_evaluation :
forall (F X Y : Ens) (f : is_a_function F) (i : IN (PAIR X Y) F),
EQ Y (EV F X).