Section function_section_1.
Variable F : Ens.
Hypothesis f : is_a_function F.
Variable X Y : Ens.
Hypothesis i : IN (PAIR X Y) F.
Let step1 : IN X (Domain F).
Proof Domain_th1 F X Y i.
Let V := EV F X.
Let step3 : IN (PAIR X V) F.
Proof EV_th1 F X step1.
Let step5 : well_definedness F.
Proof function_well_definedness F f.
Proof step5 X Y V i step3.
End function_section_1.

Theorem function_th1 :
forall (F : Ens) (f : is_a_function F) (X Y : Ens) (i : IN (PAIR X Y) F),
EQ Y (EV F X).