Proof
fun (A F : Ens) (f : is_a_function F) (X : Ens) (i : IN X A)
(j : IN X (Domain F)) =>
restricted_to_th8 A F f X (intersection_th3 A (Domain F) X i j).
(******** the empty function ******)
Section emptyfn_1.
Variable X : Ens.
Hypothesis i : IN X Zero.
Let step1 : False.
Proof EmptySet_th1 X i.

Lemma emptyfn1_Out : is_a_pair X.
Proof false_implies_everything (is_a_pair X) step1.