Proof res_to_1_Out.
Section res_to_2.
Variable A F : Ens.
Hypothesis f : is_a_function F.
Hypothesis s : SUB (Domain F) A.
Variable U : Ens.
Hypothesis i : IN U F.
Let X := PR1 U.
Let Y := PR2 U.
Let M := PAIR X Y.
Let step1 : is_a_relation F.
Proof function_is_a_relation F f.
Let step2 : is_a_pair U.
Proof step1 U i.
Let step3 : EQ U M.
Proof PAIR_proj_th1 U step2.
Let step4 : EQ M U.
Proof EQ_symm U M step3.
Let step5 : IN M F.
Proof Extensionality M U F step4 i.
Let step6 : IN X (Domain F).
Proof Domain_th1 F X Y step5.
Let step7 : IN X A.
Proof s X step6.

Lemma res_to_2_Out : IN U (restricted_to A F).
Proof restricted_to_th1 A F U i step7.