Proof inv_eval_in_domain_Out.
(******** functions ***********)
Definition is_a_relation (R : Ens) :=
forall (X : Ens) (i : IN X R), is_a_pair X.
Section relation_1.
Variable R : Ens.
Hypothesis r : is_a_relation R.
Variable S : Ens.
Hypothesis h : SUB S R.
Variable X : Ens.
Hypothesis i : IN X S.
Let step1 : IN X R.
Proof h X i.

(* Specifications: *)
Lemma relation_1_Out : is_a_pair X.
Proof r X step1.