Proof relation_1_Out.
Definition well_definedness (F : Ens) :=
forall (X Y Z : Ens) (i : IN (PAIR X Y) F) (j : IN (PAIR X Z) F), EQ Y Z.
Section well_def_1.
Variable F : Ens.
Hypothesis w : well_definedness F.
Variable R : Ens.
Hypothesis h : SUB R F.
Variable X Y Z : Ens.
Hypothesis i : IN (PAIR X Y) R.
Hypothesis j : IN (PAIR X Z) R.
Let step1 : IN (PAIR X Y) F.
Proof h (PAIR X Y) i.
Let step2 : IN (PAIR X Z) F.
Proof h (PAIR X Z) j.

Lemma well_def_1_Out : EQ Y Z.
Proof w X Y Z step1 step2.