(** By collecting the last three lemmas, we obtain the desired result: *)

Theorem pairing :
forall (X : Typ1) (A : Rel X) (a : X) (Y : Typ1) (B : Rel Y)
(b : Y) (Z : Typ1) (C : Rel Z) (c : Z),
iff (ELT Z C c (sum X Y) (PAIR X A a Y B b) (out X Y))
(or (EQV Z C c X A a) (EQV Z C c Y B b)).