(** And conversely, (X, A, a) and (Y, B, b) are the only elements of the pair, up to bisimulation. *)

Lemma pairing_elim :
forall (X : Typ1) (A : Rel X) (a : X) (Y : Typ1) (B : Rel Y)
(b : Y) (Z : Typ1) (C : Rel Z) (c : Z),
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).