(** And the elimination rule comes easily: *)

Lemma powerset_elim :
forall (X : Typ1) (A : Rel X) (a : X) (Y : Typ1) (B : Rel Y) (b : Y),
ELT Y B b (sum X (X -> Prop)) (POWER X A a) (out X (X -> Prop)) ->
SUB Y B b X A a.