Definition pures_bound2 n pi2 :=
      pures_bound n (fst pi2) /\ pures_bound n (snd pi2).

(* Specifications: *)
Lemma finite_unimage : forall {U V} (X:Ensemble U) (f:U -> V), injective f -> Finite (Im X f) -> Finite X.
Lemma Power_set_finite: forall U (F: Ensemble U), Finite F -> Finite (Power_set F).