Definition Inter' (E : Ens) : Ens :=
  Comp (Union E) (fun e : Ens => forall a : Ens, IN a E -> IN e a).

(* Specifications: *)
Theorem IN_Inter'_all : forall E E' : Ens, IN E' (Inter' E) -> forall E'' : Ens, IN E'' E -> IN E' E''.
Theorem all_IN_Inter' : forall E E' E'' : Ens, IN E'' E -> (forall E'' : Ens, IN E'' E -> IN E' E'') -> IN E' (Inter' E).