Definition PRED := forall X : Typ1, Rel X -> X -> Prop.Definition Compat (P : PRED) : Prop :=
  forall (X : Typ1) (A : Rel X) (a : X) (Y : Typ1) (B : Rel Y) (b : Y),
  EQV X A a Y B b -> P X A a -> P Y B b.Definition SELECT (X : Typ1) (A : Rel X) (a : X) (P : PRED) : 
  Rel (opt X) :=
  fun z' z : opt X =>
  forall E : Prop,
  (forall x x' : X,
   eq (opt X) z (some X x) -> eq (opt X) z' (some X x') -> A x' x -> E) ->
  (forall x' : X,
   eq (opt X) z' (some X x') ->
   eq (opt X) z (none X) -> A x' a -> P X A x' -> E) -> E.

Theorem selection : forall (X : Typ1) (A : Rel X) (a : X) (P : PRED), Compat P -> forall (Y : Typ1) (B : Rel Y) (b : Y), iff (ELT Y B b (opt X) (SELECT X A a P) (none X)) (and (ELT Y B b X A a) (P Y B b)).