Definition rel_compatible (R : relation E) : Prop :=
  forall x x' y y' : E,
  Equal x x' -> Equal y y' -> app_rel R x y -> app_rel R x' y'.Record Relation : Type := 
  {Rel_fun :> relation E; Rel_compatible_prf : rel_compatible Rel_fun}.Record Relation : Type := 
  {Rel_fun :> relation E; Rel_compatible_prf : rel_compatible Rel_fun}.Record Relation : Type := 
  {Rel_fun :> relation E; Rel_compatible_prf : rel_compatible Rel_fun}.

(* Specifications: *)
Lemma Rel_comp : forall (R : Relation) (x x' y y' : E), Equal x x' -> Equal y y' -> app_rel R x y -> app_rel R x' y'.