(* Equality of the images... *)
(* ... implies dependent equality: *)
Proof fun X Y Z x1 x2 y1 y2 R1 R2 z1 z2 H P p => H (fun f => f P) p.
(** This kind of equality is actually sufficent for proving the expected delocation property (cf Lemma FOO_deloc below). *)
(** ** Building the edge relation of the super-powerset **)
(** The edge relation (FOO X Y Z) : (Rel (foo X Y Z)) of the super-powerset of the family of small types Z : X->Y->Typ1 is defined by the following clauses:  1.  (Delocation of pointed graph structures:)  For any x:X and y:Y, for any relation R:(Rel (Z x y)) and for any pair of vertices z,z':(Z x y),  (R z' z) implies (FOO X Y Z (foo_in X Y Z x y R z') (foo_in X Y Z x y R z))  2.  (Connecting vertices to the new root:)  For any x:X and y:Y, for any relation R:(Rel (Z x y)) and for any vertex z:(Z x y), then (FOO X Y Z (foo_in X Y Z x y R z) (foo_rt X Y Z)).  Formally, we define this relation as follows: *)
Definition FOO (X Y : Typ1) (Z : X -> Y -> Typ1) (u' u : foo X Y Z) : Prop :=
forall E : Prop,
(forall (x : X) (y : Y) (R : Rel (Z x y)) (z' z : Z x y),
eq (foo X Y Z) u' (foo_in X Y Z x y R z') ->
eq (foo X Y Z) u (foo_in X Y Z x y R z) -> R z' z -> E) ->
(forall (x : X) (y : Y) (R : Rel (Z x y)) (z' : Z x y),
eq (foo X Y Z) u' (foo_in X Y Z x y R z') ->
eq (foo X Y Z) u (foo_rt X Y Z) -> E) -> E.
(** Both clauses are given by the following constructors: *)

Lemma FOO_in :
forall (X Y : Typ1) (Z : X -> Y -> Typ1) (x : X) (y : Y)
(R : Rel (Z x y)) (z' z : Z x y),
R z' z -> FOO X Y Z (foo_in X Y Z x y R z') (foo_in X Y Z x y R z).