Scheme eq_indd := Induction for eq Sort Prop.
(** Equivalent definition of dependent equality as a dependent pair of equalities *)
Inductive eq_dep1 (p:U) (x:P p) (q:U) (y:P q) : Prop :=
eq_dep1_intro : forall h:q = p, x = rew h in y -> eq_dep1 p x q y.

(* Specifications: *)
Lemma eq_dep1_dep :
forall (p:U) (x:P p) (q:U) (y:P q), eq_dep1 p x q y -> eq_dep p x q y.
Lemma eq_dep_dep1 :
forall (p q:U) (x:P p) (y:P q), eq_dep p x q y -> eq_dep1 p x q y.