Module N:=DecidableEqDep(M).
(** Invariance by Substitution of Reflexive Equality Proofs *)
Proof eq_rect_eq_dec eq_dec.
(** Injectivity of Dependent Equality *)
Proof (eq_rect_eq__eq_dep_eq U eq_rect_eq).
(** Uniqueness of Identity Proofs (UIP) *)
Proof (eq_dep_eq__UIP U eq_dep_eq).
(** Uniqueness of Reflexive Identity Proofs *)

Lemma UIP_refl : forall (x:U) (p:x = x), p = eq_refl x.