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) *)

Lemma UIP : forall (x y:U) (p1 p2:x = y), p1 = p2.