Section Corollaries.
Variable U:Type.
(** UIP implies the injectivity of equality on dependent pairs in Type *)
Definition Inj_dep_pair_on (P : U -> Type) (p : U) (x : P p) :=
forall (y : P p), existT P p x = existT P p y -> x = y.
Definition Inj_dep_pair := forall P p x, Inj_dep_pair_on P p x.

Lemma eq_dep_eq__inj_pair2 : Eq_dep_eq U -> Inj_dep_pair.
Proof (fun eq_dep_eq P p x =>
@eq_dep_eq_on__inj_pair2_on P p x (eq_dep_eq P p x)).