(** 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.