Remark nu_left_inv_on (y:A) (u:x = y) : nu_inv (nu u) = u.

Theorem inj_right_pair A (eq_dec : forall x y : A, x = y \/ x <> y) (x : A) : forall (P:A -> Prop) (y y':P x), ex_intro P x y = ex_intro P x y' -> y = y'.