proposition: Lemma eq_sigT_snd X P (x1 x2:X) H1 H2 (H:existT P x1 H1 = existT P x2 H2) : rew (eq_sigT_fst H) in H1 = H2.
proof: Proof. unfold eq_sigT_fst. change x2 with (projT1 (existT P x2 H2)). change H2 with (projT2 (existT P x2 H2)) at 3. destruct H. reflexivity. Defined.