proposition: Lemma eq_sig_snd X P (x1 x2:X) H1 H2 (H:exist P x1 H1 = exist P x2 H2) : rew (eq_sig_fst H) in H1 = H2.
proof: Proof. unfold eq_sig_fst, eq_ind. change x2 with (proj1_sig (exist P x2 H2)). change H2 with (proj2_sig (exist P x2 H2)) at 3. destruct H. reflexivity. Defined.