proposition: Lemma lt2_per2__lt : forall A B C A' B' C', Per A B C -> Per A' B' C' -> Lt A B A' B' -> Lt B C B' C' -> Lt A C A' C'.
proof: Proof. intros. destruct (eq_dec_points B C). subst C. apply lt_transitivity with A' B'; auto. assert_diffs. apply lt_comm, l11_46; Perp. apply lt_comm in H1. assert (HC0 := H2). destruct HC0 as [[C0 []] HNCong]. assert (Per A' B' C0). assert_diffs; apply per_col with C'; Col. apply lt_transitivity with A' C0. apply lt_comm, cong_lt_per2__lt with B B'; Cong; Perp. apply cong_lt_per2__lt with B' B'; Cong. apply (cong2_lt__lt B C B' C'); Cong. Qed.
imports: GeoCoq.Tarski_dev.Ch10_line_reflexivity_2