proposition: Lemma out2_out1 : forall A B C D, C <> D -> BetH A B C -> BetH A B D -> BetH A C D \/ BetH A D C.
proof: Proof. intros. apply betH_expand in H0. apply betH_expand in H1. spliter. assert(ColH A C D). apply(colH_trans A B); Col. apply between_one in H10; auto. induction H10. intuition. induction H10. apply between_comm in H10. intuition. assert(BetH B A D /\ BetH C B D). apply(betH_trans0 C B A D); Bet. spliter. apply between_only_one' in H1. spliter. contradiction. Qed.
imports: Setoid Coq.Classes.Morphisms Coq.Classes.Morphisms_Prop GeoCoq.Axioms.gupta_inspired_variant_axioms GeoCoq.Meta_theory.Models.gupta_inspired_to_tarski GeoCoq.Tactics.Coinc.tactics_axioms GeoCoq.Tactics.Coinc.ColR GeoCoq.Meta_theory.Dimension_axioms.upper_dim_3 GeoCoq.Meta_theory.Parallel_postulates.parallel_postulates GeoCoq.Axioms.hilbert_axioms