proposition: Lemma out2_out : forall A B C D, C <> D -> BetH A B C -> BetH A B D -> BetH B C D \/ BetH B D C.
proof: Proof. intros. apply betH_expand in H0. apply betH_expand in H1. spliter. assert(ColH B C D). apply (colH_trans A B); Col. apply between_one in H10; auto. induction H10. left; auto. induction H10. right. Bet. assert(ColH A C D). apply (colH_trans A B); Col. apply between_one in H11; auto. induction H11. assert(BetH B C D /\ BetH A B D). apply betH_trans0; assumption. Bet. spliter. apply between_only_one' in H12. spliter. contradiction. induction H11. assert(BetH B D C /\ BetH A B C). apply betH_trans0; Bet. Bet. spliter. apply between_only_one' in H12. spliter. apply between_comm in H10. contradiction. assert(BetH B A C /\ BetH D B C). apply betH_trans0; Bet. Bet. spliter. apply between_only_one' in H12. 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