proposition: Lemma betH_outH2__betH: forall A B C A' C', BetH A B C -> outH B C C' -> outH B A A' -> BetH A' B C'.
proof: Proof. intros. destruct H0. - destruct H1. assert (BetH A B C') by (apply (betH_trans A B C C');auto). apply between_comm. apply (betH_trans C' B A A'); auto using between_comm. destruct H1. assert (BetH C B A') by (apply between_comm; apply (betH_trans0 A A' B C);auto using between_comm). apply (betH_trans A' B C C');auto using between_comm. spliter;subst. apply between_comm; apply (betH_trans C' C B A');auto using between_comm. - destruct H0. destruct H1. assert (BetH A B C'). apply between_comm; apply (betH_trans0 C C' B A);auto using between_comm. apply between_comm; apply (betH_trans C' B A A');auto using between_comm. destruct H1. assert (BetH A B C'). apply between_comm; apply (betH_trans0 C C' B A);auto using between_comm. apply (betH_trans0 A A' B C');auto using between_comm. spliter;subst. apply between_comm;apply (betH_trans0 C C' B A');auto using between_comm. spliter;subst. destruct H1. apply (betH_trans A' A B C');auto using between_comm. destruct H1. apply (betH_trans0 A A' B C');auto using between_comm. spliter;subst. assumption. 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