proposition: Lemma betH_not_congH : forall A B C, BetH A B C -> ~ CongH A B A C.
proof: Proof. intros. intro. apply betH_expand in H. spliter. apply H2. assert(exists P : Point, BetH B A P). apply(between_out B A); auto. destruct H5 as [P]. apply between_comm in H5. assert(HB:BetH P A C). eapply (betH_trans2 _ _ B); auto. apply construction_uniqueness with P A; auto. 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