proposition: Lemma congH_colH_betH: forall A B I, A<>B -> A<>I -> B<>I -> CongH I A I B -> ColH I A B -> BetH A I B.
proof: Proof. intros. apply between_one in H3;auto. decompose [or] H3;clear H3. - exfalso. apply (betH_not_congH I A B);auto. - exfalso. assert (T:=betH_not_congH I B A). apply T; auto using between_comm, congH_sym. - 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