proposition: Lemma betH_distincts : forall A B C, BetH A B C -> A <> B /\ B <> C /\ A <> C.
proof: Proof. intros. assert(HH:= H). assert(HP:= between_comm A B C H). assert (HQ:=HP). apply between_only_one in H. apply between_only_one in HP. split. intro. subst B. contradiction. split. intro. subst B. contradiction. intro. subst C. apply between_diff in HH. tauto. 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