proposition: Lemma betH_dec : forall A B C, A <> B -> B <> C -> A <> C -> BetH A B C \/ ~ BetH A B C.
proof: Proof. intros. induction(colH_dec A B C). assert(HH:=between_one A B C H H1 H0 H2). induction HH. left; assumption. induction H3. apply between_only_one' in H3. spliter. right. intro. apply H4. apply between_comm. assumption. apply between_only_one' in H3. spliter. right. assumption. right. intro. apply H2. apply between_col in H3. spliter. 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