proposition: Lemma same_side_prime_not_colH : forall A B C D, same_side' A B C D -> ~ ColH A C D /\ ~ ColH B C D.
proof: Proof. intros. unfold same_side' in *. spliter. elim (line_existence C D H). intros l [HL1 HL2]. assert (same_side A B l) by auto. apply same_side_not_incid in H1. spliter. split. assert (~ ColH C D A). apply IncidL_not_IncidL__not_colH with l;auto. intuition Col. assert (~ ColH C D B). apply IncidL_not_IncidL__not_colH with l;auto. intuition Col. 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