proposition: Lemma same_side_refl :forall l A, ~ IncidL A l -> same_side A A l.
proof: Proof. intros. unfold same_side. unfold cut. destruct (two_points_on_line l). destruct s. spliter. destruct (between_out A x). intro;subst. intuition. exists x1. repeat split;try assumption. intro. apply betH_expand in H3. spliter. destruct H8. spliter. assert (EqL x2 l). apply line_uniqueness with x x1;try assumption. apply H. rewrite H11 in H8;auto. exists x;auto. intro. apply betH_expand in H3. spliter. destruct H8. spliter. assert (EqL x2 l). apply line_uniqueness with x x1;try assumption. apply H. rewrite H11 in H8;auto. exists x;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