proposition: Lemma point3_online_exists : forall A B l, IncidL A l -> IncidL B l -> exists C, IncidL C l /\ C <> A /\ C <> B.
proof: Proof. intros. induction(eq_dec_pointsH A B). subst B. two_points l A0 B0. induction(eq_dec_pointsH A A0). subst A0. exists B0. split; auto. exists A0. split; auto. assert(HH:= between_out A B H1). destruct HH as [C HH]. apply betH_colH in HH. spliter. exists C. unfold ColH in H3. destruct H2 as [m H2]. spliter. lines_eq l m. split; 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