proposition: Lemma betS_inner_transitivity a b c d (k1 := betR a b d) (k2 := betR b c d) : betS a b d -> betS b c d -> betS a b c.
proof: Proof. rewrite betS_neq12=> /andP[/betSP[k1P ? ?] ?] /betSP[k2P ? ?]. apply ratio_betS with k1 k2 k1=> //. by rewrite -addrA -ltr_subl_addl subrr subr_gt0 gtr_pmull. rewrite eq_inv_scale ?bet_neq0 // -addrA addrC -addf_divrr ?divff ?lt0r_neq0 //. rewrite scalerDl scale1r eq_sym -subr_eq opprB eq_div_scale ?lt0r_neq0 //. rewrite addrBDB k2P scalerA scalerBl eq_sym subr_eq. by rewrite -scalerDr addrBDB k1P scalerA mulrC. Qed.
imports: mathcomp.ssreflect.ssreflect mathcomp(ssrbool ssrfun ssrnat eqtype seq order) mathcomp(fintype bigop) mathcomp(ssralg ssrnum matrix) mathcomp(zmodp) GeoCoq.Axioms.tarski_axioms GeoCoq.Axioms.gupta_inspired_variant_axioms GeoCoq.Meta_theory.Models.gupta_inspired_to_tarski GeoCoq.Meta_theory.Counter_models.coplanarity GeoCoq.Meta_theory.Parallel_postulates.parallel_postulates GeoCoq.Tarski_dev.Ch12_parallel_inter_dec