proposition: Lemma betR_cab : betR c a b = 0 \/ betR c a b = 1.
proof: Proof. rewrite /c a_eq0 /b /betR /ratio add0r; case: pickP => /= [x|/all_v_neq0 H]; last by exfalso; apply H; rewrite subr_eq0 bc_neq. rewrite /to_nD /row_mx !mxE; case: (split x)=> [x0|x1]; rewrite ?mxE ?subrr; rewrite ?eqxx //; case: x0 => [] [|[|//]] p /=; rewrite ?add0r ?subr0 ?divr1; by rewrite ?oppr0 ?divff ?oppr_eq0 ?oner_eq0; tauto. 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