proposition: Lemma betS_sym a b c : betS a b c = betS c b a.
proof: Proof. rewrite /betS /betR !andb_assoc -(addrBDB b c a) -[c - a]opprB ratiorN oppr_gt0. case (a - c =P 0)=> [->|/eqP ?]; first by rewrite !ratiop0 ltxx !andbF. rewrite -add_ratio ?ratioNr ?ratiovv // andbAC subr_lt0 ltr_oppl ltr_subr_addl. by rewrite scaleNr -scalerN scalerBl scale1r -subr_eq !opprB !addrBDB subrr. 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