proposition: Lemma bet_ratio a b k1 : 0 < k1 -> k1 < 1 -> bet a b (k1^-1 *: (b - a) + a).
proof: Proof. case: (a =P b)=> [->|/eqP neq_ab L G]; rewrite ?bet_xxa // /bet /betS /betR. apply /orP; right; rewrite -addrA subrr addr0 -ratio_bet' //. by rewrite scalerA mulfV ?lt0r_neq0 // scale1r L G eqxx !andbb. 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