proposition: Lemma ratio_betS a b c k1 k2 k3 : a != b -> 0 < k1 -> 0 < k2 -> k1 < 1 -> 0 < k3 -> k3 < k1 + k2 - k1 * k2 -> b - a == ((k1 + k2 - k1 * k2) / k3)^-1 *: (c - a) -> betS a b c.
proof: Proof. move=> H ? ? ? ? ?; rewrite /betS/ betR; move: H. case: (a =P c)=> [->|]; rewrite ?subrr ?scaler0 ?subr_eq0; [by move=> /eqP HF /eqP H; rewrite H in HF|]. move=> ? /eqP ? k3_eq. suff: (ratio (b - a) (c - a) = ((k1 + k2 - k1 * k2) / k3)^-1). by move=> ->; rewrite bet_gt0' ?bet_lt1 // k3_eq. by apply ratio_eq => //; rewrite subr_eq0 eq_sym; apply /eqP. 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