proposition: Lemma ratio_bet'' v1 v2 k1 : 0 < k1 -> v1 != v2 -> k1^-1 = ratio (v2 - v1) (k1 *: (v2 - v1)).
proof: Proof. move=> ? ?; rewrite /ratio. have: (k1 *: (v2 - v1) != 0) by rewrite scalemx_eq0 negb_or lt0r_neq0 // subr_eq0 eq_sym. move=> neq; case: pickP => /=[x px|/all_v_neq0 //]. rewrite !mxE -{1}[(v2 0 x - v1 0 x)]mul1r -mulf_div mulfV ?mulr1 ?div1r //. by apply /eqP; move=> eq; move/eqP: px; rewrite !mxE eq mulr0. 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