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Abelian group assume asymptote to g Axiom of Pasch calculus of variations carries circle coincide conclude condition congruent conjugate point contains a point converges coordinates curve defined Desargues Desarguesian distance elliptic finitely compact Finsler spaces fixed point follows from Theorem Furthermore g and h geodesic g geodesics geometry given point group of motions hence homeomorphic homothetic hyperbolic hyperbolic geometry interior point intersect g isometric representation Lemma length Let g limit bisector limit spheres line g linear m-space mapping metric space Minkowski metric Minkowski space Minkowskian open S.L. plane open S.L. space oriented asymptote parallel axiom parallel to g perpendicular to g point of g point Q points A,B prove Riemann space satisfies Axioms Section segment sequence subset supporting planes symmetric tangent tend topological translations along g traverses triangle inequality unique