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Summary

Timothy Makarios (with Isabelle/HOL1) and John Harrison (with HOL-Light2) shown that “the Klein-Beltrami model of the hyperbolic plane satisfy all of Tarski’s axioms except his Euclidean axiom” [2],[3],[4],[5].

With the Mizar system [1] we use some ideas taken from Tim Makarios’s MSc thesis [10] to formalize some definitions (like the absolute) and lemmas necessary for the verification of the independence of the parallel postulate. In this article we prove that our constructed model (we prefer “Beltrami-Klein” name over “Klein-Beltrami”, which can be seen in the naming convention for Mizar functors, and even MML identifiers) satisfies the congruence symmetry, the congruence equivalence relation, and the congruence identity axioms formulated by Tarski (and formalized in Mizar as described briefly in [8]).

Summary

Timothy Makarios (with Isabelle/HOL1) and John Harrison (with HOL-Light2) shown that “the Klein-Beltrami model of the hyperbolic plane satisfy all of Tarski’s axioms except his Euclidean axiom” [2],[3],[4, 5].

With the Mizar system [1] we use some ideas taken from Tim Makarios’s MSc thesis [10] to formalize some definitions and lemmas necessary for the verification of the independence of the parallel postulate. In this article, which is the continuation of [8], we prove that our constructed model satisfies the axioms of segment construction, the axiom of betweenness identity, and the axiom of Pasch due to Tarski, as formalized in [11] and related Mizar articles.

Summary

Tim Makarios (with Isabelle/HOL1) and John Harrison (with HOL-Light2) have shown that “the Klein-Beltrami model of the hyperbolic plane satisfy all of Tarski’s axioms except his Euclidean axiom” [, , , ].

With the Mizar system [], [] we use some ideas are taken from Tim Makarios’ MSc thesis [] for formalized some definitions (like the tangent) and lemmas necessary for the verification of the independence of the parallel postulate. This work can be also treated as a further development of Tarski’s geometry in the formal setting [9].

Summary

Tim Makarios (with Isabelle/HOL1) and John Harrison (with HOL-Light2) shown that “the Klein-Beltrami model of the hyperbolic plane satisfy all of Tarski’s axioms except his Euclidean axiom” [], [], [], [].

With the Mizar system [], [] we use some ideas are taken from Tim Makarios’ MSc thesis [] for the formalization of some definitions (like the absolute) and lemmas necessary for the verification of the independence of the parallel postulate. This work can be also treated as further development of Tarski’s geometry in the formal setting []. Note that the model presented here, may also be called “Beltrami-Klein Model”, “Klein disk model”, and the “Cayley-Klein model” [].

, revealed a good adjustment to the general sample as well as gender and sports samples ( Byrne, 2010 ; Hair et al., 2014 ; Marsh et al., 2004 ) Table 2 Fit indexes of the measurement model of the MCSYS: males & females, soccer, swimming, handball, basketball, and futsal. Samples N Ages Gender Training experience male female (years) Soccer 109 12 -20 1098 - 1 - 14 8 (M = 14.15; SD = 2.51) (M = 10.89; SD = 3.78) Swimming 104 12 - 20 - 714 335 6 -14 9 (M = 15.08; SD = 2.47) (M = 9.22; SD = 2.87) Basketball 175 11 -20 800 954 1 - 13 4 (M = 14.61; SD = 1.54) (M = 4.42; SD