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Coghetto. Klein-Beltrami model. Part I. Formalized Mathematics , 26( 1 ):21–32, 2018. doi:10.2478/forma-2018-0003. [8] Roland Coghetto. Klein-Beltrami model. Part III. Formalized Mathematics , 28( 1 ):1–7, 2020. doi:10.2478/forma-2020-0001. [9] Kanchun, Hiroshi Yamazaki, and Yatsuka Nakamura. Cross products and tripple vector products in 3-dimensional Euclidean space. Formalized Mathematics , 11( 4 ):381–383, 2003. [10] Timothy James McKenzie Makarios. A mechanical verification of the independence of Tarski’s Euclidean Axiom. Victoria University of Wellington, New

):55–62, 2017. doi:10.1515/forma-2017-0005. [7] Roland Coghetto. Klein-Beltrami model. Part I. Formalized Mathematics , 26( 1 ):21–32, 2018. doi:10.2478/forma-2018-0003. [8] Roland Coghetto. Pascal’s theorem in real projective plane. Formalized Mathematics , 25 ( 2 ):107–119, 2017. doi:10.1515/forma-2017-0011. [9] Adam Grabowski. Tarski’s geometry modelled in Mizar computerized proof assistant. In Proceedings of the 2016 Federated Conference on Computer Science and Information Systems, FedCSIS 2016, Gdańsk, Poland, September 11–14, 2016 , pages 373–381, 2016. doi

-2017-0005. [7] Roland Coghetto. Klein-Beltrami model. Part II. Formalized Mathematics , 26( 1 ):33–48, 2018. doi:10.2478/forma-2018-0004. [8] Adam Grabowski and Roland Coghetto. Tarski’s geometry and the Euclidean plane in Mizar. In Joint Proceedings of the FM4M, MathUI, and ThEdu Workshops, Doctoral Program, and Work in Progress at the Conference on Intelligent Computer Mathematics 2016 co-located with the 9th Conference on Intelligent Computer Mathematics (CICM 2016), Białystok, Poland, July 25–29, 2016 , volume 1785 of CEUR-WS , pages 4–9. CEURWS.org, 2016. [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” [].

R eferences [1] Grzegorz Bancerek, Czesław Byliński, Adam Grabowski, Artur Korniłowicz, Roman Matuszewski, Adam Naumowicz, Karol Pak, and Josef Urban. Mizar: State-of-the-art and beyond. In Manfred Kerber, Jacques Carette, Cezary Kaliszyk, Florian Rabe, and Volker Sorge, editors, Intelligent Computer Mathematics , volume 9150 of Lecture Notes in Computer Science , pages 261–279. Springer International Publishing, 2015. ISBN 978-3-319-20614-1. doi:10.1007/978-3-319-20615-8_17. [2] Roland Coghetto. Klein-Beltrami model. Part I. Formalized Mathematics , 26( 1