{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,2,14]],"date-time":"2026-02-14T10:03:00Z","timestamp":1771063380684,"version":"3.50.1"},"reference-count":14,"publisher":"Walter de Gruyter GmbH","issue":"2","license":[{"start":{"date-parts":[[2017,7,1]],"date-time":"2017-07-01T00:00:00Z","timestamp":1498867200000},"content-version":"unspecified","delay-in-days":0,"URL":"http:\/\/creativecommons.org\/licenses\/by-sa\/4.0\/"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":[],"published-print":{"date-parts":[[2017,7,1]]},"abstract":"<jats:title>Summary<\/jats:title>\n               <jats:p>In this article we check, with the Mizar system [2], Pascal\u2019s theorem in the real projective plane (in projective geometry Pascal\u2019s theorem is also known as the Hexagrammum Mysticum Theorem)<jats:sup>1<\/jats:sup>. Pappus\u2019 theorem is a special case of a degenerate conic of two lines.<\/jats:p>\n               <jats:p>For proving Pascal\u2019s theorem, we use the techniques developed in the section \u201cProjective Proofs of Pappus\u2019 Theorem\u201d in the chapter \u201cPappus\u2019 Theorem: Nine proofs and three variations\u201d [11]. We also follow some ideas from Harrison\u2019s work. With HOL Light, he has the proof of Pascal\u2019s theorem<jats:sup>2<\/jats:sup>. For a lemma, we use PROVER9<jats:sup>3<\/jats:sup> and OTT2MIZ by Josef Urban<jats:sup>4<\/jats:sup> [12, 6, 7]. We note, that we don\u2019t use Skolem\/Herbrand functions (see \u201cSkolemization\u201d in [1]).<\/jats:p>","DOI":"10.1515\/forma-2017-0011","type":"journal-article","created":{"date-parts":[[2017,9,25]],"date-time":"2017-09-25T10:01:13Z","timestamp":1506333673000},"page":"107-119","source":"Crossref","is-referenced-by-count":3,"title":["Pascal\u2019s Theorem in Real Projective Plane"],"prefix":"10.1515","volume":"25","author":[{"given":"Roland","family":"Coghetto","sequence":"first","affiliation":[{"name":"Rue de la Brasserie 5, 7100 La Louvi\u00e8re , Belgium"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"374","published-online":{"date-parts":[[2017,9,23]]},"reference":[{"key":"2021040812094307073_j_forma-2017-0011_ref_001_w2aab3b7b3b1b6b1ab1b1b1Aa","unstructured":"[1] Jesse Alama. Escape to Mizar for ATPs. arXiv preprint arXiv:1204.6615, 2012."},{"key":"2021040812094307073_j_forma-2017-0011_ref_002_w2aab3b7b3b1b6b1ab1b1b2Aa","unstructured":"[2] Grzegorz Bancerek, Czes\u0142aw Byli\u0144ski, Adam Grabowski, Artur Korni\u0142owicz, Roman Matuszewski, Adam Naumowicz, Karol P\u0105k, 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\u2013279. Springer International Publishing, 2015. ISBN 978-3-319-20614-1. doi: 10.1007\/978-3-319-20615-817.10.1007\/978-3-319-20615-817"},{"key":"2021040812094307073_j_forma-2017-0011_ref_003_w2aab3b7b3b1b6b1ab1b1b3Aa","unstructured":"[3] Roland Coghetto. Homography in \u211d \u21192. Formalized Mathematics, 24(4):239\u2013251, 2016. doi: 10.1515\/forma-2016-0020.10.1515\/forma-2016-0020"},{"key":"2021040812094307073_j_forma-2017-0011_ref_004_w2aab3b7b3b1b6b1ab1b1b4Aa","unstructured":"[4] Roland Coghetto. Group of homography in real projective plane. Formalized Mathematics, 25(1):55\u201362, 2017. doi: 10.1515\/forma-2017-0005.10.1515\/forma-2017-0005"},{"key":"2021040812094307073_j_forma-2017-0011_ref_005_w2aab3b7b3b1b6b1ab1b1b5Aa","unstructured":"[5] Agata Darmochwa\u0142. The Euclidean space. Formalized Mathematics, 2(4):599\u2013603, 1991."},{"key":"2021040812094307073_j_forma-2017-0011_ref_006_w2aab3b7b3b1b6b1ab1b1b6Aa","unstructured":"[6] Adam Grabowski. Solving two problems in general topology via types. In Types for Proofs and Programs, International Workshop, TYPES 2004, Jouy-en-Josas, France, December 15-18, 2004, Revised Selected Papers, pages 138\u2013153, 2004. doi: 10.1007\/116179909.10.1007\/116179909"},{"key":"2021040812094307073_j_forma-2017-0011_ref_007_w2aab3b7b3b1b6b1ab1b1b7Aa","unstructured":"[7] Adam Grabowski. Mechanizing complemented lattices within Mizar system. Journal of Automated Reasoning, 55:211\u2013221, 2015. doi: 10.1007\/s10817-015-9333-5.10.1007\/s10817-015-9333-5"},{"key":"2021040812094307073_j_forma-2017-0011_ref_008_w2aab3b7b3b1b6b1ab1b1b8Aa","unstructured":"[8] Kanchun, Hiroshi Yamazaki, and Yatsuka Nakamura. Cross products and tripple vector products in 3-dimensional Euclidean space. Formalized Mathematics, 11(4):381\u2013383, 2003."},{"key":"2021040812094307073_j_forma-2017-0011_ref_009_w2aab3b7b3b1b6b1ab1b1b9Aa","unstructured":"[9] Wojciech Leo\u0144czuk and Krzysztof Pra\u017cmowski. A construction of analytical projective space. Formalized Mathematics, 1(4):761\u2013766, 1990."},{"key":"2021040812094307073_j_forma-2017-0011_ref_010_w2aab3b7b3b1b6b1ab1b1c10Aa","unstructured":"[10] Wojciech Leo\u0144czuk and Krzysztof Pra\u017cmowski. Projective spaces \u2013 part I. Formalized Mathematics, 1(4):767\u2013776, 1990."},{"key":"2021040812094307073_j_forma-2017-0011_ref_011_w2aab3b7b3b1b6b1ab1b1c11Aa","unstructured":"[11] J\u00fcrgen Richter-Gebert. Pappos\u2019s Theorem: Nine Proofs and Three Variations, pages 3\u201331. Springer Berlin Heidelberg, 2011. ISBN 978-3-642-17286-1. doi: 10.1007\/978-3-642-17286-11.10.1007\/978-3-642-17286-11"},{"key":"2021040812094307073_j_forma-2017-0011_ref_012_w2aab3b7b3b1b6b1ab1b1c12Aa","unstructured":"[12] Piotr Rudnicki and Josef Urban. Escape to ATP for Mizar. In First International Workshop on Proof eXchange for Theorem Proving-PxTP 2011, 2011."},{"key":"2021040812094307073_j_forma-2017-0011_ref_013_w2aab3b7b3b1b6b1ab1b1c13Aa","unstructured":"[13] Wojciech Skaba. The collinearity structure. Formalized Mathematics, 1(4):657\u2013659, 1990."},{"key":"2021040812094307073_j_forma-2017-0011_ref_014_w2aab3b7b3b1b6b1ab1b1c14Aa","unstructured":"[14] Nobuyuki Tamura and Yatsuka Nakamura. Determinant and inverse of matrices of real elements. 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