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Formalization of Probability and Random Sampling on a Finite and Discrete Sample Space Using the Mizar Proof Checker., , and . FCS, page 81-86. CSREA Press, (2010)Real Vector Space and Related NotionsThis study was supported in part by JSPS KAKENHI Grant Numbers 17K00182 and 20K19863., , and . Formaliz. Math., 29 (3): 117-127 (2021)Formal definition of probability on finite and discrete sample space for proving security of cryptographic systems using Mizar., , and . Artif. Intell. Res., 2 (4): 37-48 (2013)Cooperative Control Over Logical and Physical Networks for Multiservice Environments, and . page 365-370. (1991)Digital Telephone Junction Network Planning System Considering Logical and Physical Network Aspects, and . page 2.1-2.7. (1985)Privacy Preserving Logic Formula Calculation in Cloud., , , and . ICSC, page 329-332. IEEE Computer Society, (2016)Formalization of Gaussian integers, Gaussian rational numbers, and their algebraic structures with Mizar., , and . ISITA, page 591-595. IEEE, (2012)On the design and an implementation of broadband access management systems., , , , , and . NOMS, page 382-391. IEEE, (1996)Analysis of a Gradual Input Model for Bursty Traffic in ATM, , and . page 795-800. (1991)Application to Women's Healthcare of Health Management System Using a Tablet Phone., , , , and . BIH, volume 9250 of Lecture Notes in Computer Science, page 262-271. Springer, (2015)