<abstract xmlns="http://www.w3.org/1999/xhtml">In contrast to other proving systems Mizar Mathematical Library, considered as one of the largest formal mathematical libraries [4], is maintained as a single base of theorems, which allows the users to benefit from earlier formalized items [3], [2]. This eventually leads to a development of certain branches of articles using common notation and ideas. Such formalism for finite sequences has been developed since 1989 [1] and further developed despite of the controversy over indexing which excludes zero [6], also for some advanced and new mathematics [5].The article aims to add some new machinery for dealing with finite sequences, especially those of short length.</abstract>

Arithmetic Operations on Short Finite Sequences

<abstract xmlns="http://www.w3.org/1999/xhtml">This article covers some technical aspects about the product topology which are usually not given much of a thought in mathematics and standard literature like [7] and [6], not even by Bourbaki in [4].Let <italic>{Ti}i∈I</italic> be a family of topological spaces. The prebasis of the product space <italic>T</italic> = ∏<italic>i∈I Ti</italic> is defined in [5] as the set of all <italic>π</italic>−1<italic>i</italic>(<italic>V</italic>) with <italic>i ∈ I</italic> and <italic>V</italic> open in <italic>Ti</italic>. Here it is shown that the basis generated by this prebasis consists exactly of the sets ∏<italic>i∈I Vi</italic> with <italic>Vi</italic> open in <italic>Ti</italic> and for all but finitely many <italic>i ∈ I</italic> holds <italic>Vi</italic> = <italic>Ti</italic>. Given <italic>I</italic> = <italic>{a}</italic> we have <italic>T</italic> ≅ <italic>Ta</italic>, given <italic>I</italic> = <italic>{a, b}</italic> with <italic>a</italic>≠ <italic>b</italic> we have <italic>T</italic> ≅ <italic>Ta ×Tb</italic>. Given another family of topological spaces <italic>{Si}i∈I</italic> such that <italic>Si</italic> ≅ <italic>Ti</italic> for all <italic>i ∈ I</italic>, we have <italic>S</italic> = ∏<italic>i∈I Si</italic> ≅ <italic>T</italic>. If instead <italic>Si</italic> is a subspace of <italic>Ti</italic> for each <italic>i ∈ I</italic>, then <italic>S</italic> is a subspace of <italic>T</italic>.These results are obvious for mathematicians, but formally proven here by means of the Mizar system [3], [2].</abstract>

Some Remarks about Product Spaces

<abstract xmlns="http://www.w3.org/1999/xhtml">Binary representation of integers <xref ref-type="bibr" rid="j_forma-2018-0020_ref_005_w2aab3b7b2b1b6b1ab1ab5Aa">[5]</xref>, <xref ref-type="bibr" rid="j_forma-2018-0020_ref_003_w2aab3b7b2b1b6b1ab1ab3Aa">[3]</xref> and arithmetic operations on them have already been introduced in Mizar Mathematical Library <xref ref-type="bibr" rid="j_forma-2018-0020_ref_008_w2aab3b7b2b1b6b1ab1ab8Aa">[8</xref>, <xref ref-type="bibr" rid="j_forma-2018-0020_ref_007_w2aab3b7b2b1b6b1ab1ab7Aa">7</xref>, <xref ref-type="bibr" rid="j_forma-2018-0020_ref_006_w2aab3b7b2b1b6b1ab1ab6Aa">6</xref>, <xref ref-type="bibr" rid="j_forma-2018-0020_ref_004_w2aab3b7b2b1b6b1ab1ab4Aa">4</xref>]. However, these articles formalize the notion of integers as mapped into a certain length tuple of boolean values.In this article we formalize, by means of Mizar system <xref ref-type="bibr" rid="j_forma-2018-0020_ref_002_w2aab3b7b2b1b6b1ab1ab2Aa">[2]</xref>, <xref ref-type="bibr" rid="j_forma-2018-0020_ref_001_w2aab3b7b2b1b6b1ab1ab1Aa">[1]</xref>, the binary representation of natural numbers which maps ℕ into bitstreams.</abstract>

Binary Representation of Natural Numbers

<abstract xmlns="http://www.w3.org/1999/xhtml">In this article, using the Mizar system <xref ref-type="bibr" rid="j_forma-2018-0021_ref_001_w2aab3b7b3b1b6b1ab1ab1Aa">[1]</xref>, <xref ref-type="bibr" rid="j_forma-2018-0021_ref_002_w2aab3b7b3b1b6b1ab1ab2Aa">[2]</xref>, we discuss the continuity of bounded linear operators on normed linear spaces. In the first section, it is discussed that bounded linear operators on normed linear spaces are uniformly continuous and Lipschitz continuous. Especially, a bounded linear operator on the dense subset of a complete normed linear space has a unique natural extension over the whole space. In the next section, several basic currying properties are formalized.In the last section, we formalized that continuity of bilinear operator is equivalent to both Lipschitz continuity and local continuity. We referred to <xref ref-type="bibr" rid="j_forma-2018-0021_ref_004_w2aab3b7b3b1b6b1ab1ab4Aa">[4]</xref>, <xref ref-type="bibr" rid="j_forma-2018-0021_ref_013_w2aab3b7b3b1b6b1ab1ac13Aa">[13]</xref>, and <xref ref-type="bibr" rid="j_forma-2018-0021_ref_003_w2aab3b7b3b1b6b1ab1ab3Aa">[3]</xref> in this formalization.</abstract>

Continuity of Bounded Linear Operators on Normed Linear Spaces

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Formalized Mathematics

  Pioneering Journal in the field of formalized and mechanically checked mathematics. Excellent articles on latest research in formal methods and computer-checked mathematics. Full-length original research articles in English. Archival issues available at the journal's home page fm.mizar.org/.   To build a uniform and extensible computer-checked repository of mathematical knowledge.  Reconstruction of mathematics based on mechanization of mathematical reasoning.  To check a wide range of mathematical theorems using a computer in an automatic fashion.  General mathematics: especially topology, lattice theory, functional analysis.  Applications: computer-aided verification of software and hardware, computer-aided teaching of mathematics.   Why subscribe and read   Pioneering source of high quality research in formalized and mechanically checked mathematics.  The oldest journal in the field, a medium for the publication of research reporting pioneering findings of broad mathematical significance that are likely to advance the direction and thinking of the field.  Long-term experience in publishing original articles on mechanizing mathematics and various aspects of automated reasoning, since 1990.   Why submit   Fair and constructive peer review by international experts in the field (three referees are designated for the assessment of each manuscript) - double-blind review process.  New publishing technology: articles automatically translated from a formal language into English and automatically typeset, available in both forms - electronic and printed.  Articles abstracted by online indexing services and databases.   Rejection Rate   59% of submitted articles.  The rejection percentage is described in detail in the following paper, pp.239-240: dx.doi.org/10.1007/978-3-540-73086-6_20 Archiving  Sciendo archives the contents of this journal in Portico - digital long-term preservation service of scholarly books, journals and collections.  Plagiarism Policy  The editorial board is participating in a growing community of Similarity Check System's users in order to ensure that the content published is original and trustworthy. Similarity Check is a medium that allows for comprehensive manuscripts screening, aimed to eliminate plagiarism and provide a high standard and quality peer-review process. Back Issues fm.mizar.org/   Other Files  Addenda 2008Formalized Mathematics Bibliography File.Formalized Mathematics External Bibliography File.  "In years: 2016-2017, the English version of Formalized Mathematics was financed under agreements 548/P-DUN/2016 and 548/1/P-DUN/2016 with the funds from the Polish Minister of Science and Higher Education for the dissemination of science." 