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        <title>Formalized Mathematics Feed</title>
        <link>https://sciendo.com/journal/FORMA</link>
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            <title>Formalized Mathematics Feed</title>
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            <link>https://sciendo.com/journal/FORMA</link>
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        <copyright>All rights reserved 2026, University of Białystok</copyright>
        <item>
            <title><![CDATA[Triangular Fuzzy Set Composed of Two Intersecting Affine Maps]]></title>
            <link>https://sciendo.com/article/10.2478/forma-2025-0011</link>
            <guid>https://sciendo.com/article/10.2478/forma-2025-0011</guid>
            <pubDate>Wed, 31 Dec 2025 00:00:00 GMT</pubDate>
            <description><![CDATA[

In this article, various properties of triangular membership functions are formally proven, including the relationship between a triangular membership function composed of two straight lines and a MAX function, as well as a triangular membership function defined by the horizontal coordinates of the triangle’s vertices. Furthermore, we formalize defuzzified value of a triangular membership function and the integration of two connected functions.
]]></description>
            <category>ARTICLE</category>
        </item>
        <item>
            <title><![CDATA[Higher Order Partial Differentiable Functions]]></title>
            <link>https://sciendo.com/article/10.2478/forma-2025-0009</link>
            <guid>https://sciendo.com/article/10.2478/forma-2025-0009</guid>
            <pubDate>Wed, 31 Dec 2025 00:00:00 GMT</pubDate>
            <description><![CDATA[

This article formalizes higher-order partial differentiable functions in Mizar: it introduces the concept of partial differentiation for functions between real normed spaces and develops the theory of partial derivatives of arbitrary order. Key results include properties of partial derivatives, their relationship with total derivatives, and criteria for the continuity of higher-order partial derivatives.
]]></description>
            <category>ARTICLE</category>
        </item>
        <item>
            <title><![CDATA[Measure for Product Space of Real Normed Spaces]]></title>
            <link>https://sciendo.com/article/10.2478/forma-2025-0010</link>
            <guid>https://sciendo.com/article/10.2478/forma-2025-0010</guid>
            <pubDate>Wed, 31 Dec 2025 00:00:00 GMT</pubDate>
            <description><![CDATA[

This paper deals with Cartesian product spaces of real linear spaces and Cartesian product spaces of real normed spaces. In principle, both are attributed to the direct product of real linear spaces, but the direct product of normed spaces is needed to deal with the calculus of n-dimensional spaces. We also prove that the Lebesgue measure on tuple-type sets introduced in [6] is σ-finite.
]]></description>
            <category>ARTICLE</category>
        </item>
        <item>
            <title><![CDATA[Formalization of Separable Version of Banach–Alaoglu Theorem]]></title>
            <link>https://sciendo.com/article/10.2478/forma-2025-0018</link>
            <guid>https://sciendo.com/article/10.2478/forma-2025-0018</guid>
            <pubDate>Wed, 31 Dec 2025 00:00:00 GMT</pubDate>
            <description><![CDATA[

In this article, we first formalize the weak sequential compactness in dual normed spaces; then we prove the separable version of Banach– Alaoglu theorem.
]]></description>
            <category>ARTICLE</category>
        </item>
        <item>
            <title><![CDATA[Integrability of Multivariable Continuous Functions]]></title>
            <link>https://sciendo.com/article/10.2478/forma-2025-0015</link>
            <guid>https://sciendo.com/article/10.2478/forma-2025-0015</guid>
            <pubDate>Wed, 31 Dec 2025 00:00:00 GMT</pubDate>
            <description><![CDATA[

In this article, we prove the integrability of continuous functions on n-dimensional real normed spaces, using the Mizar formalism. Generalizing selected theorems from the Mizar Mathematical Library, we prove the integrability of continuous real n-variable functions and then, using the correspondence between product-type and tuple-type spaces, we demonstrate the integrability of continuous functions on the desired multidimensional spaces.
]]></description>
            <category>ARTICLE</category>
        </item>
        <item>
            <title><![CDATA[The Lattice of Intermediate Fields and Other Preliminaries to Galois Theory]]></title>
            <link>https://sciendo.com/article/10.2478/forma-2025-0013</link>
            <guid>https://sciendo.com/article/10.2478/forma-2025-0013</guid>
            <pubDate>Wed, 31 Dec 2025 00:00:00 GMT</pubDate>
            <description><![CDATA[

This article initiates a series devoted to the formalization of the Fundamental Theorem of Galois Theory. It presents several preliminaries required for the formal development of Galois theory. In particular, as a main result, we define the lattice of intermediate fields of an extension E/F; we also treat sets of functions, groups, and intermediate fields in order to provide the necessary cluster registrations that enable effective Mizar automation.
]]></description>
            <category>ARTICLE</category>
        </item>
        <item>
            <title><![CDATA[Introduction to Galois Theory]]></title>
            <link>https://sciendo.com/article/10.2478/forma-2025-0014</link>
            <guid>https://sciendo.com/article/10.2478/forma-2025-0014</guid>
            <pubDate>Wed, 31 Dec 2025 00:00:00 GMT</pubDate>
            <description><![CDATA[

This article continues a series devoted to the formalization of the Fundamental Theorem of Galois Theory using the Mizar proof assistant. We define groups of automorphisms and fixed fields and establish their fundamental properties. We also introduce an encoding of conjugates for groups of automorphisms and Galois extensions, and present the classical example demonstrating that the field of complex numbers is a Galois extension of the field of real numbers.
]]></description>
            <category>ARTICLE</category>
        </item>
        <item>
            <title><![CDATA[Elementary Number Theory Problems. Part XVIII]]></title>
            <link>https://sciendo.com/article/10.2478/forma-2025-0017</link>
            <guid>https://sciendo.com/article/10.2478/forma-2025-0017</guid>
            <pubDate>Wed, 31 Dec 2025 00:00:00 GMT</pubDate>
            <description><![CDATA[

In this paper another seven problems from Wacław Sierpiński’s book “250 Problems in Elementary Number Theory” are formalized, using the Mizar formalism, namely: 53, 61, 81, 90, 100, 156, and 167.
]]></description>
            <category>ARTICLE</category>
        </item>
        <item>
            <title><![CDATA[Characterization of Finite Galois Extensions]]></title>
            <link>https://sciendo.com/article/10.2478/forma-2025-0019</link>
            <guid>https://sciendo.com/article/10.2478/forma-2025-0019</guid>
            <pubDate>Wed, 31 Dec 2025 00:00:00 GMT</pubDate>
            <description><![CDATA[

In this article we prove the well-known characterization of finite Galois extensions: a finite extension E of F is a Galois extension of F i E is both normal and separable i E is the splitting field of a separable polynomial p ∈ F [X]. We also prove some applications of the characterization, so for example that F (a1, . . ., an) is a separable extension of F if and only if all the ai are separable, or that every finite separable extension of F is contained in a Galois extension of F .
]]></description>
            <category>ARTICLE</category>
        </item>
        <item>
            <title><![CDATA[Elementary Number Theory Problems. Part XIX]]></title>
            <link>https://sciendo.com/article/10.2478/forma-2025-0020</link>
            <guid>https://sciendo.com/article/10.2478/forma-2025-0020</guid>
            <pubDate>Wed, 31 Dec 2025 00:00:00 GMT</pubDate>
            <description><![CDATA[

In this paper, we present formal solutions to twelve problems selected from Wacław Sierpiński’s book 250 Problems in Elementary Number Theory. The selected problems are: 108, 112–114, 118–119, 127, 129, 130, and 132–134 formalized in the Mizar system.
]]></description>
            <category>ARTICLE</category>
        </item>
        <item>
            <title><![CDATA[Application of Complex Classes to Number Theory]]></title>
            <link>https://sciendo.com/article/10.2478/forma-2025-0016</link>
            <guid>https://sciendo.com/article/10.2478/forma-2025-0016</guid>
            <pubDate>Wed, 31 Dec 2025 00:00:00 GMT</pubDate>
            <description><![CDATA[

The paper illustrates how the formal framework proposed earlier for complex numbers can be applied to integer and natural numbers, facilitating proofs by means of clusters’ registrations for Mizar adjectives. The contents of the article are closely related to the series of Mizar articles “Elementary Number Theory Problems” (with MML identifiers NUMBER*), but it is not solving any of “250 Problems in Elementary Number Theory” by Wacław Sierpiński book.
]]></description>
            <category>ARTICLE</category>
        </item>
        <item>
            <title><![CDATA[Elementary Number Theory Problems. Part XVII]]></title>
            <link>https://sciendo.com/article/10.2478/forma-2025-0012</link>
            <guid>https://sciendo.com/article/10.2478/forma-2025-0012</guid>
            <pubDate>Wed, 31 Dec 2025 00:00:00 GMT</pubDate>
            <description><![CDATA[

This paper furthers the formalization of problems from Wacław Sierpiński book “250 Problems in Elementary Number Theory” in the Mizar system. The selected twelve items are 37, 101, 115, 117, 145, 157, 159, 161–163, 165, and 169.
]]></description>
            <category>ARTICLE</category>
        </item>
        <item>
            <title><![CDATA[Formalization of Wallis Infinite Product Formula for π and the Wallis Integral]]></title>
            <link>https://sciendo.com/article/10.2478/forma-2025-0007</link>
            <guid>https://sciendo.com/article/10.2478/forma-2025-0007</guid>
            <pubDate>Tue, 23 Dec 2025 00:00:00 GMT</pubDate>
            <description><![CDATA[

In this article, we develop the proof of the so-called Wallis formula using the Mizar formalism. The purpose of this formalization is to complete the proof of Stirling’s formula using elementary techniques of calculus; however, the formal encoding of the Wallis-type integral is also included.
]]></description>
            <category>ARTICLE</category>
        </item>
        <item>
            <title><![CDATA[A Formal Proof of Stirling’s Formula]]></title>
            <link>https://sciendo.com/article/10.2478/forma-2025-0008</link>
            <guid>https://sciendo.com/article/10.2478/forma-2025-0008</guid>
            <pubDate>Tue, 23 Dec 2025 00:00:00 GMT</pubDate>
            <description><![CDATA[

In this article, we formalized the proof of the Stirling’s formula, which is considered an essential item in the field of statistics, as shown below:



limn→∞n!nn+12e-n=2π
\mathop {\lim }\limits_{n \to \infty } {{n!} \over {{n^{n + {1 \over 2}}}{e^{ - n}}}} = \sqrt {2\pi }


using the Mizar formalism.
]]></description>
            <category>ARTICLE</category>
        </item>
        <item>
            <title><![CDATA[Semidirect Products of Groups]]></title>
            <link>https://sciendo.com/article/10.2478/forma-2025-0006</link>
            <guid>https://sciendo.com/article/10.2478/forma-2025-0006</guid>
            <pubDate>Wed, 17 Dec 2025 00:00:00 GMT</pubDate>
            <description><![CDATA[

We formalize the semidirect product of groups in Mizar, following §10 of Aschbacher’s Finite Group Theory [2]. We also prove the universal property for semidirect products as found in Bourbaki [5, III §2.10] Proposition 27. In an appendix, we define the dihedral group of the regular n-gon and the infinite dihedral group.
]]></description>
            <category>ARTICLE</category>
        </item>
        <item>
            <title><![CDATA[Extensions of Languages in Polish Notation]]></title>
            <link>https://sciendo.com/article/10.2478/forma-2025-0005</link>
            <guid>https://sciendo.com/article/10.2478/forma-2025-0005</guid>
            <pubDate>Wed, 17 Dec 2025 00:00:00 GMT</pubDate>
            <description><![CDATA[

This article prepares a collection of basic tools for adding new operators to a language in Polish notation, a parenthesis-free logical system where operators precede their operands. A need for this arose while attempting to formalize some extensions of Roman Suszko’s basic non-Fregean logic SCI, namely WB and WH.
]]></description>
            <category>ARTICLE</category>
        </item>
        <item>
            <title><![CDATA[Free Product of Groups]]></title>
            <link>https://sciendo.com/article/10.2478/forma-2025-0004</link>
            <guid>https://sciendo.com/article/10.2478/forma-2025-0004</guid>
            <pubDate>Mon, 08 Dec 2025 00:00:00 GMT</pubDate>
            <description><![CDATA[

In this article the free product of groups is formalized in the Mizar system.
]]></description>
            <category>ARTICLE</category>
        </item>
        <item>
            <title><![CDATA[Surreal Dyadic and Real Numbers: A Formal Construction]]></title>
            <link>https://sciendo.com/article/10.2478/forma-2025-0002</link>
            <guid>https://sciendo.com/article/10.2478/forma-2025-0002</guid>
            <pubDate>Tue, 30 Sep 2025 00:00:00 GMT</pubDate>
            <description><![CDATA[

The concept of surreal numbers, as postulated by John Conway, represents a complex and multifaceted structure that encompasses a multitude of familiar number systems, including the real numbers, as integral components. In this study, we undertake the construction of the real numbers, commencing with the integers and dyadic rationals as preliminary steps. We proceed to contrast the resulting set of real numbers derived from our construction with the axiomatically defined set of real numbers based on Conway’s axiom. Our findings reveal that both approaches culminate in the same set.
]]></description>
            <category>ARTICLE</category>
        </item>
        <item>
            <title><![CDATA[Conway’s Normal Form in the Mizar System]]></title>
            <link>https://sciendo.com/article/10.2478/forma-2025-0003</link>
            <guid>https://sciendo.com/article/10.2478/forma-2025-0003</guid>
            <pubDate>Tue, 30 Sep 2025 00:00:00 GMT</pubDate>
            <description><![CDATA[

This paper presents a formal definition of the Conway normal form, a structured representation uniquely suited to characterising surreal numbers by expressing them as sums within a hierarchically ordered group. To this end, we formalise the first sections of the chapter The Structure of the General Surreal Number in Conway’s book. In particular, we define omega maps and prove the existence and uniqueness of the Conway name for surreal numbers.
]]></description>
            <category>ARTICLE</category>
        </item>
        <item>
            <title><![CDATA[Surreal Numbers: A Study of Square Roots]]></title>
            <link>https://sciendo.com/article/10.2478/forma-2025-0001</link>
            <guid>https://sciendo.com/article/10.2478/forma-2025-0001</guid>
            <pubDate>Tue, 30 Sep 2025 00:00:00 GMT</pubDate>
            <description><![CDATA[

This paper sets out to formalize the concept of the square root as proposed by Clive Bach in the section entitled Properties of Division in Conway’s book. The proposed construction extends the classical approach to the square root of real numbers to include both infinitely large and infinitely small numbers.
]]></description>
            <category>ARTICLE</category>
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