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• Author: Artur Kornilowicz
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## Collective Operations on Number-Membered Sets

The article starts with definitions of sets of opposite and inverse numbers of a given number membered set. Next, collective addition, subtraction, multiplication and division of two sets are defined. Complex numbers cases and extended real numbers ones are introduced separately and unified for reals. Shortcuts for singletons cases are also defined.

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## Summary

The paper introduces Cartesian products in categories without uniqueness of cod and dom. It is proven that set-theoretical product is the product in the category Ens [7].

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## Summary

In this article, we formalize in the Mizar system [3] the notion of the derivative of polynomials over the field of real numbers [4]. To define it, we use the derivative of functions between reals and reals [9].

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## Cayley's Theorem

The article formalizes the Cayley's theorem saying that every group G is isomorphic to a subgroup of the symmetric group on G.

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## Mazur-Ulam Theorem

The Mazur-Ulam theorem [15] has been formulated as two registrations: cluster bijective isometric -> midpoints-preserving Function of E, F; and cluster isometric midpoints-preserving -> Affine Function of E, F; A proof given by Jussi Väisälä [23] has been formalized.

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## Arithmetic Operations on Functions from Sets into Functional Sets

In this paper we introduce sets containing number-valued functions. Different arithmetic operations on maps between any set and such functional sets are later defined.

MML identifier: VALUED 2, version: 7.11.01 4.117.1046

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## On the Continuity of Some Functions

We prove that basic arithmetic operations preserve continuity of functions.

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## Summary

Cayley-Dickson construction produces a sequence of normed algebras over real numbers. Its consequent applications result in complex numbers, quaternions, octonions, etc. In this paper we formalize the construction and prove its basic properties.

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## The Correspondence Between n-dimensional Euclidean Space and the Product of n Real Lines

In the article we prove that a family of open n-hypercubes is a basis of n-dimensional Euclidean space. The equality of the space and the product of n real lines has been proven.

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## Summary

In this article we prove that fundamental groups based at the unit point of topological groups are commutative [11].