Volume 13, 2001

University of Bialystok

Copyright (c) 2001 Association of Mizar Users

**Barbara Dzienis**- University of Bialystok

- The main result of the paper is, that the ring of polynomials with $o_1$ variables and coefficients in the ring of polynomials with $o_2$ variables and coefficient in a ring $L$ is isomorphic with the ring with $o_1+o_2$ variables, and coefficients in $L$.

- Preliminaries
- About Bags
- Main Results

- [1]
Grzegorz Bancerek.
The fundamental properties of natural numbers.
*Journal of Formalized Mathematics*, 1, 1989. - [2]
Grzegorz Bancerek.
The ordinal numbers.
*Journal of Formalized Mathematics*, 1, 1989. - [3]
Grzegorz Bancerek.
Sequences of ordinal numbers.
*Journal of Formalized Mathematics*, 1, 1989. - [4]
Grzegorz Bancerek.
Ordinal arithmetics.
*Journal of Formalized Mathematics*, 2, 1990. - [5]
Grzegorz Bancerek and Krzysztof Hryniewiecki.
Segments of natural numbers and finite sequences.
*Journal of Formalized Mathematics*, 1, 1989. - [6]
Grzegorz Bancerek and Piotr Rudnicki.
On defining functions on trees.
*Journal of Formalized Mathematics*, 5, 1993. - [7]
Jozef Bialas.
Group and field definitions.
*Journal of Formalized Mathematics*, 1, 1989. - [8]
Czeslaw Bylinski.
Functions and their basic properties.
*Journal of Formalized Mathematics*, 1, 1989. - [9]
Czeslaw Bylinski.
Functions from a set to a set.
*Journal of Formalized Mathematics*, 1, 1989. - [10]
Czeslaw Bylinski.
Finite sequences and tuples of elements of a non-empty sets.
*Journal of Formalized Mathematics*, 2, 1990. - [11]
Agata Darmochwal.
Finite sets.
*Journal of Formalized Mathematics*, 1, 1989. - [12]
Jaroslaw Kotowicz.
Monotone real sequences. Subsequences.
*Journal of Formalized Mathematics*, 1, 1989. - [13]
Eugeniusz Kusak, Wojciech Leonczuk, and Michal Muzalewski.
Abelian groups, fields and vector spaces.
*Journal of Formalized Mathematics*, 1, 1989. - [14]
Beata Madras.
Product of family of universal algebras.
*Journal of Formalized Mathematics*, 5, 1993. - [15]
Robert Milewski.
Associated matrix of linear map.
*Journal of Formalized Mathematics*, 7, 1995. - [16]
Robert Milewski.
The ring of polynomials.
*Journal of Formalized Mathematics*, 12, 2000. - [17]
Michal Muzalewski.
Construction of rings and left-, right-, and bi-modules over a ring.
*Journal of Formalized Mathematics*, 2, 1990. - [18]
Piotr Rudnicki and Andrzej Trybulec.
Multivariate polynomials with arbitrary number of variables.
*Journal of Formalized Mathematics*, 11, 1999. - [19]
Christoph Schwarzweller.
The field of quotients over an integral domain.
*Journal of Formalized Mathematics*, 10, 1998. - [20]
Andrzej Trybulec.
Binary operations applied to functions.
*Journal of Formalized Mathematics*, 1, 1989. - [21]
Andrzej Trybulec.
Tarski Grothendieck set theory.
*Journal of Formalized Mathematics*, Axiomatics, 1989. - [22]
Andrzej Trybulec.
Many-sorted sets.
*Journal of Formalized Mathematics*, 5, 1993. - [23]
Andrzej Trybulec.
Subsets of real numbers.
*Journal of Formalized Mathematics*, Addenda, 2003. - [24]
Wojciech A. Trybulec.
Vectors in real linear space.
*Journal of Formalized Mathematics*, 1, 1989. - [25]
Wojciech A. Trybulec.
Groups.
*Journal of Formalized Mathematics*, 2, 1990. - [26]
Wojciech A. Trybulec.
Pigeon hole principle.
*Journal of Formalized Mathematics*, 2, 1990. - [27]
Zinaida Trybulec.
Properties of subsets.
*Journal of Formalized Mathematics*, 1, 1989. - [28]
Edmund Woronowicz.
Relations and their basic properties.
*Journal of Formalized Mathematics*, 1, 1989. - [29]
Katarzyna Zawadzka.
Sum and product of finite sequences of elements of a field.
*Journal of Formalized Mathematics*, 4, 1992.

[ Download a postscript version, MML identifier index, Mizar home page]