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The field of complex numbers
 Journal of Formalized Mathematics
"... [14], [5], [10], and [9] provide the notation and terminology for this paper. The following propositions are true: 1. PRELIMINARIES (1) For every element z of C holds z  = z. (2) For all real numbers x1, y1, x2, y2 holds (x1 + y1i) ·(x2 + y2i) = (x1 · x2 − y1 · y2)+(x1 · y2 + x2 · y1)i. (3) ..."
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Cited by 26 (1 self)
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[14], [5], [10], and [9] provide the notation and terminology for this paper. The following propositions are true: 1. PRELIMINARIES (1) For every element z of C holds z  = z. (2) For all real numbers x1, y1, x2, y2 holds (x1 + y1i) ·(x2 + y2i) = (x1 · x2 − y1 · y2)+(x1 · y2 + x2 · y1)i. (3) For every real number r holds (r+0i) · i = 0+ri. (4) For every real number r holds r+0i  = r. (5) For every element z of C such that z  � = 0 holds z+0i = z z+0i · z. 2. SOME FACTS ON THE FIELD OF COMPLEX NUMBERS Let x, y be real numbers. The functor x+yiCF yielding an element of CF is defined as follows: (Def. 1) x+yiCF = x+yi. The element iCF of CF is defined as follows: (Def. 2) iCF = i. We now state several propositions: (6) iCF = 0+1i and iCF = 0+1iCF.
On Equivalents of Wellfoundedness  An experiment in Mizar
, 1998
"... Four statements equivalent to wellfoundedness (wellfounded induction, existence of recursively defined functions, uniqueness of recursively defined functions, and absence of descending omegachains) have been proved in Mizar and the proofs mechanically checked for correctness. It seems not to be w ..."
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Cited by 13 (3 self)
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Four statements equivalent to wellfoundedness (wellfounded induction, existence of recursively defined functions, uniqueness of recursively defined functions, and absence of descending omegachains) have been proved in Mizar and the proofs mechanically checked for correctness. It seems not to be widely known that the existence (without the uniqueness assumption) of recursively defined functions implies wellfoundedness. In the proof we used regular cardinals, a fairly advanced notion of set theory. The theory of cardinals in Mizar was developed earlier by G. Bancerek. With the current state of the Mizar system, the proofs turned out to be an exercise with only minor additions at the fundamental level. We would like to stress the importance of a systematic development of a mechanized data base for mathematics in the spirit of the QED Project.
The Hahn Banach theorem in the vector space over the field of complex numbers
 Journal of Formalized Mathematics
"... [14], [5], [10], and [9] provide the notation and terminology for this paper. The following propositions are true: 1. PRELIMINARIES (1) For every element z of C holds z  = z. (2) For all real numbers x1, y1, x2, y2 holds (x1 + y1i) · (x2 + y2i) = (x1 · x2 − y1 · y2) + (x1 · y2 + x2 · y1)i. ..."
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Cited by 8 (0 self)
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[14], [5], [10], and [9] provide the notation and terminology for this paper. The following propositions are true: 1. PRELIMINARIES (1) For every element z of C holds z  = z. (2) For all real numbers x1, y1, x2, y2 holds (x1 + y1i) · (x2 + y2i) = (x1 · x2 − y1 · y2) + (x1 · y2 + x2 · y1)i. (3) For every real number r holds (r + 0i) · i = 0 + ri. (4) For every real number r holds r + 0i  = r. (5) For every element z of C such that z  � = 0 holds z  + 0i = z z+0i · z. 2. SOME FACTS ON THE FIELD OF COMPLEX NUMBERS Let x, y be real numbers. The functor x + yiCF yielding an element of CF is defined as follows: (Def. 1) x + yiCF = x + yi. The element iCF of CF is defined as follows: (Def. 2) iCF = i. We now state several propositions: (6) iCF = 0 + 1i and iCF = 0 + 1iCF.
Bounding Boxes for Compact Sets in Ε²
, 1997
"... this paper. 1. Preliminaries Let X be a set. Let us observe that X has non empty elements if and only if: (Def. 1) 0 / # X. We introduce X is without zero as a synonym of X has non empty elements. We introduce X has zero as an antonym of X has non empty elements. Let us note that R has zero and ..."
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this paper. 1. Preliminaries Let X be a set. Let us observe that X has non empty elements if and only if: (Def. 1) 0 / # X. We introduce X is without zero as a synonym of X has non empty elements. We introduce X has zero as an antonym of X has non empty elements. Let us note that R has zero and N has zero. Let us note that there exists a set which is non empty and without zero and there exists a set which is non empty and has zero. Let us observe that there exists a subset of R which is non empty and without zero and there exists a subset of R which is non empty and has zero. The following proposition is true (1) For every set F such that F is non empty and #linear and has non empty elements holds F is centered. Let
On Some Properties of Real Hilbert Space. Part I Hiroshi Yamazaki
"... Summary. In this paper, we first introduce the notion of summability of an infinite set of vectors of real Hilbert space, without using index sets. Further we introduce the notion of weak summability, which is weaker than that of summability. Then, several statements for summable sets and weakly sum ..."
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Summary. In this paper, we first introduce the notion of summability of an infinite set of vectors of real Hilbert space, without using index sets. Further we introduce the notion of weak summability, which is weaker than that of summability. Then, several statements for summable sets and weakly summable ones are proved. In the last part of the paper, we give a necessary and sufficient condition for summability of an infinite set of vectors of real Hilbert space as our main theorem. The last theorem is due to [8].
Quotient Vector Spaces and Functionals 1
"... Summary. The article presents well known facts about quotient vector spaces and functionals (see [8]). There are repeated theorems and constructions with either weaker assumptions or in more general situations (see [11], [7], [10]). The construction of coefficient functionals and non degenerated fun ..."
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Summary. The article presents well known facts about quotient vector spaces and functionals (see [8]). There are repeated theorems and constructions with either weaker assumptions or in more general situations (see [11], [7], [10]). The construction of coefficient functionals and non degenerated functional in quotient vector space generated by functional in the given vector space are the only new things which are done.
A DEFENCE OF MATHEMATICAL PLURALISM
, 2004
"... We approach the philosophy of mathematics via a discussion of the differences between classical mathematics and constructive mathematics, arguing that each is a valid activity within its own context. ..."
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We approach the philosophy of mathematics via a discussion of the differences between classical mathematics and constructive mathematics, arguing that each is a valid activity within its own context.