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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.
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.
Isomorphisms of categories
 Journal of Formalized Mathematics
, 1991
"... [8] (compare also [7]). We define the concept of isomorphic categories and prove basic facts related, e.g. that the Cartesian product of categories is associative up to the isomorphism. We introduce the composition of a functor and a transformation, and of transformation and a functor, and afterward ..."
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[8] (compare also [7]). We define the concept of isomorphic categories and prove basic facts related, e.g. that the Cartesian product of categories is associative up to the isomorphism. We introduce the composition of a functor and a transformation, and of transformation and a functor, and afterwards we define again those concepts for natural transformations. Let us observe, that we have to duplicate those concepts because of the permissiveness: if a functor F is not naturally transformable to G, then natural transformation from F to G has no fixed meaning, hence we cannot claim that the composition of it with a functor as a transformation results in a natural transformation. We define also the so called horizontal composition of transformations ([8], p. 140, exercise 4.2,5(C)) and prove interchange law ([7], p.44). We conclude with the definition of equivalent categories.
Injective Spaces. Part II
"... this paper. 1. Injective Spaces The following two propositions are true: (1) For every point p of the Sierpi'nski space such that p = 0 holds fpg is closed. (2) For every point p of the Sierpi'nski space such that p = 1 holds fpg is non closed. Let us observe that the Sierpi'nski space is non T 1 . ..."
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this paper. 1. Injective Spaces The following two propositions are true: (1) For every point p of the Sierpi'nski space such that p = 0 holds fpg is closed. (2) For every point p of the Sierpi'nski space such that p = 1 holds fpg is non closed. Let us observe that the Sierpi'nski space is non T 1 . One can verify that every toplattice which is complete and Scott is also discernible. One can verify that there exists a T 0 space which is injective and strict. Let us note that there exists a toplattice which is complete, Scott, and strict. One can prove the following propositions: (3) Let I be a non empty set and T be a Scott topological augmentation of Q (I 7\Gamma! 2 1 ` ): Then t