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On the rectangular finite sequences of the points of the plane
 Journal of Formalized Mathematics
, 1997
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Introduction to MeetContinuous Topological Lattices
"... Introduction to MeetContinuous Topological Lattices 1 Artur Korni#lowicz University of Bia#lystok MML Identifier: YELLOW13. WWW: http://mizar.org/JFM/Vol10/yellow13.html The articles [23], [29], [7], [28], [30], [6], [10], [19], [26], [20], [31], [27], [9], [15], [13], [14], [1], [21], [4], [24], ..."
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Introduction to MeetContinuous Topological Lattices 1 Artur Korni#lowicz University of Bia#lystok MML Identifier: YELLOW13. WWW: http://mizar.org/JFM/Vol10/yellow13.html The articles [23], [29], [7], [28], [30], [6], [10], [19], [26], [20], [31], [27], [9], [15], [13], [14], [1], [21], [4], [24], [5], [22], [2], [3], [12], [11], [8], [32], [16], [17], [25], and [18] provide the notation and terminology for this paper. 1. Preliminaries Let S be a finite 1sorted structure. One can check that the carrier of S is finite. Let S be a trivial 1sorted structure. Observe that the carrier of S is trivial. Let us observe that every set which is trivial is also finite. One can check that every 1sorted structure which is trivial is also finite. Let us note that every 1sorted structure which is non trivial is also non empty. One can verify the following observations:<F17.2
Tietze Extension Theorem
"... notation and terminology for this paper. We adopt the following rules: r, s denote real numbers, X denotes a set, and f, g, h denote realyielding functions. The following propositions are true: (1) For all real numbers a, b, c such that a − b  ≤ c holds b − c ≤ a and a ≤ b + c. (2) If r < s, ..."
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notation and terminology for this paper. We adopt the following rules: r, s denote real numbers, X denotes a set, and f, g, h denote realyielding functions. The following propositions are true: (1) For all real numbers a, b, c such that a − b  ≤ c holds b − c ≤ a and a ≤ b + c. (2) If r < s, then]−∞,r] misses [s,+∞[. (3) If r ≤ s, then]−∞,r [ misses]s,+∞[. (4) If f ⊆ g, then h − f ⊆ h − g. (5) If f ⊆ g, then f − h ⊆ g − h.
Lines on planes in ndimensional Euclidean spaces
 Formalized Mathematics
, 2005
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Projections in ndimensional Euclidean space to each coordinates
 Journal of Formalized Mathematics
, 1997
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Determinant and Inverse of Matrices of Real Elements
, 2007
"... In this paper the classic theory of matrices of real elements (see e.g. [12], [13]) is developed. We prove selected equations that have been proved previously for matrices of field elements. Similarly, we introduce in this special context the determinant of a matrix, the identity and zero matrices, ..."
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In this paper the classic theory of matrices of real elements (see e.g. [12], [13]) is developed. We prove selected equations that have been proved previously for matrices of field elements. Similarly, we introduce in this special context the determinant of a matrix, the identity and zero matrices, and the inverse matrix. The new concept discussed in the case of matrices of real numbers is the property of matrices as operators acting on finite sequences of real numbers from both sides. The relations of invertibility of matrices and the “onto” property of matrices as operators are discussed.
The product space of real normed spaces and its properties
 Formalized Mathematics
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The Fashoda Meet Theorem for Rectangles
, 2005
"... Here, the so called Fashoda Meet Theorem is proven in the case of rectangles. All cases of proper location of arcs are listed up, and it is shown that the theorem is valid in each case. Such a list of cases will be useful when one wants to apply the theorem. ..."
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Here, the so called Fashoda Meet Theorem is proven in the case of rectangles. All cases of proper location of arcs are listed up, and it is shown that the theorem is valid in each case. Such a list of cases will be useful when one wants to apply the theorem.