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33
Functional Sequence from a Domain to a Domain
, 1992
"... this paper. For simplicity, we use the following convention: D, D 1 , D 2 denote non empty sets, n, k ..."
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this paper. For simplicity, we use the following convention: D, D 1 , D 2 denote non empty sets, n, k
On the Subcontinua of a Real Line
, 2003
"... In [11] we showed that the only proper subcontinua of the simple closed curve are arcs and single points. In this article we prove that the only proper subcontinua of the real line are closed intervals. We introduce some auxiliary notions such as]a,b[Q,]a,b[IQ – intervals consisting of rational an ..."
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Cited by 8 (5 self)
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In [11] we showed that the only proper subcontinua of the simple closed curve are arcs and single points. In this article we prove that the only proper subcontinua of the real line are closed intervals. We introduce some auxiliary notions such as]a,b[Q,]a,b[IQ – intervals consisting of rational and irrational numbers respectively. We show also some basic topological properties of intervals.
The limit of a real function at a point
 Journal of Formalized Mathematics
, 1990
"... Summary. We introduce the leftside and the rightside limit of a real function at a point. We prove a few properties of the operations on the proper and improper oneside limits and show that Cauchy and Heine characterizations of the oneside limit are equivalent. ..."
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Summary. We introduce the leftside and the rightside limit of a real function at a point. We prove a few properties of the operations on the proper and improper oneside limits and show that Cauchy and Heine characterizations of the oneside limit are equivalent.
On Constructing Topological Spaces and Sorgenfrey Line
, 2005
"... ... the book [19] by Engelking. In the article the formalization of Section 1.2 is almost completed. Namely, we formalize theorems on introduction of topologies by bases, neighborhood systems, closed sets, closure operator, and interior operator. The Sorgenfrey line is defined by a basis. It is prov ..."
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Cited by 4 (2 self)
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... the book [19] by Engelking. In the article the formalization of Section 1.2 is almost completed. Namely, we formalize theorems on introduction of topologies by bases, neighborhood systems, closed sets, closure operator, and interior operator. The Sorgenfrey line is defined by a basis. It is proved that the weight of it is continuum. Other techniques are used to demonstrate introduction of discrete and antidiscrete topologies.
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.
Monotonic and continuous real function
 Formalized Mathematics
, 1991
"... Summary. A continuation of [13] and [11]. We prove a few theorems about real functions monotonic and continuous on interval, on halfline and on the set of real numbers and continuity of the inverse function. At the beginning of the paper we show some facts about topological properties of the set of ..."
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Summary. A continuation of [13] and [11]. We prove a few theorems about real functions monotonic and continuous on interval, on halfline and on the set of real numbers and continuity of the inverse function. At the beginning of the paper we show some facts about topological properties of the set of real numbers, halflines and intervals which rather belong to [14].
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.
The properties of supercondensed sets, subcondensed sets and condensed sets
 353–359, MML Id: ISOMICHI
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Y.: Combining of Multi Cell Circuits
 Formalized Mathematics
"... Summary. In this article we continue the investigations from [10] and [2] of verification of a circuit design. We concentrate on the combination of multi cell circuits from given cells (circuit modules). Namely, we formalize a design of the form 0 1 2 n and prove its stability. The formalization pro ..."
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Summary. In this article we continue the investigations from [10] and [2] of verification of a circuit design. We concentrate on the combination of multi cell circuits from given cells (circuit modules). Namely, we formalize a design of the form 0 1 2 n and prove its stability. The formalization proposed consists in a series of schemes which allow to define multi cells circuits and prove their properties. Our goal is to achive mathematical formalization which will allow to verify designs of real circuits.