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85
The lattice of natural numbers and the sublattice of it. The set of prime numbers
 Journal of Formalized Mathematics
, 1991
"... Summary. Basic properties of the least common multiple and the greatest common divisor. The lattice of natural numbers (LN) and the lattice of natural numbers greater than zero (L N +) are constructed. The notion of the sublattice of the lattice of natural numbers is given. Some facts about it are p ..."
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Summary. Basic properties of the least common multiple and the greatest common divisor. The lattice of natural numbers (LN) and the lattice of natural numbers greater than zero (L N +) are constructed. The notion of the sublattice of the lattice of natural numbers is given. Some facts about it are proved. The last part of the article deals with some properties of prime numbers and with the notions of the set of prime numbers and the nth prime number. It is proved that the set of prime numbers is infinite. MML Identifier:NAT_LAT. WWW:http://mizar.org/JFM/Vol3/nat_lat.html
Trigonometric Functions and Existence of Circle Ratio
 Journal of Formalized Mathematics
, 1998
"... this article, we defined sinus and cosine as real part and imaginary part of exponential function on complex, and gave thier series expression either. Then we proved the differentiablity of sin, cos and exponential function of real. At last, we showed the existence of circle ratio, and some formulas ..."
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Cited by 13 (1 self)
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this article, we defined sinus and cosine as real part and imaginary part of exponential function on complex, and gave thier series expression either. Then we proved the differentiablity of sin, cos and exponential function of real. At last, we showed the existence of circle ratio, and some formulas of sin, cos. MML Identifier: SINCOS.
Several differentiation formulas of special functions
 Part V. Formalized Mathematics
"... Summary. In this article, we give several differentiation formulas of special and composite functions including trigonometric, polynomial and logarithmic functions. ..."
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Cited by 11 (7 self)
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Summary. In this article, we give several differentiation formulas of special and composite functions including trigonometric, polynomial and logarithmic functions.
The Binomial Theorem for Algebraic Structures
 Journal of Formalized Mathematics
, 1999
"... Summary. In this paper we prove the wellknown binomial theorem for algebraic structures. In doing so we tried to be as modest as possible concerning the algebraic properties of the underlying structure. Consequently, we proved the binomial theorem for “commutative rings ” in which the existence of ..."
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Summary. In this paper we prove the wellknown binomial theorem for algebraic structures. In doing so we tried to be as modest as possible concerning the algebraic properties of the underlying structure. Consequently, we proved the binomial theorem for “commutative rings ” in which the existence of an inverse with respect to addition is replaced by a weaker property of cancellation.
Difference and Difference Quotient
"... Summary. In this article, we give the definitions of forward difference, backward difference, central difference and difference quotient, and some of their important properties. ..."
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Summary. In this article, we give the definitions of forward difference, backward difference, central difference and difference quotient, and some of their important properties.
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.