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Page "Axiom of choice" ¶ 53
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** and theorem
** Well-ordering theorem: Every set can be well-ordered.
** Tarski's theorem: For every infinite set A, there is a bijective map between the sets A and A × A.
** Tychonoff's theorem stating that every product of compact topological spaces is compact.
** If S is a set of sentences of first-order logic and B is a consistent subset of S, then B is included in a set that is maximal among consistent subsets of S. The special case where S is the set of all first-order sentences in a given signature is weaker, equivalent to the Boolean prime ideal theorem ; see the section " Weaker forms " below.
** The Vitali theorem on the existence of non-measurable sets which states that there is a subset of the real numbers that is not Lebesgue measurable.
** Stone's representation theorem for Boolean algebras needs the Boolean prime ideal theorem.
** The Nielsen – Schreier theorem, that every subgroup of a free group is free.
** The Hahn – Banach theorem in functional analysis, allowing the extension of linear functionals
** The theorem that every Hilbert space has an orthonormal basis.
** The Banach – Alaoglu theorem about compactness of sets of functionals.
** The Baire category theorem about complete metric spaces, and its consequences, such as the open mapping theorem and the closed graph theorem.
** Gödel's completeness theorem for first-order logic: every consistent set of first-order sentences has a completion.
** The numbers and are not algebraic numbers ( see the Lindemann – Weierstrass theorem ); hence they are transcendental.
** Hilbert's basis theorem
** Bayes ' theorem
** More generally, Rademacher's theorem extends the differentiability result to Lipschitz mappings between Euclidean spaces: a Lipschitz map ƒ: U → R < sup > m </ sup >, where U is an open set in R < sup > n </ sup >, is almost everywhere differentiable.
** Lyapunov's central limit theorem
** Superposition theorem, in electronics
** " Kelvin's vorticity theorem for incompressible or barotropic flow ".
** Artin reciprocity law, a general theorem in number theory that provided a partial solution to Hilbert's ninth problem
** Various proofs of the four colour theorem.

** and sum
** The Lebesgue measure of a countable disjoint union of measurable sets is equal to the sum of the measures of the individual sets.
** Atkinson independently discovered the Midpoint circle algorithm for fast drawing of circles by using the sum of consecutive odd numbers.
** < sub > 1 </ sub >( n ) = ( n ), the sum of all the positive divisors of n.
** the smash product and wedge sum ( sometimes called the wedge product ) in homotopy.
** Note: The sum of the individual digits is usually compared with a previously computed value.
** Bézout domain, an integral domain in which the sum of two principal ideals is again a principal ideal
** Closure axiom for addition: Given two integers a and b, their sum, a + b is also an integer.
** Signal-to-noise and distortion ratio ( SNDR ) indicates in dB the ratio between the powers of the converted main signal and the sum of the noise and the generated harmonic spurs
** Total harmonic distortion ( THD ) is the sum of the powers of all HDi
** it is proved on oath, or in such other manner as may be prescribed, that he has been duly served with the summons, and that a reasonable sum has been paid or tendered to him for costs and expenses, and
** Khaliq-O-Vision, Ray Bardani ( engineers ), David Isaac ( producer ) & Marcus Miller ( producer & artist ) for M < sum >
** Composition ( number theory ), a way of writing a positive integer as a sum of positive integers
** Quid sum miser ( soprano, mezzo-soprano, tenor )
** Roger Frye used experimental mathematics techniques to find the smallest counterexample to Euler's sum of powers conjecture.
** July 10-He discovers that every positive integer is representable as a sum of at most three triangular numbers, noting in his diary " Heureka!
** The Equivalent, a sum paid from England to Scotland at their Union in 1707.
** In a variation on the basic definition of a bioship, the Star Trek: The Next Generation episode " Emergence ", features a story in which the Enterprise-D develops an artificial intelligence from the sum of the ship's experiences, for the sole purpose of creating a biological life form which is released into space.
** Attendance: 32000 ( figures indicate the sum of both days )
** Consequently, the sum of distances d ( F < sub > 1 </ sub >, P ) + d ( F < sub > 2 </ sub >, P ) must be constant as P moves along the curve because the sum of distances d ( P < sub > 1 </ sub >, P ) + d ( P < sub > 2 </ sub >, P ) also remains constant.
** Compute the number of forest descendants for this node, by adding one to the sum of its children's descendants.
** the sum of votes remaining in the multi-seat constituencies after the distribution of the seats, plus
** the sum of votes cast for losing candidates of each party in the first valid round of each single-seat constituency ( similar to the scorporo system ).
** Connected sum

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