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Page "Axiom of choice" ¶ 67
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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.
** König's theorem: Colloquially, the sum of a sequence of cardinals is strictly less than the product of a sequence of larger cardinals.
** 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 stating
** Renowned medium, Derek Acorah visited Manning's home, stating that he was able to communicate with the spirit, and that it was called " Jim " and had died from a heart attack at the age of 58 around 1900.
** The Reichstag passes a statement stating that Hitler, s second-in-command Reichsmarshall Hermann Göring should be appointed as Hitler, s succesor as Führer should Hitler die in the middle of the Second World War. Rudolf Hess is to be appointed in Göring, s place should anything befall Göring.
** Canadian Prime Minister Mackenzie King, in English, and Justice Minister Ernest Lapointe, in French, give an international radio address, stating its intentions to declare war against Nazi Germany.
** On some maps intended for wilderness or navigational use, including the topographic maps of the U. S. Geological Survey ( USGS ), a diagram shows the relationship between magnetic north in the area concerned ( with an arrow marked " MN ") and true north ( a vertical line with a five-pointed star at its top ), with a label near the angle between the MN arrow and the vertical line, stating the size of the declination and of that angle, in degrees, mils, or both.
** Pip kept his between his toes, stating that " a gem on the forehead ain't my look.
** NDP: The NDP respond to the Liberal platform stating it is " An empty health package ; a broken promise ; a small but good start ; and a misplaced priority.
** The Conservatives release press statements stating that the Liberals ( and later, the NDP ) are in favour of child pornography.
** EUROCONTROL puts a disclaimer on its data ( stating that USCG data takes precedence ), whereas the FAA certifies its website as meeting regulatory requirements.
It ’ s stating that I ’ m not gorgeous ,, I ’ m not some f ** cking beauty.
** Village Pound marked with a plaque stating in existence since at least 1686
** Marvel Comics issue a press release stating that Chris Claremont is suffering from cardiac stress and that this will result in delays to two books, Exiles and GeNext, which Claremont was due to take over in May.

** and every
** Zorn's lemma: Every non-empty partially ordered set in which every chain ( i. e. totally ordered subset ) has an upper bound contains at least one maximal element.
** Hausdorff maximal principle: In any partially ordered set, every totally ordered subset is contained in a maximal totally ordered subset.
** For every non-empty set S there is a binary operation defined on S that makes it a group.
** On every infinite-dimensional topological vector space there is a discontinuous linear map.
** for every object
** for every object,
** The microcode can employ both pipelines to enable auto-repeating instructions such as rep movsw perform one iteration every clock cycle, while the 80486 needed three clocks per iteration ( and the earliest x86-chips significantly more than the 486 ).
** Jane was responsible for the elevator in every respect
** Online Variorum, showing every change between the six British editions.
** hypernyms: Y is a hypernym of X if every X is a ( kind of ) Y ( canine is a hypernym of dog )
** hyponyms: Y is a hyponym of X if every Y is a ( kind of ) X ( dog is a hyponym of canine )
** United Kingdom Census held, the first to record names and approximate ages of every household member and to be administered nationally.
** Circuit analysis, the process of finding the voltages across, and the currents through, every component in an electrical circuit
** Principal ideal domain, an integral domain in which every ideal is principal
** Unique factorization domain, an integral domain in which every non-zero element can be written as a product of irreducible elements in essentially a unique way
** Atomic domain, an integral domain in which every non-zero non-unit is a finite product of irreducible elements
** Dedekind domain, an integral domain in which every nonzero proper ideal factors into a product of prime ideals
** GCD domain, an integral domain in which every two non-zero elements have a greatest common divisor
** The Pythian Games ( founded 527 BC ) held in Delphi every four years
** The Nemean Games ( founded 516 BC ) held in Argolid every two years

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