: Proof: Let A be infinite RE.

: Proof: Let D be the midpoint of AC.

Proof: Let ( v, λ ) be an eigenvector-eigenvalue pair for a matrix A.

Proof: Let p < sub > 1 </ sub >⊆ p < sub > 2 </ sub > be prime ideals of A and let q < sub > 2 </ sub > be a prime ideal of B such that q < sub > 2 </ sub > ∩ A

Proof: Let S, T be primitive polynomials in R, and assume that their product ST is not primitive, so that some noninvertible element d of R divides all coefficients of ST.

: Proof: Let S be a set, and let T be the union of the elements of S. Then every element of S is a subset of T, and hence is of cardinality less than or equal to the cardinality of T. Cantor's theorem then implies that every element of S is of cardinality strictly less than the cardinality of 2 < sup > lTl </ sup >.

Proof of his popular and critical appeal, Franquin was awarded the very first Grand Prix de la ville d ' Angoulême in 1974.

Proof for the existence of a common Germanic goddess once known as * Fraujon does not exist, but scholars have commented that this may simply be due to lack of evidence.

Lessing outlined the concept of the religious " Proof of Power ": How can miracles continue to be used as a base for Christianity when we have no proof of miracles?

#* Proof: suppose that p is composite, hence can be written with a and Then is prime, but and contradicting statement 1.

* ( German ), reprinted in English translation as " Proof that every set can be well-ordered ", van Heijenoort 1976, pp. 139 – 141.

Proof of this must be supplied with one's application for naturalisation.

Proof: Since p is a prime number the only possible values of gcd ( p < sup > k </ sup >, m ) are 1, p, p < sup > 2 </ sup >, ..., p < sup > k </ sup >, and the only way for gcd ( p < sup > k </ sup >, m ) to not equal 1 is for m to be a multiple of p. The multiples of p that are less than or equal to p < sup > k </ sup > are p, 2p, 3p, ..., p < sup > k − 1 </ sup > p = p < sup > k </ sup >, and there are p < sup > k − 1 </ sup > of them.

* Proof by verbosity, sometimes colloquially referred to as argumentum verbosum-a rhetorical technique that tries to persuade by overwhelming those considering an argument with such a volume of material that the argument sounds plausible, superficially appears to be well-researched, and it is so laborious to untangle and check supporting facts that the argument might be allowed to slide by unchallenged.

This informal analysis can be formalized to make a rigorous proof of the incompleteness theorem, as described in the section " Proof sketch for the first theorem " below.

Proof beyond a reasonable doubt, therefore, is proof of such a convincing character that you would be willing to rely and act upon it without hesitation in the most important of your own affairs.

In the shortest of them ( 43 pages as of 2009 ), which he titles " Apology for the Proof of the Riemann Hypothesis " ( using the word " apology " in the rarely used sense of apologia ), he claims to use his tools on the theory of Hilbert spaces of entire functions to prove the Riemann Hypothesis for Dirichlet L-functions ( thus proving GRH ) and a similar statement for the Euler zeta function, and even to be able to assert that zeros are simple.

Proof that competitors have shared prices can be used as part of the evidence of an illegal price fixing agreement.

Proof: Fix a first-order language L, and let Σ be a collection of L-sentences such that every finite subcollection of L-sentences, i ⊆ Σ of it has a model.

1922: Proof by Abraham Fraenkel that the axiom of choice cannot be proved from the axioms of Zermelo's set theory with urelements.

Proof theory is important in philosophical logic, where the primary interest is in the idea of a proof-theoretic semantics, an idea which depends upon technical ideas in structural proof theory to be feasible.

As there is no explicit Federal requirement for a two person train crew the Utah Transit Authority's FrontRunner service was originally planned to be operated by an operator only with revenue collection being handled by a Proof of Payment system.

Proof of two way contacts, either in the form of QSL cards or via digital entry into Logbook of The World ( LoTW ), must be submitted to qualify.

When a second child is cut, the node itself needs to be cut from its parent and becomes the root of a new tree ( see Proof of degree bounds, below ).

Proof by exhaustion, also known as proof by cases, perfect induction, or the brute force method, is a method of mathematical proof in which the statement to be proved is split into a finite number of cases and each case is checked to see if the proposition in question holds.

Proof coins of the early 19th century even appear to be scratched, but it was part of the production process.

Proof coins through 1955 could be ordered individually from the mint.

In protest, he resigned as Eminem's hypeman, with the position being taken up by Proof.

Among Moore's most famous works are his book Principia Ethica, and his essays, " The Refutation of Idealism ", " A Defence of Common Sense ", and " A Proof of the External World ".

: Proof: each minimal polynomial has degree at most.

It uses an exterior Water and Boil Proof ( WBP ) glue similar to most exterior plywoods.

The short story volume, Final Proof, was recognized by Ellery Queen as one of Queen's Quorum -- the most important collections of detective short stories.

Proof sets issued in 2009 contain 18 coins – the most ever included – as that year marked the Lincoln Birth Bicentennial and featured four different reverses for the Lincoln Cent.

Proof of this was that even though Cheoljong ruled the country for 13 years, until his very last days he had not yet learned how to move with dignity or how to wear royal clothes, so that in even the most luxurious of robes he still looked like a fisherman.

Proof of concept trials have indicated that Ebola-targeted siRNAs may be effective as post-exposure prophylaxis in humans, with 100 % of non-human primates surviving a lethal dose of Zaire Ebolavirus, the most lethal strain.

Proof of concept centers provide " seed funding to novel, early stage research that most often would not be funded by any other conventional source ".

Clifford Vivian Devon " Cliff " Curtis ( born 27 July 1968 ) is a New Zealand actor who has had major roles in film, including The Piano, Whale Rider, and Blow, and most recently has appeared in NBC's television series Trauma, and Body of Proof, and starred in ABC's TV series, Missing which was cancelled after its first season.

Proof of the devastation can be seen from the Domesday survey of 1086 most of the lands in Cheshire were recorded as ' wasta ', or wasteland, as " abandoned or useless lands ".

* Proof that Lingula ( Brachiopoda ) is not a living-fossil, and emended diagnoses of the Family Lingulidae A paper that argues that genus Lingula is not as old as most think.

The Republic of Tecala, where most of Proof of Life is set, is a fictional South American country.

Critic David Ansen gave the film a mixed review, writing, " Taylor Hackford's thriller Proof of Life leaves a lot to be desired, but it's got its hands on a fascinating subject ... To be fair, Tony Gilroy's screenplay keeps the romance on the back burner ... Thorne is the most compelling aspect of Proof of Life, thanks to Crowe's quiet, hard-bitten charisma.