: Proof: Let A be infinite RE.

Proof: Let d be the position of the leftmost ( most significant ) nonzero bit in the binary representation of s, and choose k such that the dth bit of x < sub > k </ sub > is also nonzero.

: 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.

The U. S. Mint began producing silver quarters again in 1992 for inclusion in the annual Silver Proof set.

*** Washington District of Columbia and U. S. territories 2009 ( Proof Only )

* Body of Proof, U. S. ( ABC ), 2011

Proof of this is the legacy " coral " that led to the U. S., especially since the founding of St. Olaf College Choir by F. Melius Christiansen in 1912.

After leaving the U. S. Attorney's office, Turow became a novelist, writing legal thrillers such as The Burden of Proof, Presumed Innocent, Pleading Guilty, and Personal Injuries, which Time magazine named as the Best Fiction Novel of 1999.

Proof: It is straightforward to prove that ∇ satisfies the Leibniz identity and is C < sup >∞</ sup >( S < sup > 2 </ sup >) linear in the first variable.

Starting in 1992, the U. S. Mint began issuing Silver Proof Sets annually, which contain dimes composed of the pre-1965 standard of 90 percent silver and 10 percent copper.

In Handbook of Proof Theory, S. Buss, ed.

In: Handbook of Proof Theory, S. Buss, ed.

* Japaridze, G., and De Jongh, D. ( 1998 ) " The logic of provability " in Buss, S., ed., Handbook of Proof Theory.

In: Handbook of Proof Theory, S. Buss, ed., North-Holland, 1998, pages 475-545.

* R. Schoen and S .- T. Yau, " Proof of the positive mass theorem.

* R. Schoen and S. T. Yau, " Proof that the Bondi mass is positive ", Phys.

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 2000, Crow became romantically involved with his co-star Meg Ryan on the set of Proof of Life.

* Proof of the uncountability of the set of all subsets of the set of natural numbers ( Cantor's theorem 1891 )

* Proof ( 2003-2004 ) set in Dublin and starring Orla Brady.

Proof by Contradiction: Assume that there is a non-empty set of natural numbers that are not interesting.

The Birmingham pen trade prospered and areas such as the Gun Quarter and Jewellery Quarter existed with high standards of production set out by institutions such as the Birmingham Proof House and Birmingham Assay Office.

: Proof: In a size -( n + 1 ) set, choose one distinguished element.

Some issues sell for significant sums, for example the 1995-W proof ($ 3, 500 as of 2011 ) and the 2006 20th anniversary set containing a special " Reverse Proof " coin along with a regular proof coin and the new " Burnished Uncirculated " coin ( over $ 300 as of 2011 ).

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