: Proof: By mathematical induction.

By 1990, Banks went on to produce Spice 1's 187 Proof E. P., another successful independent project that he put his signature production on, selling more than 200, 000 units.

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

Any proof by contrapositive can also be trivially formulated in terms of a Proof by contradiction: To prove the proposition, we consider the opposite,.

* Proof: try to prove that the problem cannot be solved.

* Proof of Concept, a test to prove something intended is actually possible, conceptually identical to the intended final product

* Proof of concept, a term used in system development to prove the viability of a potential system

He famously put the point into dramatic relief with his 1939 essay " Proof of an External World ", in which he gave a common sense argument against scepticism by raising his right hand and saying " Here is one hand ," and then raising his left and saying " And here is another ," then concluding that there are at least two external objects in the world, and therefore that he knows ( by this argument ) that an external world exists.

Proof: If there is no possible move, then the lemma is vacuously true ( and the first player loses the normal play game by definition ).

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.

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

In 1924, there were rumours that gold had been discovered close to the Rabbit Proof fence and the town of Holleton was established about 50 km from Narembeen, but Narembeen prospered as it was the closest train station to Holleton and offered a motor transport service to the miners.

The ranch was also featured as a supposedly " haunted house " in an episode of Proof Positive and is frequently mentioned on the Howard Stern Show, usually to announce that another porn star has started to work there.

Proof: Suppose the product of two primitive polynomials f ( x ) and g ( x ) is not primitive, so there exists a prime number p that is a common divisor of all the coefficients of the product.

Generally this is in a very small detail, such as the number of leaves on the ear of corn on the recent US Wisconsin state quarter: File: 2004 WI Proof. png.

From the criterion it also follows that Carmichael numbers are cyclic .< ref > Proof sketch: If is square-free but not cyclic, for two prime factors and of.

* Demonstratio evangelica ( Proof of the Gospel ) is closely connected to the Praeparatio and comprised originally twenty books of which ten have been completely preserved as well as a fragment of the fifteenth.

Proof of second property: since a * b and b * b < sup >− 1 </ sup > are defined, so is a * b * b < sup >− 1 </ sup >=

#* Proof: Then so Thus However, is prime, so or In the former case, hence ( which is a contradiction, as neither 1 nor 0 is prime ) or In the latter case, or If however, which is not prime.

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

#* Proof:, so is a square root of 2 modulo.

* In Hal Clement's short story Proof ( 1942 ), neutronium is the only form of solid matter known to Solarians, the inhabitants of the Sun's interior.

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: either 1 is positive or − 1 is positive.

Proof by contradiction is also known as indirect proof, apagogical argument, proof by assuming the opposite, and reductio ad impossibilem.

* Proof that R is uncountable

Proof: Suppose G = ⟨ H | R ⟩ is a finitely presented, residually finite group.

* Proof of Fermat's Last Theorem is discovered by Andrew Wiles.

Proof: We only give the proof in the simplified case ; the general case is similar.

A Proof of concept prototype is used to test some aspect of the intended design without attempting to exactly simulate the visual appearance, choice of materials or intended manufacturing process.

Proof that conservation does work, some species have returned home for continuation of programmes to reintroduce them to their own environment

Proof: If is diagonalizable, then A is annihilated by some polynomial, which has no multiple root ( since ) and is divided by the minimal polynomial of A.

Quentin Tarantino's Death Proof ( 2007 ) is a modern homage to this genre ( containing some references to Vanishing Point ), as well as to slasher films and the films of Russ Meyer.

Also included are some original pieces such as " Fate to Despair ", " Awake ", and " Proof of Blood ".

Usually a demonstration is less than a Proof of concept, but can come some of the way to showing how a business project may be justified.

Proof this fact some families that later used Korwin coat of arms but in the fifteenth century were still considered to be in Ślepowron clan, ( Kochanowski family, for example ).

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 entire video is in black-and-white, apart from some cutaway sequences, including Bizarre living in a run-down house, Eminem as an angry old man, a cut scene where Eminem is at his old house and when Proof is with the chicks.

This is touted as a more accurate processing of the rebate, reducing the potential for human or mechanical error and in many cases eliminating the postage costs associated with traditional mail-in rebates, although some require the UPC or Proof Of Purchase to be mailed in.