Proof of claim: Consider a graph on t vertices and colour its edges with c colours.

: Proof: Consider a language L from PSPACE.

Proof: Suppose that and are two identity elements of.

: 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 first property: from 3. we obtain ( a < sup >− 1 </ sup >)< sup >− 1 </ sup >=

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: If q divides 2 < sup > p </ sup > − 1 then 2 < sup > p </ sup > ≡ 1 ( mod q ).

( Proof: either 1 is positive or − 1 is positive.

< div class =" NavHead " style =" background :# ccf ; text-align: left ; font-size: larger ;"> Proof of ( 1 )</ div >

Fourth Session ( October 1, 787 ) -- Proof of the lawfulness of the veneration of icons was drawn from Exodus 25: 19 sqq.

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.

* Arora, Sanjeev ; Barak, Boaz, " Complexity Theory: A Modern Approach ", Cambridge University Press, 2009, ISBN 978-0-521-42426-4, section 1. 4, " Machines as strings and the universal Turing machine " and 1. 7, " Proof of theorem 1. 9 "

*: Puncture Proof: 1

Proof of Lemma 1:

* 3200th Proof Test Group, 1 July 1948-1 July 1953

* 3200th Proof Test Wing, 1 April 1951-1 July 1952

* Proof that the Sample Bivariate Correlation Coefficient has Limits ± 1

# is a polynomial ( of degree ≤ n − 1 ) in .< ref > Proof: When A is normal, use Lagrange's interpolation formula to construct a polynomial P such that, where are the eigenvalues of A .</ ref >

Proof: Notice that the nim-sum (⊕) obeys the usual associative and commutative laws of addition (+), and also satisfies an additional property, x ⊕ x = 0 ( technically speaking, the nonnegative integers under ⊕ form an Abelian group of exponent 2 ).

* Proof sketch of the approximation of the number of roots of ξ ( s ) whose imaginary parts lie between 0 and T.

Proof sheets were the means by which syndicates provided newspapers with black-and-white line art for the reproduction of strips ( which they arranged to have colored in the case of Sunday strips ).

* Gödel's Proof ( 2002 revised edition ) by Ernest Nagel and James R. Newman, edited by Hofstadter ( ISBN 0-8147-5816-9 ).

* Benson, Donald C., The Moment of Proof: Mathematical Epiphanies, Oxford University Press, USA ; New Ed edition ( December 14, 2000 ).

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

His films include Reservoir Dogs ( 1992 ), Pulp Fiction ( 1994 ), Jackie Brown ( 1997 ), Kill Bill ( 2003, 2004 ), Death Proof ( 2007 ), and Inglourious Basterds ( 2009 ).

In fact, Thurston resolved so many outstanding problems in foliation theory in such a short period of time that, according to Thurston, it led to a kind of exodus from the field, where advisors counselled students against going into foliation theory because Thurston was " cleaning out the subject " ( see " On Proof and Progress in Mathematics ", especially section 6 ).

* Luis Moreno & Bharath Sriraman ( 2005 ). Structural Stability and Dynamic Geometry: Some Ideas on Situated Proof.

* Gödel's Proof by Ernest Nagel and James R. Newman ( 1959 ).

Fusion shred guitarist, and session musician Greg Howe has released solo albums such as Introspection ( 1993 ), Uncertain Terms ( 1994 ), Parallax ( 1995 ), Five ( 1996 ), Ascend ( 1999 ), Hyperacuity ( 2000 ), Extraction ( 2003 ) with electric bassist Victor Wooten and drummer Dennis Chambers, and Sound Proof ( 2008 ).

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 ).

In further TV movies, he played the title role in the biopic Living Proof: The Hank Williams, Jr. Story ( 1983 ); Will Mossup in Hobson ’ s Choice ( 1983 ); Henry Durrie in The Master of Ballantrae ( 1984 ); Martin Campbell in Final Jeopardy ( 1985 ); and William Denbrough in Stephen King ’ s It ( 1990 ).

Discharging these proof obligations is performed using the SPADE Simplifier ( an automated theorem prover ) and the SPADE Proof Checker ( a manual theorem prover, used for those VCs too thorny for the Simplifier to automatically discharge ).

It is a version by Andrei Frolkin of a problem by Ernest Clement Mortimer, and was published in Shortest Proof Games ( 1991 ).

& Knipping, C. ( 2010 ). Proof in Mathematics Education: Research, Learning, and Teaching Sense Publishers.

Proof coinage means special early samples of a coin issue, historically made for checking the dies and for archival purposes, but nowadays often struck in greater numbers specially for coin collectors ( numismatists ).

The mixtape lacked appearance from Proof and Eminem ( who was working on his solo album, Relapse ).

Grand Marnier Cordon Rouge is 40 % alcohol ( 70 Proof in UK, 80 Proof in US ).

Proof of Mechanism or PoM relates to the earliest stages of drug development, often pre-clinical ( i. e., before the drug is given to humans, or before given to research animals ).

It has historically been a Kentucky Straight Bourbon whiskey ( 80 Proof ).

The book professes to be " A Short and Easy Method with the Deists wherein the certainty of the Christian Religion is Demonstrated by Infallible Proof from Four Rules, which are incompatible to any imposture that ever yet has been, or that can possibly be " ( 1697 ).

# Proof ( a rough thought-experiment or argument, decomposing the primitive conjecture into subconjectures ).