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 ).

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 ).

* Elusive Proof, Elusive Prover: A New Mathematical Mystery, by Dennis Overbye, New York Times, Science, August 15, 2006.

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

* Proof by contrapositive

meaning " quod erat demonstrandum " or by one of the tombstone marks "" or "" meaning " End of Proof ", introduced by Paul Halmos following their usage in magazine articles.

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

* 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.

* Proof, 1980 B / W student film by Black-and-white, expanded to Fandango

* Proof ( 1991 film ), an Australian film by Jocelyn Moorhouse

* Proof ( play ), a play by David Auburn

** Proof ( 2005 film ), a film by John Madden, based on the play

* " Proof ", a 1942 hard SF short story by Hal Clement

* " Proof ", a song by Coldplay from " Speed of Sound "

* " Proof " ( I Am Kloot song ), a song by I Am Kloot from their eponymous second album

* " Proof ", a song by Paul Simon from The Rhythm of the Saints

* " Proof ", a song by Happy Rhodes from Many Worlds Are Born Tonight

Proof: Assume that a signal f ( t ) which has finite support in both domains exists.

: Proof: Assume the contrary, and let C be the largest cardinal number.

Proof: By symmetry, it suffices to prove that there is some constant c such that for all bitstrings s

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.

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 that R is uncountable

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

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

( Proof: If m < n, then we can view R < sup > m </ sup > as a subspace of R < sup > n </ sup >, and the non-empty open subsets of R < sup > m </ sup > are not open when considered as subsets of R < sup > n </ sup >.