#* Proof: We show if p

A closed densely defined operator T is bounded if and only if T < sup >∗</ sup > is bounded .< ref > Proof: We have:.

*“ Mechanical Mathematicians: The End of Proof as We Know It ,” review essay, The Information Society, 20, 1 ( January, 2004 ).

: Proof: We use mathematical induction.

Are We Knowledge Proof?

* 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.

Note that the relationships above are only valid for the form of the Fourier transform shown in the Proof section below.

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: Using the same sum for R ( p, n ) as in Lemma 2, we see that since p < sup > 2 </ sup > > 2n, in fact only the first term is nonzero ; this term is exactly the right-hand side of the above equation.

In 2002, Bracegirdle co-wrote two songs on Cher's album Living Proof, " You Take It All " and " Alive Again ", the latter which was released as a single in Europe only.

In 1997, Pantera released their first and only official live album titled Official Live: 101 Proof.

Proof: The argument invokes only Fuglede's theoerm.

Proof of this is that he only won fifteen fights by knockout, out of more than seventy professional bouts.

The unusual repudiation of torture is reflected not only in our criminal laws, see e. g. §§ 2340-2340A, but also in international agreements, in centuries of Anglo-American law, see e. g., John H. Langbeen, Torture and the Law of Proof: Europe and England in the Ancien Régime ( 1977 ) (“ Torture and the Law of Proof ”), and in the long standing policy of the United States, repeatedly and recently reaffirmed by the President.

Echo had originally planned to release " Proof " as a single to promote the second album, but after agreeing on artwork and a track listing ( and commissioning a video featuring Christopher Eccleston, directed by Krishna Stott ), the label shelved these ideas and subsequently issued the single as a download only ( though a few copies were issued in Europe on the PIAS imprint ).

Proof of Zocor's efficacy in reducing cardiovascular disease was only presented five years after its original introduction, and then only for secondary prevention.

Birmingham Proof House was built in 1813, then one of only two such proof houses in England, the other being in London.

Proof, the only group member not to rap in the original song, also utilised the beat in a solo cover version called " Shoot At You " in 2002, available from his website.

Proof of this are the results of UDI in the 2000 municipal elections, the parliamentary elections of 2001, and the Lagos-Longueira agreement of January 17, 2003 to modernize the State administration and give a consensual political solution to Inverlink case and MOP-Gate case, which affected the institutional stability of the Lagos administration.

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

Proof examples from 1936 to 1942 and 1950 to 1967 were struck at the Philadelphia Mint ; in 1968 proof production was shifted to the San Francisco Mint.

 He made another major contribution to the field in his 1979 paper, joint with his student Robert A. Reckhow, " The Relative Efficiency of Propositional Proof Systems ", in which they formalized the notions of p-simulation and efficient propositional proof system, which started an area now called propositional proof complexity.

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.

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.

* Metamath Proof explorer a repository of mathematical theorems along with their proofs using the Metamath proof assistant.

* Proof by exhaustion or brute force method, a method of mathematical proof

Proof methods that are not direct include proof by contradiction, including proof by infinite descent.

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 ).

:: Sketch of proof: Proof by contradiction.

* Proof by exhaustion, proof by examining all individual cases

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.

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

A popular front-end for proof assistants is the Emacs-based Proof General, developed at the University of Edinburgh.

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 LeMats which found their way through the Union blockade were stamped with British proof marks from the Birmingham Proof House, leading to the misapprehension that the pistols were actually manufactured in the UK.

Becker's Postulate later played a role in the formalization given, by Charles Hartshorne, the American process theologian, of the Ontological Proof of God's existence, stimulated by conversations with the logical positivist and opponent of the alleged proof, Rudolf Carnap.

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 ).

#* 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.

Thus, boundedness of T < sup >∗</ sup > on its domain does not imply boundedness of T. On the other hand, if T < sup >∗</ sup > is defined on the whole space then T is bounded on its domain and therefore can be extended by continuity to a bounded operator on the whole space .< ref > Proof: being closed, the everywhere defined T < sup >∗</ sup > is bounded, which implies boundedness of T < sup >∗∗</ sup >, the latter being the closure of T. See also for the case of everywhere defined T .</ ref > If the domain of T < sup >∗</ sup > is dense, then it has its adjoint T < sup >∗∗</ sup >.