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.

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 that competitors have shared prices can be used as part of the evidence of an illegal price fixing agreement.

Proof that eccentric films can survive in today's off-the-rack Hollywood!

One such free community, Collectors Proof enables manufacturers and users alike to associate unique identification numbers to virtually any item so that each new owner can update its chain of custody.

( 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 >.

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 of this can be found on platform Five of Sheffield railway station, because the Sheaf also runs under that, and it also flooded at the same time.

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.

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 of the devastation can be seen from the Domesday survey of 1086 most of the lands in Cheshire were recorded as ' wasta ', or wasteland, as " abandoned or useless lands ".

Proof of Mexican citizenship is required to attend public schools for free, but foreigners can attend public schools by paying a tuition.

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 >.

Proof: If we have a description in, we can convert it into a description in our optimal language by first describing as a computer program ( part 1 ), and then using the original description as input to that program ( part 2 ).

Proof of this can be seen at the Parker Flats at ( the former ) Gage School.

Proof may be rendered invalid if the firearm is damaged or modified significantly ; at this point it is described as " out of proof " and must be re-proofed before it can be sold or transferred.

The theme can also be heard in Quentin Tarantino's movie Death Proof as Rosario Dawson's character's ringtone.

The last equation in Proof 1 above can be expressed as

: Proof: Let A be infinite RE.

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.

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 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.

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: 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: Let D be the midpoint of AC.

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.

Proof: Let ( v, λ ) be an eigenvector-eigenvalue pair for a matrix A.

Proof that the system did not remain foolproof forever was seen in GM's problems of the 1980s, 1990s, and 2000s.

Proof of this may be seen namely in remains such as the Punic Tower, Xarolla Catacombs, Cart Ruts at ' Tal-Bakkari ', ' Tal-Hlantun Tower ' and many others.

Spielberg had seen a copy of Proof and offered to produce the project, which became Fandango, through his company, Amblin Entertainment.

Although Proof does not appear in the song, he provides a spoken introduction to the video, and is seen alongside his bandmates.