In this case, if a proof uses this statement, researchers will often look for a new proof that doesn't require the hypothesis ( in the same way that it is desirable that statements in Euclidean geometry be proved using only the axioms of neutral geometry, i. e. no parallel postulate.

Empire magazine praised the film saying " the gaudily gory, virtuoso, hyper-kinetic horror sequel / remake uses every trick in the cinematic book " and confirms that " Bruce Campbell and Raimi are gods " and Caryn James of The New York Times called it " genuine, if bizarre, proof of Sam Raimi's talent and developing skill.

The proof uses Euclid's lemma ( Elements VII, 30 ): if a prime p divides the product of two natural numbers a and b, then p divides a or p divides b ( or perhaps both ).

The proof uses modal logic, which distinguishes between necessary truths and contingent truths.

The proof of Gödel's completeness theorem given by Kurt Gödel in his doctoral dissertation of 1929 ( and a rewritten version of the dissertation, published as an article in 1930 ) is not easy to read today ; it uses concepts and formalism that are outdated and terminology that is often obscure.

George Boolos has since sketched an alternative proof of the first incompleteness theorem that uses Berry's paradox rather than the liar paradox to construct a true but unprovable formula.

The proof of this statement uses the linearity of classical probability, and has exactly the same structure as the proof of the quantum no-cloning theorem.

Newman's proof is arguably the simplest known proof of the theorem, although it is non-elementary in the sense that it uses Cauchy's integral theorem from complex analysis.

Perelman's proof uses a modified version of a Ricci flow program developed by Richard Hamilton.

The proof uses the classification of finite simple groups.

Although BB84 does not use entanglement, Ekert's protocol uses the violation of a Bell's inequality as a proof of security.

The proof of Euler's identity uses only the formula for the geometric series and the fundamental theorem of arithmetic.

A sig block is easily forged, whereas a digital signature uses cryptographic techniques to provide verifiable proof of authorship.

* Centipede game, the Nash equilibrium of which uses a similar mechanism as its proof.

The alternative is formal software verification, which uses mathematical proof techniques to show the absence of bugs.

An alternative proof uses Prüfer sequences.

While the proof that is a Noetherian ring uses the order structure of, typical proofs in ring theory in general do not assume such additional structure on the ring.

One proof of the impossibility of finding a planar embedding of K < sub > 3, 3 </ sub > uses a case analysis involving the Jordan curve theorem, in which one examines different possibilities for the locations of the vertices with respect to the 4-cycles of the graph and shows that they are all inconsistent with a planar embedding.

The original proof uses hypergeometric functions and innovative tools from the theory of Hilbert spaces of entire functions, largely developed by de Branges.

In logic, a formal proof is not written in a natural language, but instead uses a formal language consisting of certain strings of symbols from a fixed alphabet.

This proof uses definition of even integers, as well as distribution law.

* H. G. Wells wrote The Croquet Player, which uses croquet as a metaphor for the way in which man confronts the very problem of his own existence.

Rader's algorithm, exploiting the existence of a generator for the multiplicative group modulo prime, expresses a DFT of prime size as a cyclic convolution of ( composite ) size, which can then be computed by a pair of ordinary FFTs via the convolution theorem ( although Winograd uses other convolution methods ).

Discussion of political correctness, to the extent that it uses the term and accepts the existence of political correctness as a significant force, is almost invariably critical.

Aquinas uses this principle to construct an objection to God's existence, an objection which he in turn answers and refutes generally ( cf.

Later Hesiod uses a lot of eastern material in his cosmology and in the genealogical trees of the gods, and he introduces the idea of the existence of something else behind the gods, which was more powerful than they.

The philosopher Ken Wilber uses the term kosmos to refer to all of manifest existence, including various realms of consciousness.

Hubbard uses the existence of body thetans to explain many of the physical and mental ailments of humanity which, he says, prevent people from achieving their highest spiritual levels.

J. J. C. Smart, a proponent of this view, uses the term " tenselessness " to describe the simultaneous existence of past, present, and future.

Somewhat surprisingly, given their wide range of practical uses, mathematicians have not yet proven that in three dimensions solutions always exist ( existence ), or that if they do exist, then they do not contain any singularity ( smoothness ).

The earliest uses of the term appear to have been commercial — for example, the Delmarva Heat, Light, and Refrigerating Corp. of Chincoteague, Virginia, was in existence by 1913 — but general use of the term did not occur until the 1920s.

After arguing that evolution is capable of explaining the origin of complexity, near the end of the book Dawkins uses this to argue against the existence of God: " a deity capable of engineering all the organized complexity in the world, either instantaneously or by guiding evolution ... must already have been vastly complex in the first place ..." He calls this " postulating organized complexity without offering an explanation.

* Super-Girl — In Superman # 123 ( August 1958 ), Jimmy Olsen uses a magic totem to wish a " Super-Girl " into existence as a companion and helper for Superman ; however, the two frequently get in each other's way until she is fatally injured protecting Superman from a Kryptonite meteor that a criminal has dropped towards him.

This approach has been unpopular, presumably due to the requirement of a far call which uses x86 memory segmentation and the resulting lack of portability it causes, and existence of the faster instructions mentioned above.

In a plot very similar to that of " Time ", " The Mind and the Matter " tells of a man who uses his mind to erase humanity, only to find that existence without other people is unbearable.

The company producing this liquid was incorporated in 1905 and is still in existence, though the recommended uses are now more modest: principally as a mouthwash and topical antiseptic.

( While the railroad line is no longer in existence, the Cardinal Greenway uses the old trail bed as a biking and hiking trail.

The building, the first the company had built new in its 13-year existence, uses an array of 3, 248 solar panels, spanning, and providing 650 kilowatts of power to eBay's campus.

In addition to the general Storyteller rules, it uses a number of specific mechanics aimed towards simulating the vampiric existence.

:* The non-custodial parent viewing the custodial parent as someone who uses the law for financial gain (' using ' the child's existence as an income stream ).

He buys an Inverted Jenny from a stamp dealer ( who claims-inaccurately-that it is " the only known copy in existence ") for $ 1. 25 million, and then uses it to mail a postcard to the crooked lawyers " Granville & Baxter " plotting to steal his money from him.

Graves ' original theory uses a double helix model to show the interrelatedness of an individual's perception of life conditions with their inner neuronal systems, producing a level of psychological existence.

The FCC was susequently charged by Congress to conduct a study into the uses and capabilities of Amateur Radio Service communications in emergencies and disaster relief, and into the existence of impediments thereto, including the effects of unreasonable or unnecessary private land use restrictions on residential antenna installations.

* Josip Plemelj solves the Riemann problem about the existence of a differential equation with a given monodromic group and uses Sokhotsky-Plemelj formulae.

The homotopy theory of defects uses the fundamental group of the order parameter space of a medium to discuss the existence, stability and classifications of topological defects in that medium.