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proof and theorem
The reader will find it helpful to think of the special case when the primes are of degree 1, and even more particularly, to think of the proof of Theorem 10, a special case of this theorem.
The axiom of choice was formulated in 1904 by Ernst Zermelo in order to formalize his proof of the well-ordering theorem.
It is possible, however, that there is a shorter proof of a theorem from ZFC than from ZF.
In fact, Zermelo initially introduced the axiom of choice in order to formalize his proof of the well-ordering theorem.
Mordell's theorem had an ad hoc proof ; Weil began the separation of the infinite descent argument into two types of structural approach, by means of height functions for sizing rational points, and by means of Galois cohomology, which was not to be clearly named as that for two more decades.
He had introduced the adele ring in the late 1930s, following Claude Chevalley's lead with the ideles, and given a proof of the Riemann – Roch theorem with them ( a version appeared in his Basic Number Theory in 1967 ).
The " heuristic " approach of the Logic Theory Machine tried to emulate human mathematicians, and could not guarantee that a proof could be found for every valid theorem even in principle.
* Metamath-a language for developing strictly formalized mathematical definitions and proofs accompanied by a proof checker for this language and a growing database of thousands of proved theorems ; while the Metamath language is not accompanied with an automated theorem prover, it can be regarded as important because the formal language behind it allows development of such a software ; as of March, 2012, there is no " widely " known such software, so it is not a subject of " automated theorem proving " ( it can become such a subject ), but it is a proof assistant.
Following Desargues ' thinking, the sixteen-year-old Pascal produced, as a means of proof, a short treatise on what was called the " Mystic Hexagram ", Essai pour les coniques (" Essay on Conics ") and sent it — his first serious work of mathematics — to Père Mersenne in Paris ; it is known still today as Pascal's theorem.
A more general binomial theorem and the so-called " Pascal's triangle " were known in the 10th-century A. D. to Indian mathematician Halayudha and Persian mathematician Al-Karaji, in the 11th century to Persian poet and mathematician Omar Khayyam, and in the 13th century to Chinese mathematician Yang Hui, who all derived similar results .< ref > Al-Karaji also provided a mathematical proof of both the binomial theorem and Pascal's triangle, using mathematical induction.
If one integrates this picture, which corresponds to applying the fundamental theorem of calculus, one obtains Cavalieri's quadrature formula, the integral – see proof of Cavalieri's quadrature formula for details.
SAT was the first known NP-complete problem, as proved by Stephen Cook in 1971 ( see Cook's theorem for the proof ).
In mathematics, a Gödel code was the basis for the proof of Gödel's incompleteness theorem.
A copyright certificate for proof of the Fermat theorem, issued by the State Department of Intellectual Property of Ukraine
The full significance of Bolzano's theorem, and its method of proof, would not emerge until almost 50 years later when it was rediscovered by Karl Weierstrass.
* ( Purely analytic proof of the theorem that between any two values which give results of opposite sign, there lies at least one real root of the equation ).
The first proof relies on a theorem about products of limits to show that the derivative exists.
In his 1799 doctorate in absentia, A new proof of the theorem that every integral rational algebraic function of one variable can be resolved into real factors of the first or second degree, Gauss proved the fundamental theorem of algebra which states that every non-constant single-variable polynomial with complex coefficients has at least one complex root.
* 1799: Doctoral dissertation on the Fundamental theorem of algebra, with the title: Demonstratio nova theorematis omnem functionem algebraicam rationalem integram unius variabilis in factores reales primi vel secundi gradus resolvi posse (" New proof of the theorem that every integral algebraic function of one variable can be resolved into real factors ( i. e., polynomials ) of the first or second degree ")

proof and consists
A basic negligence claim consists of proof of
The Elements of Theology, which consists of 211 propositions, each followed by a proof, beginning from the existence of the One ( the first principle of all things ) and ending with the descent of individual souls into the material world.
Hall and Popkin defend Mill against this accusation pointing out that he begins Chapter Four by asserting that “ that questions of ultimate ends do not admit of proof, in the ordinary acceptation of the term ” and that this is “ common to all first principles .” According to Hall and Popkin, therefore, Mill does not attempt to “ establish that what people do desire is desirable but merely attempts to make the principles acceptable .” The type ofproof ” Mill is offering " consists only of some considerations which, Mill thought, might induce an honest and reasonable man to accept utilitarianism ".
Essentially, her ' proof ' that he murdered his nephews consists of two skeletons discovered in the Tower of London in 1674, some inferences wholly unsupported by the ' evidence ' she offers and the opinions and assertions of ' contemporary ' sources such as John Rous and Thomas More, which Weir is inclined to treat as proven fact.
For the proof, assume that the graph consists of infinitely many vertices and is connected.
The remainder of the proof consists of a verification that F is recursively enumerable and is the least fixed point of Φ.
The Mizar system consists of a formal language for writing mathematical definitions and proofs, a proof assistant which is able to mechanically check proofs written in this language, and a library of formalized mathematics which can be used in the proof of new theorems.
This proof consists of ' completing ' the right triangle to form a rectangle and noticing that the center of that rectangle is equidistant from the vertices and so is the center of the circumscribing circle of the original triangle, it utilizes two facts:
The five main banking halls were based on the same basic layout, starting with the Bank Stock Office of 1791 – 96, consists of a rectangular room, the centre with a large lantern light supported by piers and pendentives, then the four corners of the rectangle have low vaulted spaces, and in the centre of each side compartments rising to the height of the arches supporting the central lantern, the room is vaulted in brick and windows are iron framed to ensure the rooms are as fire proof as possible.
: Overview: The proof consists of two claims about, which we outline, before proving.
This set consists of all wffs for which there is a proof.
The proof of correctness for AKS consists of showing that there exists a suitably small r and suitably small set of integers A such that, if the congruence holds for all such a in A, then n must be prime.
The proof consists of these four steps.
Here the prize-problem consists, especially, in a proof of the conjecture that the lowest excitations of a pure Yang – Mills theory ( i. e. without matter fields ) have a finite mass-gap with regard to the vacuum state.
The proof consists of two parts — first, the proof of the existence of q and r, and second, the proof of the uniqueness of q and r.
An interactive proof system consists of two machines, a prover, P, which presents a proof that a given string n is a member of some language, and a verifier, V, that checks that the presented proof is correct.
Last part that is part 3 consists of chapter 7 to chapter 11. chapter 7 talks about the burden of proof. Chapter 8 talks about estoppel, chapter 9 talks about witnesses, chapter 10 talks about examination of witnesses, and last chapter which is chapter 11 talks about improper admission and rejection of evidence.

proof and tens
The classification of finite simple groups is regarded by some to be the longest proof of a theorem ; it comprises tens of thousands of pages in 500 journal articles by some 100 authors.
John Schlight, in his A War Too Long, said of the PAVN's logistical apparatus :" This sustained effort, requiring the full-time activities of tens of thousands of soldiers, who might otherwise have been fighting in South Vietnam, seems proof positive that the bombing of the Ho Chi Minh Trail had disrupted the North Vietnamese war effort.
The coin was normally issued in cased " proof " condition, and rarely circulated, as well as being issued in small quantities which today often result in extremely high values of many tens of thousands of pounds being achieved when a coin appears at auction.

proof and thousands
Modern tests and contemporary accounts agree therefore that well-made plate armour could protect against longbows, however there are a number of caveats to this point ; not all plate armour was well-made or well looked after, and there were also weak points in the eye and air holes and joints where arrows could penetrate, meaning that even if the armour was proof against nearly all arrows, being shot at by thousands of longbowmen would have been an uncomfortable experience, physically and mentally.
However, analysis of thousands of high-quality games ( most of them computer-generated ) appears to lead to a reliable conclusion ( pending actual proof if true ) that, on the standard 8-by-8 board, perfect play on both sides results in a draw.
As proof that authors and publishers will continue to produce without copyrights, he cites the fact that hundreds of thousands of articles are uploaded onto the Internet by their authors every day, available to anyone in the world for free and that nearly all works written before 1900 are in the public domain, yet pre-1900 works are still published, and still sell.
Shockley offers no additional scientific proof of love-shyness except to reference the thousands of members of online discussion forums devoted to love-shyness.
He seeks out the Hulk and attempts to talk him down, telling his friend that, while he recognised that the Illuminati had been out of line in their decision to exile him to Sakaar and their apparent involvement with the deaths of thousands of innocent people on that planet, including the Hulk's wife and unborn child, his current blind quest for vengeance wasn't him, using Hulk's willingness to protect innocent people caught in the fight between him and a Zom-possessed Doctor Strange as proof that the Hulk was still a hero rather than a man blindly seeking vengeance.
Archeological discoveries throughout the Seward Peninsula show proof that Inupiat Eskimos have been living in the region for thousands of years.
The British correspondent of The Times, Crawford Price, reported: " Eloquent proof of the size and gravity of the Turkish defeat lies in the thousands of the refugees, who come to Salonika like swarms of locusts.
* Metamath – a language for developing strictly formalized mathematical definitions and proofs accompanied by a proof checker for this language and a growing database of thousands of proved theorems
For example, the Library of Congress facility in Culpeper, VA, holds thousands of its titles on U-matic video, as a means of providing access copies and proof for copyright deposit of old television broadcasts and films.
By going through hundreds or thousands of potential similarities between John and Mary, each having a low probability of being true, we may eventually find proof of virtually any hypothesis.
Adler spent long hours poring through thousands of files and eventually found eighty incidents where Wilson met on a professional basis with the CIA and proof that the CIA had indirectly used Wilson after his retirement.
There she leaves a message for Miranda Sharifi, as have thousands of supplicants before her, pouring out her heart and soul, her views on self, her views on pain and its necessity as proof of life, her views on the red syringes — And, at the news of these syringes, Miranda Sharifi opens a comlink.
Generally, the bullion versions have been minted in the millions, while the proof and uncirculated versions were issued in the hundreds of thousands each.

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