Since the change to better nutrition, he feels he can report on improvements in health, though he considers the following statements observations and not scientific proof.

The proof of the independence result also shows that a wide class of mathematical statements, including all statements that can be phrased in the language of Peano arithmetic, are provable in ZF if and only if they are provable in ZFC.

We can call a person, a house, a symphony, a fragrance, and a mathematical proof beautiful.

* 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.

* California Penal Code Section 159: " No person can be convicted of common barratry except upon proof that he has excited suits or proceedings at law in at least three instances, and with a corrupt or malicious intent to vex and annoy.

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An " elementary " proof can be given using the fact that geometric mean of positive numbers is less than arithmetic mean

The basic idea of his proof is that a proposition that holds of x if x = n for some natural number n can be called a definition for n, and that the set

The higher-dimensional chain rule can be proved using a technique similar to the second proof given above.

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 ")

The proof for the fact that no co-NP-complete problem can be in NP, if NP is unequal co-NP, is symmetrical.

The proof of this fact relies on an algorithm which, given the first n digits of Ω, solves Turing's halting problem for programs of length up to n. Since the halting problem is undecidable, Ω can not be computed.

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Convictions can only be made when proof beyond a reasonable doubt is achieved.

The burden of proof for civil contempt, however, is a preponderance of the evidence, and theoretically punitive sanctions ( punishment ) can only be imposed after due process but the due process is unpublished.

The proof can be broken up into several major pieces as follows:

Indeed, following, suppose ƒ is a complex function defined in an open set Ω ⊂ C. Then, writing for every z ∈ Ω, one can also regard Ω as an open subset of R < sup > 2 </ sup >, and ƒ as a function of two real variables x and y, which maps Ω ⊂ R < sup > 2 </ sup > to C. We consider the Cauchy – Riemann equations at z = 0 assuming ƒ ( z ) = 0, just for notational simplicity – the proof is identical in general case.

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where the rule is that whenever instances of "" appear on lines of a proof, "" can be placed on a subsequent line.

where the rule is that whenever instances of "", and "" and "" appear on lines of a proof, "" can be placed on a subsequent line.

They also furnish proof that, in modern war, message sending must be monitored.

Both Alfred Harcourt and Donald Brace had written him enthusiastic praise of Elmer Gantry ( any changes could be made in proof, which was already coming from the printer ) and they had ordered 140,000 copies -- the largest first printing of any book in history.

The appointment of U Thant of Burma as the U.N.'s Acting Secretary General -- at this writing, the choice appears to be certain -- offers further proof that in politics it is more important to have no influential enemies than to have influential friends.

Most of them, the world over, operate on the same principle by which justice is administered in France and some other Latin countries: the customer is to be considered guilty of abysmal ignorance until proven otherwise, with the burden of proof on the customer himself.

Narayanan – AIR India Reporter 1988 Court Page No. 1381 ; 1988 Volume No. 3 SCC Court Cases Page No. 366 ; 1988 PLJR 78 – Although an affidavit may be taken as proof of the facts stated therein, the Courts have no jurisdiction to admit evidence by way of affidavit.

Because of independence, the decision whether to use of the axiom of choice ( or its negation ) in a proof cannot be made by appeal to other axioms of set theory.

The grammarian Athenaeus quoted some verses about perfumed ointments to prove just how unwarlike Alcaeus could be and he quoted his description of the armour adorning the walls of his house as proof that he could be unusually warlike for a lyric poet.

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.

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.

The work which black women have been forced to perform, either in slavery or in a discriminatory work place, that would be non-gender conforming for white women has been used against black women as a proof of their emasculating behaviour.

It was at this time that ` Abdu ' l-Bahá, in order to provide proof of the falsity of the accusations leveled against him, in tablets to the West, stated that he was to be known as "` Abdu ' l-Bahá " an Arabic phrase meaning the Servant of Bahá to make it clear that he was not a Manifestation of God, and that his station was only servitude.

Reason makes equal claim to each proof, since they are both correct, so the question of the limits of time must be regarded as meaningless.

As the problem of P ≟ PSPACE has not yet been solved, the proof of inequality between BQP and classes mentioned above is supposed to be difficult.

The burden of proof should be on the people who make these statements, to show where they got their information from, to see if their conclusions and interpretations are valid, and if they have left anything out.

It follows that the proof that follows may be adapted for any Euclidean domain.

Maimonides argued that executing a defendant on anything less than absolute certainty would lead to a slippery slope of decreasing burdens of proof, until we would be convicting merely " according to the judge's caprice ".

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.

In these papers he sketched a proof of the Poincaré conjecture and a more general conjecture, Thurston's geometrization conjecture, completing the Ricci flow program outlined earlier by Richard Hamilton.

Grigori Perelman sketched a proof of the full geometrization conjecture in 2003 using Ricci flow with surgery.

In 2003 Grigori Perelman sketched a proof of the geometrization conjecture by showing that the Ricci flow can indeed be continued past the singularities, and has the behavior described above.

Haken sketched out a proof of an algorithm to check if two Haken manifolds were homeomorphic or not.

The proof sketched in the previous paragraph that the consistency of ZFC + " there is an inaccessible cardinal " implies the consistency of ZFC + " there is not an inaccessible cardinal " can be formalized in ZFC.

The proof sketched in this announcement was never published by them, though it appears in the book.

A proof for the homotopy invariance of singular homology groups can be sketched as follows.

The consequence ((( A ) B ) C ) = ( AC )(( B ) C ), C7 in LoF, enables an algorithm, sketched in LoFs proof of T14, that transforms an arbitrary pa formula to an equivalent formula whose depth does not exceed two.

The theorem is named after the mathematician Friedrich Engel, who sketched a proof of it in a letter to Wilhelm Killing dated 20 July 1890.

Thurston's geometrization conjecture whose proof was sketched