Walton dropped everything to serve as a district co-ordinator in the hard-fought Wisconsin primary and proved so useful that he was promoted to be liaison officer to critically important New York City.

* 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.

Two 2008 case studies show that the layout of a system provides a strong correlation to the ease of use for a system and that it proved more beneficial for the design to focus on presenting information in a way that reflected the most important goals and tasks of the user, rather than the structure of the organization.

Maximian's consistent loyalty to Diocletian proved an important component of the Tetrarchy's early successes.

Ignatius's letters proved to be important testimony to the development of Christian theology, since the number of extant writings from this period of Church history is very small.

The book proved to be an enormous success and established Wyndham as an important exponent of science fiction.

Despite having received no military education, Julian proved to be an able military commander, obtaining an battle of Strasbourg | important victory in Gaul and leading a Roman army under the walls of the Sassanid Empire's capital.

Newman made important contributions leading to an invitation to present his work at the 1962 International Congress of Mathematicians in Stockholm at the age of 65, and proved a Generalized Poincaré conjecture for topological manifolds in 1966.

His 1853 trip to Europe to buy arms was undoubtedly the most important experience of his life ; his stay in Paris proved to be a turning point for him.

Perhaps its most important and prominent usage in scientific literature was Nobel laureate Theodor W. Hänsch's reference in a 1979 Scientific American article on spectroscopy where he says that " the spectrum of the hydrogen atoms has proved to be the Rosetta stone of modern physics: once this pattern of lines had been deciphered much else could also be understood ".

Separability is especially important in numerical analysis and constructive mathematics, since many theorems that can be proved for nonseparable spaces have constructive proofs only for separable spaces.

These informal structures proved to be very important in the functioning of the Uzbek state.

Namco's Tekken ( released in arcades in 1994 and on the PlayStation in 1995 ) proved critical to the PlayStation's early success, with its sequels also becoming some of the console's most important titles.

Submarines, which had proved to be an effective weapon during the First World War were anticipated by all sides to be important in the second.

In domestic affairs King strengthened the Liberal policy of increasing the powers of the provincial governments by transferring to the governments of Manitoba, Alberta, and Saskatchewan the ownership of the crown lands within those provinces, as well as the subsoil rights ; these in particular would become increasingly important, as petroleum and other natural resources proved very abundant.

The curriculum guidelines he developed proved important progressive innovations in the field of higher education.

Yale proved important as a site for this research.

Campbell's work on loading coils provided the theoretical basis for his subsequent work on filters which proved to be so important for frequency-division multiplexing.

More recently, Dutilleux's L ' Arbre des Songes has proved an important addition to the repertoire and a fine example of the composer's atonal yet melodic style.

The method has become important in forensic science to assist police detective work, and it has also proved useful in resolving paternity and immigration disputes.

In France, where the term cercle is most usual, the first was Le Club Politique ( 1782 ), and during the French Revolution such associations proved important political forces ( see Jacobins, Feuillants, Cordeliers ).

The last war waged by Louis XIV proved to be one of the most important to dynastic Europe.

This feature proved important in evaluations of entrained fish in water diversions and for studies of migratory fish in rivers.

However it has proved that teen models whether male or female play an important roles also in fashion industries like Akash Soukhee who is only sixteen and a half has been one of the male supermodels in Mauritius.

In emerging markets, such as China and Vietnam, bonds have become increasingly important as term financing options, since access to traditional bank credits has always been proved to be limited, especially if borrowers are non-state corporates.

However, that particular case is a theorem of Zermelo – Fraenkel set theory without the axiom of choice ( ZF ); it is easily proved by mathematical induction.

Note that " completeness " has a different meaning here than it does in the context of Gödel's first incompleteness theorem, which states that no recursive, consistent set of non-logical axioms of the Theory of Arithmetic is complete, in the sense that there will always exist an arithmetic statement such that neither nor can be proved from the given set of axioms.

His first ( pre-IHÉS ) breakthrough in algebraic geometry was the Grothendieck – Hirzebruch – Riemann – Roch theorem, a far-reaching generalisation of the Hirzebruch – Riemann – Roch theorem proved algebraically ; in this context he also introduced K-theory.

The Cook – Levin theorem states that the Boolean satisfiability problem is NP-complete, and in fact, this was the first decision problem proved to be NP-complete.

SAT was the first known NP-complete problem, as proved by Stephen Cook in 1971 ( see Cook's theorem for the proof ).

The term Bayesian refers to Thomas Bayes ( 1702 – 1761 ), who proved a special case of what is now called Bayes ' theorem in a paper titled " An Essay towards solving a Problem in the Doctrine of Chances ".

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.

Gauss proved the method under the assumption of normally distributed errors ( see Gauss – Markov theorem ; see also Gaussian ).

He also studied and proved some theorems on perfect powers, such as the Goldbach – Euler theorem, and made several notable contributions to analysis.

He also proved a result concerning Fermat numbers that is called Goldbach's theorem.

Groups of rank at least 3 are further subdivided into 3 classes by the trichotomy theorem, proved by Aschbacher for rank 3 and by Gorenstein and Lyons for rank at least 4.

J. Chernick proved a theorem in 1939 which can be used to construct a subset of Carmichael numbers.

In graph theory, much research was motivated by attempts to prove the four color theorem, first stated in 1852, but not proved until 1976 ( by Kenneth Appel and Wolfgang Haken, using substantial computer assistance ).

Gödel's second incompleteness theorem, proved in 1931, showed that this was not possible – at least not within arithmetic itself.

Euclid's axioms seemed so intuitively obvious that any theorem proved from them was deemed true in an absolute, often metaphysical, sense.

The Pythagorean theorem is proved.

Cantor supposed that Thales proved his theorem by means of Euclid book I, prop 32 after the manner of Euclid book III, prop 31.

It improves Dirichlet's theorem on prime numbers in arithmetic progressions, by showing that by averaging over the modulus over a range, the mean error is much less than can be proved in a given case.

It was the first major theorem to be proved using a computer.

In 1890, in addition to exposing the flaw in Kempe's proof, Heawood proved the five color theorem and generalized the four color conjecture to surfaces of arbitrary genus — see below.

The existence of this functor can be proved by realizing that it is the right-adjoint to the diagonal functor and invoking the Freyd adjoint functor theorem.