The latter was a result of the Pythagorean theorem and the former the assumption that a + b ≤ c. The contradiction means that it is impossible for both to be true and it is known that the Pythagorean theorem holds.

One might expect that by the Hahn-Banach theorem for bounded linear functionals, every bounded linear functional on C < sub > c </ sub >( X ) extends in exactly one way to a bounded linear functional on C < sub > 0 </ sub >( X ), the latter being the closure of C < sub > c </ sub >( X ) in the supremum norm, and that for this reason the first statement implies the second.

The latter can be computed as a determinant, by the matrix tree theorem, giving a polynomial time algorithm.

In the latter case, the Gelfand-Naimark representation theorem is one avenue in the development of spectral theory for normal operators, and generalizes the notion of diagonalizing a normal matrix.

The latter statement gives original theorem for or ( in Eisenstein's formulation ) for.

However, this is not the case ; this misconception is the result of confusing countable choice with finite choice for a finite set of size n ( for arbitrary n ), and it is this latter result ( which is an elementary theorem in combinatorics ) that is provable by induction.

Since K ( Z, 2 ) is a complex projective space and the latter admits a skeleton structure with no cells in odd dimensions, we can apply the cellular approximation theorem to conclude that the map f can be pushed down to the 2-skeleton, which happens to be the 2-sphere.

This implies that there is a well-defined norm-one linear form on the subspace Ru + Y taking tu + y to t. By the Hahn – Banach theorem the latter admits a norm-one linear extension on ℓ < sup >∞</ sup >( Z ), which is by construction a shift-invariant finitely additive probability measure on Z.

We claim that without loss of generality, the latter inequality is always strict ; once we do this the theorem can be proved as follows.

One can also prove in absolute geometry the exterior angle theorem ( an exterior angle of a triangle is larger than either of the remote angles ), as well as the Saccheri-Legendre theorem, which states that a triangle has at most 180 < sup >°</ sup >.< ref > One again sees the incompatibility of absolute geometry with elliptic geometry, because in the latter theory all triangles have more than 180 < sup >°</ sup >.</ ref >

That is the relevance of Fuglede's theorem: The latter hypothesis is not really necessary.

Other related activities include attempts to create lossless data compression algorithms which will compress all possible inputs or to disprove the four-color theorem ; both of these belong to the first category of problems proven to be impossible ( assuming that there is no significant error in the accepted proof of the latter ).

The latter types of results are known as the Début theorem.

The Hammersley – Clifford theorem shows that other probabilistic models such as Markov networks and Bayesian networks can be represented as factor graphs ; the latter representation is frequently used when performing inference over such networks using belief propagation.

If the latter case holds for even one of these pairs, then we have the first conclusion of Beck's theorem.

The latter question is answered by Sperner's theorem, which gave rise to much of extremal set theory.

As it is, they consider that the North is now reaping the fruits of excess egalitarianism, that in spite of its high standard of living the `` American way '' has been proved inferior to the English and Scandinavian ways, although they disapprove of the socialistic features of the latter.

the former contains no poem dated before 1909 - 10 -- that is, no poem from a period covered by a previous volume -- and the latter has only a few such.

As a matter of fact, this latter approach has already been tried, and with pleasing results.

The latter has been shown to be extensively paraphyletic, and has now been divided into 11 subfamilies, but the former still stands.

The latter quote has provided some controversy because it is technically speaking of wisdom.

This has led to a dramatic depletion in numbers since the latter half of the 19th century, and ' ormering ' is now strictly regulated in order to preserve stocks.

The latter etymology has resulted in a number of theories.

The latter manuscript was severely damaged in the 18th and 19th centuries, and the authorship of the verse has been much disputed ; but likely it also is by Alfred.

The elephant ( though its range has become restricted through the attacks of hunters ) is found both in the savannas and forest regions, the latter being otherwise poor in large game, though the special habitat of the chimpanzee and gorilla.

The latter has not been out of print since it was first published and has sold more than 30 million copies worldwide.

There is also the strange myth of the brothers Aegyptus and Danaus, sons of Belus, with the latter supposedly coming from Egypt, that Marianne Luban has suggested may date to this time.

It is this latter version, which has some syntax differences from the first as well as support for the Apple II high-resolution graphics modes, that most people mean by the term " Applesoft.

For example, armoured personnel carriers were generally replaced by infantry fighting vehicles in a very similar role, but the latter has some capabilities lacking in the former.

However, the Church declared that "' Extreme unction ' ... may also and more fittingly be called ' anointing of the sick '" ( emphasis added ), and has itself adopted the latter term, while not outlawing the former.

Other well-known Berg compositions include the Lyric Suite ( 1926 ), which was later shown to employ elaborate cyphers to document a secret love affair ; the extraordinarily elaborate post-Mahlerian Three Pieces for Orchestra ( completed in 1915 but not performed until after Wozzeck ); and the Chamber Concerto ( Kammerkonzert, 1923 – 25 ) for violin, piano and 13 wind instruments: this latter is written so conscientiously that Pierre Boulez has called it " Berg's strictest composition " and it, too, is permeated by cyphers and posthumously disclosed hidden programs.

The latter in particular has spread widely, and there are now over a hundred Camphill communities and other anthroposophical homes for children and adults in need of special care in about 22 countries around the world.

* Special equipment for locating hostile artillery: flash spotting and notably sound ranging appeared in World War I the latter has been undergone increasing refinement as technology has improved.

In the latter units fire as tactically necessary and replenish to maintain or reach their authorised holding ( which can vary ), so the logistic system has to be able to cope with surge and slack.

This suggestion has been recently repeated by Jerome Murphy-O ' Connor: " It is difficult to imagine that an Alexandrian Jew ... could have escaped the influence of Philo, the great intellectual leader ... particularly since the latter seems to have been especially concerned with education and preaching.

The latter has a pronounced effect on vehicle aerodynamics including stability.

Wedge-shaped polycrystals were identified by transmission electron microscopy to grow out of the amorphous phase only after the latter has exceeded a certain thickness, the precise value of which depends on deposition temperature, background pressure and various other process parameters.

Of the latter sample, 944 persons had been studied two years earlier ; ;

The latter jobs include major management responsibilities and have been filled by those who have come up primarily through the engineering-management side.

At the central level the scrutin uninominal voting system was selected over some form of the scrutin de liste system, even though the latter had been recommended by Duverger and favored by all political parties.

This latter failure is more than merely bad reportage and it is distinctly more important than it would have been had the author drawn Clerfayt as, say, a tournament golfer.

The eastern Sanhaja included the Kutama Berbers, who had been the base of the Fatimid rise in the early 10th C., and the Zirid dynasty who ruled Ifriqiya as vassals of the Fatimids after the latter moved to Egypt in 972.

Andrew's report to his sovereign, whom he rejoined in 1251 at Caesarea in the Palestine, appears to have been a mixture of history and fable ; the latter affects his narrative of the Mongols ' rise to greatness, and the struggles of their leader Genghis Khan with Prester John ; it is still more evident in the position assigned to the Mongols ' homeland, close to the prison of Gog and Magog.

These have been heavily armoured vehicles, the former providing close fire-support for infantry and the latter acting as specialized anti-tank vehicles.

He was such a favourite with the latter, that, when Greece was visited by a drought in consequence of a murder which had been committed, the oracle of Delphi declared that the calamity would not cease unless Aeacus prayed to the gods that it might.

When James Bradley and Samuel Molyneux entered this sphere of astronomical research in 1725, there consequently prevailed much uncertainty whether stellar parallaxes had been observed or not ; and it was with the intention of definitely answering this question that these astronomers erected a large telescope at the house of the latter at Kew.

* Several US Navy ships have been named the or the, the latter being a French translation of his penname " Poor Richard ".

After the latter had been crushed, Philip planned to settle Bastarnae families in Dardania ( southern Kosovo / Skopje region ), to ensure that the region was permanently subdued.

In the 20th century the first part of the prologue ( chapters 1: 1-2: 5 ) and the two parts of the epilogue ( 17-21 ) were commonly seen as miscellaneous collections of fragments tacked on to the main text, and the second part of the prologue ( 2: 6-3: 6 ) as an introduction composed expressly for the book ; this view has been challenged in the latter decades of the century, and there is an increasing willingness to see Judges as the work of a single individual, working by carefully selecting, reworking and positioning his source material to introduce and conclude his themes.

Traits that have been engineered into corn include resistance to herbicides and resistance to insect pests, the latter being achieved by incorporation of a gene that codes for the Bacillus thuringiensis ( Bt ) toxin.

Other names have been used to describe this disease, such as " The Black Plague " and " The Black Death "; the latter is now used primarily by scholars to describe the second, and most devastating, pandemic of the disease.

Continuing Arab disquiet over Allied intentions also led during 1918 to the British Declaration to the Seven and the Anglo-French Declaration, the latter promising " the complete and final liberation of the peoples who have for so long been oppressed by the Turks, and the setting up of national governments and administrations deriving their authority from the free exercise of the initiative and choice of the indigenous populations.