As a consequence of the Hahn – Banach theorem, this map is injective, and isometric.

* The Hilbert cube is compact, again a consequence of Tychonoff's theorem.

In this situation, the chain rule represents the fact that the derivative of is the composite of the derivative of f and the derivative of g. This theorem is an immediate consequence of the higher dimensional chain rule given above, and it has exactly the same formula.

As a consequence of the Nyquist – Shannon sampling theorem, any spatial waveform that can be displayed must consist of at least two pixels, which is proportional to image resolution.

According to Noether's theorem, the conservation of energy is a consequence of the fact that the laws of physics do not change over time.

As a consequence of Liouville's theorem, any function that is entire on the whole Riemann sphere ( complex plane and the point at infinity ) is constant.

Although the logical consequence relation is only semidecidable, much progress has been made in automated theorem proving in first-order logic.

It is deduced from the model existence theorem as follows: if there is no formal proof of a formula then adding its negation to the axioms gives a consisten theory, which has thus a model, so that the formula is not a semantic consequence of the initial theory.

An important consequence of the completeness theorem is that it is possible to recursively enumerate the semantic consequences of any effective first-order theory, by enumerating all the possible formal deductions from the axioms of the theory, and use this to produce an enumeration of their conclusions.

The compactness theorem says that if a formula φ is a logical consequence of a ( possibly infinite ) set of formulas Γ then it is a logical consequence of a finite subset of Γ.

This is an immediate consequence of the completeness theorem, because only a finite number of axioms from Γ can be mentioned in a formal deduction of φ, and the soundness of the deduction system then implies φ is a logical consequence of this finite set.

Conversely, for many deductive systems, it is possible to prove the completeness theorem as an effective consequence of the compactness theorem.

As a consequence of the polynomial remainder theorem, the entries in the third row are the coefficients of the second-degree polynomial, the quotient of on division by.

An important consequence of the KAM theorem is that for a large set of initial conditions the motion remains perpetually quasiperiodic.

A multiplicative function is completely determined by its values at the powers of prime numbers, a consequence of the fundamental theorem of arithmetic.

That is, conservation of momentum is a consequence of the fact that the laws of physics do not depend on position ; this is a special case of Noether's theorem.

is a consequence of Stone's theorem on one-parameter unitary groups.

The embedding property is a consequence of a result known as Desargues ' theorem.

Quines are possible in any programming language that has the ability to output any computable string, as a direct consequence of Kleene's recursion theorem.

As a consequence of the noisy-channel coding theorem, noise limits the information content of an analog information carrier to be finite.

One consequence of Toda's theorem is that a polynomial-time machine with a # P oracle ( P < sup ># P </ sup >) can solve all problems in PH, the entire polynomial hierarchy.

As a consequence of the Weierstrass approximation theorem, one can show that the space C is separable: the polynomial functions are dense, and each polynomial function can be uniformly approximated by one with rational coefficients ; there are only countably many polynomials with rational coefficients.

In 1839, Michael Faraday showed that the apparent division between static electricity, current electricity, and bioelectricity was incorrect, and all were a consequence of the behavior of a single kind of electricity appearing in opposite polarities.

Clan leaders took opposite sides and as a consequence, several thousand Hmong participated in the fighting against the Pathet Lao Communists, while perhaps as many were enrolled in the People's Liberation Army.

The congruence of opposite sides and opposite angles is a direct consequence of the Euclidean Parallel Postulate and neither condition can be proven without appealing to the Euclidean Parallel Postulate or one of its equivalent formulations.

A variety of Enlightenment thinkers have alleged the opposite, by suggesting that evil is learned as a consequence of tyrannical social structures.

Because each half of the retina receives light coming from the opposite half of the visual field, the functional consequence is that visual input from the left side of the world goes to the right side of the brain, and vice versa.

Beauty was, for St. Augustine, the consequence of the benevolence and goodness of God's creation, and as a category had no opposite.

: Bakufu policy in this era was designed to marginalize the influence of foreigners in An ' ei Japan ; however, an unintended and opposite consequence of sakoku was to enhance the value and significance of a very small number of thoughtful observers like Thunberg and Titsingh, whose writings document what each scholar learned or discovered first-hand.

Lobachevsky replaced Euclid's parallel postulate with the statement that for any given point there exists more than one line that can be extended through that point and run parallel to another line of which that point is not part ; a famous consequence is that the sum of angles in a triangle must be less than 180 degrees.

A consequence of the theorem is that the order of any element a of a finite group ( i. e. the smallest positive integer number k with a < sup > k </ sup > = e, where e is the identity element of the group ) divides the order of that group, since the order of a is equal to the order of the cyclic subgroup generated by a.

This is a consequence of the fact that the recurring part of a decimal representation is, in fact, an infinite geometric series which will sum to a rational number.

For example, many asymptotic expansions are derived from the formula, and Faulhaber's formula for the sum of powers is an immediate consequence.

As a consequence, a rank-k matrix can be written as the sum of k rank-1 matrices, but not fewer.

A consequence of this is that the total pressure of a mixture of ideal gases is equal to the sum of the partial pressures of the individual gases in the mixture as stated by Dalton's law.

As a consequence, if f and g are as above, with, we can relate the complex powers of f / X and g / X: precisely, if and are non-zero complex numbers with negative integer sum,, then

Soon after, Ebers went bankrupt and Ainsworth lost a large sum as a consequence.

As a consequence, economic capital for a conglomerate is usually substantially less than economic capital is for the sum of its parts.

The Gauss sum can thus be written as a linear combination of Gaussian periods ( with coefficients χ ( a )); the converse is also true, as a consequence of the orthogonality relations for the group ( Z / nZ )< sup >×</ sup >.

Al-Alfi and his partisans failed to raise the sum promised to the Porte ; Salih Pasha received plenipotentiary powers from Istanbul, but in consequence of the letter from the ulema ; and, on the condition of Muhammad Ali's paying 4000 purses to the Porte, it was decided that he should continue in his post as governor of Egypt, and the reinstatement of the beys was abandoned.

Thus, in the motivational example, above, the fact that the sum of residual values is equal to zero it is not accidental but is a consequence of the presence of the constant term, α, in the model.

In May 1795, the antiquarian and topographer John Swete spent some time in Dawlish and reported that although not long ago it had been no more than a fishing village, and the best lodging house would not cost more than half a guinea per week, it was now so fashionable that " in the height of the season, not a house of the least consequence is to be hired for less than two guineas a week, and many of them rise to so high a sum as four or five.

It was, however, evidently impossible for a chartered company to hold its own against the state-supported protectorates of France and Germany, and in consequence, on 1 January 1900, the Royal Niger Company transferred its territories to the British government for the sum of £ 865, 000.

It was, however, evidently impossible for a chartered company to hold its own against the state-supported protectorates of France and Germany, and in consequence, on 1 January 1900, the Royal Niger Company transferred its territories to the British Government for the sum of £ 865, 000.

As a consequence of the degree sum formula, any sequence with an odd sum, such as ( 3, 3, 1 ), cannot be realized as the degree sequence of a graph.

::" In sum, there has been no showing of any threat, either to participants or the public in general, in consequence of the voluntary engagement by adults in private, discreet, sodomous conduct.

Bartholomeus V. Welser lent the Emperor Charles V a great sum of money for which in 1528 he received as security the Province of Venezuela, developing it as Klein-Venedig ( little Venice ) but in consequence of their rapacious acts the Welsers were deprived of their rule before the Emperor's reign was over.

In consequence of the delay the merchant suffered some loss of business at the fair and on his return he refused to pay the rather large sum which Zaschwitz demanded as a condition of returning the horses.

In some simple cases, and at thermodynamic equilibrium, the results can be shown to be a consequence of the additivity property of independent random variables ; namely that the variance of the sum is equal to the sum of the variances of the independent variables.