The distance formula on the plane follows from the Pythagorean theorem.

Since a maximal ideal in A is closed, is a Banach algebra that is a field, and it follows from the Gelfand-Mazur theorem that there is a bijection between the set of all maximal ideals of A and the set Δ ( A ) of all nonzero homomorphisms from A to C. The set Δ ( A ) is called the " structure space " or " character space " of A, and its members " characters.

This follows from the Heine – Borel theorem.

Then is a compact topological space ; this follows from the Tychonoff theorem.

It follows from this theorem that all Carmichael numbers are odd, since any even composite number that is square-free ( and hence has only one prime factor of two ) will have at least one odd prime factor, and thus results in an even dividing an odd, a contradiction.

In Euclid's original approach, the Pythagorean theorem follows from Euclid's axioms.

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.

The theorem follows by induction on the length of the game from these two lemmas.

from which the strategy above follows by the Sprague – Grundy theorem.

There is a classical analogue to the quantum no-cloning theorem, which we might state as follows: given only the result of one flip of a ( possibly biased ) coin, we cannot simulate a second, independent toss of the same coin.

This follows from the Chinese remainder theorem.

Another way of stating Rice's theorem that is more useful in computability theory follows.

* It follows easily from the Weierstrass approximation theorem that the set Q of polynomials with rational coefficients is a countable dense subset of the space C () of continuous functions on the unit interval with the metric of uniform convergence.

At any given x, a right-angled triangle connects x, y and r to the origin, hence it follows from the Pythagorean theorem that:

The statement of the approximation theorem as originally discovered by Weierstrass is as follows:

This follows from the Chinese remainder theorem and the fact that a ring of the form Z / kZ is a field if and only if k is a prime.

If both a 0 % rate and 100 % rate of taxation generate no revenue, it follows from the extreme value theorem that there must exist at least one rate in between where tax revenue would be a maximum.

It follows from the work-energy theorem that W also represents the change in the rotational kinetic energy E < sub > r </ sub > of the body, given by

* A corollary is a proposition that follows with little or no proof from one other theorem or definition.

A theorem and its proof are typically laid out as follows:

In the case that y < sub > i </ sub > are independent observations from a normal distribution, Cochran's theorem shows that s < sup > 2 </ sup > follows a scaled chi-squared distribution:

The well ordering theorem follows easily from Zorn's Lemma.

The first follows from direct diagonalization ( the space hierarchy theorem, ) and the fact that via Savitch's theorem.

* 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.

Antiderivatives are important because they can be used to compute definite integrals, using the fundamental theorem of calculus: if F is an antiderivative of the integrable function f, then:

In these more general contexts, the bracket does not have the meaning of an inner product, because the Riesz representation theorem does not apply.

a − λ1 is not invertible ( because the spectrum of a is not empty ) hence a = λ1: this algebra A is naturally isomorphic to C ( the complex case of the Gelfand-Mazur theorem ).

Compactness in this more general situation plays an extremely important role in mathematical analysis, because many classical and important theorems of 19th century analysis, such as the extreme value theorem, are easily generalized to this situation.

This is because if a group has sectional 2-rank at least 5 then MacWilliams showed that its Sylow 2-subgroups are connected, and the balance theorem implies that any simple group with connected Sylow 2-subgroups is either of component type or characteristic 2 type.

( For groups of low 2-rank the proof of this breaks down, because theorems such as the signalizer functor theorem only work for groups with elementary abelian subgroups of rank at least 3.

Gordan, the house expert on the theory of invariants for the Mathematische Annalen, was not able to appreciate the revolutionary nature of Hilbert's theorem and rejected the article, criticizing the exposition because it was insufficiently comprehensive.

( This doesn't violate Gödel's theorem, because Euclidean geometry cannot describe a sufficient amount of arithmetic for the theorem to apply.

That is, energy is conserved because the laws of physics do not distinguish between different instants of time ( see Noether's theorem ).

Because the four color theorem is true, this is always possible ; however, because the person drawing the map is focused on the one large region, he fails to notice that the remaining regions can in fact be colored with three colors.

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.

The uniqueness in this theorem requires excluding 1 as a prime because it is the multiplicative identity.

However, it is standard practice to assume that the sample mean from a random sample is normal, because of the central-limit theorem.

Her brilliant theorem is known only because of the footnote in Legendre's treatise on number theory, where he used it to prove Fermat's Last Theorem for p = 5 ( see Correspondence with Legendre ).

Because polynomials are among the simplest functions, and because computers can directly evaluate polynomials, this theorem has both practical and theoretical relevance, especially in polynomial interpolation.

Gödel's completeness theorem of 1930 implicitly contains a definition of a universal computer, because the logical rules acting on some axioms of arithmetic will eventually prove as a theorem the result of any computation.

Because of Savitch's theorem, NPSPACE is equivalent to PSPACE, essentially because a deterministic Turing machine can simulate a nondeterministic Turing machine without needing much more space ( even though it may use much more time ).

The infinite monkey theorem and its associated imagery is considered a popular and proverbial illustration of the mathematics of probability, widely known to the general public because of its transmission through popular culture rather than because of its transmission via the classroom.

Neither are prime numbers of the form because Fermat's theorem on sums of two squares assures us they can be written for integers and, and.