Tychonoff's theorem has been used to prove many other mathematical theorems.

The theorem which we prove is more general than what we have described, since it works with the primary decomposition of the minimal polynomial, whether or not the primes which enter are all of first degree.

They are a set of axioms strong enough to prove many important facts about number theory and they allowed Gödel to establish his famous second incompleteness theorem.

The notion of the Kolmogorov complexity can be used to state and prove impossibility results akin to Gödel's incompleteness theorem and Turing's halting problem.

* To prove that a given language is not context-free, one may employ the pumping lemma for context-free languages or a number of other methods, such as Ogden's lemma, Parikh's theorem, or using closure properties.

Furthermore, by breaking the system up into smaller components, the complexity of individual components is reduced, opening up the possibility of using techniques such as automated theorem proving to prove the correctness of crucial software subsystems.

The main problem is to prove a signalizer functor theorem for nonsolvable signalizer functors.

The fixed point theorem is often used to prove the inverse function theorem on complete metric spaces such as Banach spaces.

Much research in graph theory was motivated by attempts to prove that all maps, like this one, could be graph coloring | colored with four color theorem | only four colors.

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 ).

Liouville's theorem may be used to elegantly prove the fundamental theorem of algebra.

Although flawed, Kempe's original purported proof of the four color theorem provided some of the basic tools later used to prove it.

So it suffices to prove the four color theorem for triangulated graphs to prove it for all planar graphs, and without loss of generality we assume the graph is triangulated.

Heawood noticed Kempe's mistake and also observed that if one was satisfied with proving only five colors are needed, one could run through the above argument ( changing only that the minimal counterexample requires 6 colors ) and use Kempe chains in the degree 5 situation to prove the five color theorem.

The argument above began by giving an unavoidable set of five configurations ( a single vertex with degree 1, a single vertex with degree 2, ..., a single vertex with degree 5 ) and then proceeded to show that the first 4 are reducible ; to exhibit an unavoidable set of configurations where every configuration in the set is reducible would prove the theorem.

To prove this, one can combine a proof of the theorem for finite planar graphs with the De Bruijn – Erdős theorem stating that, if every finite subgraph of an infinite graph is k-colorable, then the whole graph is also k-colorable.

The homomorphism theorem is used to prove the isomorphism theorems.

Cantor points out that his constructions prove more — namely, they provide a new proof of Liouville's theorem: Every interval contains infinitely many transcendental numbers.

To formally state, and then prove, the completeness theorem, it is necessary to also define a deductive system.

Gödel's completeness theorem says that a deductive system of first-order predicate calculus is " complete " in the sense that no additional inference rules are required to prove all the logically valid formulas.

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

In the case of Portugal, which a few weeks ago was rumored ready to walk out of the NATO Council should critics of its Angola policy prove harsh, there has been a noticeable relaxation of tension.

To anyone who remembers Newport at its less than maximum violence, this view of what the boys and girls do in the springtime before they wing north for the Jazz Festival ought to prove entertaining.

It is possible to prove many theorems using neither the axiom of choice nor its negation ; such statements will be true in any model of Zermelo – Fraenkel set theory ( ZF ), regardless of the truth or falsity of the axiom of choice in that particular model.

The traditional definition of the role of the conservator involves the examination, conservation, and preservation of cultural heritage using " any methods that prove effective in keeping that property in as close to its original condition as possible for as long as possible.

The American explorations of the Argive Heraeum, concluded in 1895, also failed to prove that site to have been important in the prehistoric time, though, as was to be expected from its neighbourhood to Mycenae itself, there were traces of occupation in the later Aegean periods.

* Praeparatio evangelica ( Preparation for the Gospel ), commonly known by its Latin title, which attempts to prove the excellence of Christianity over every pagan religion and philosophy.

As it turned out, 2006 would prove to be the worst season for Essendon under Sheedy, and its worst in over 70 years, with a multitude of injuries and poor form affecting the team, none more so than the serious hamstring injury suffered by newly appointed captain Lloyd.

As it is a new member of the Agile family, there is not enough data to prove its usability, however it claims to be an answer to what it sees as XP's imperfections.

The new weapon would prove its worth in the Battle of Mylae, the first Roman naval victory, and would continued to do so in the following years, especially in the huge Battle of Cape Ecnomus.

This goes to prove that even minor poetry may have its practical uses.

Once in the arena morgue, the corpse would have been stripped of armour, and probably had its throat cut to prove that dead was dead.

Hershey and Chase were also able to prove that the DNA from the phage is inserted into the bacteria shortly after the virus attaches to its host.

Emperor Justinian I ensured that his new structure would not be burned down, like its predecessors, by commissioning architects that would build the church mainly out of stone, rather than wood, “ He compacted it of baked brick and mortar, and in many places bound it together with iron, but made no use of wood, so that the church should no longer prove combustible .”

Then the theory at hand, which is assumed to prove certain facts about numbers, also proves facts about its own statements.

During TLS negotiation the server sends its X. 509 certificate to prove its identity.

The client may also send a certificate to prove its identity.

In some cases, accusers may have been motivated by a desire to take the property of the accused, though this is a difficult assertion to prove in the majority of areas where the inquisition was active, as the inquisition had several layers of oversight built into its framework in a specific attempt to limit prosecutorial misconduct.

Thus, Pakistan is in fact a multi-regional state and not a South Asian state actor only ; its history if analyzed in depth would prove the point.

Gödel's second incompleteness theorem ( 1931 ) shows that no formal system extending basic arithmetic can be used to prove its own consistency.

Unfortunately, there is a conflict between stored hashed-passwords and hash-based challenge-response authentication ; the latter requires a client to prove to a server that he knows what the shared secret ( i. e., password ) is, and to do this, the server must be able to obtain the shared secret from its stored form.

In other words, to prove by contradiction that < big > P </ big >, show that or its equivalent.

Despite its harsh provisions, a RICO-related charge is considered easy to prove in court, as it focuses on patterns of behavior as opposed to criminal acts.

Like the decomposition rank characterization, this does not give an efficient way of computing the rank, but it is useful theoretically: a single non-zero minor witnesses a lower bound ( namely its order ) for the rank of the matrix, which can be useful to prove that certain operations do not lower the rank of a matrix.

Due to their sometimes greatly exaggerated nature, secondary sexual characteristics can prove to be a hindrance to an animal, thereby lowering its chances of survival.