( A formal proof for all finite sets would use the principle of mathematical induction to prove " for every natural number k, every family of k nonempty sets has a choice function.

* Drinking songs: According to the grammarian Athenaeus, Alcaeus made every occasion an excuse for drinking and he has provided posterity several quotes in proof of it.

The " heuristic " approach of the Logic Theory Machine tried to emulate human mathematicians, and could not guarantee that a proof could be found for every valid theorem even in principle.

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.

* 1799: Doctoral dissertation on the Fundamental theorem of algebra, with the title: Demonstratio nova theorematis omnem functionem algebraicam rationalem integram unius variabilis in factores reales primi vel secundi gradus resolvi posse (" New proof of the theorem that every integral algebraic function of one variable can be resolved into real factors ( i. e., polynomials ) of the first or second degree ")

Indeed, following, suppose ƒ is a complex function defined in an open set Ω ⊂ C. Then, writing for every z ∈ Ω, one can also regard Ω as an open subset of R < sup > 2 </ sup >, and ƒ as a function of two real variables x and y, which maps Ω ⊂ R < sup > 2 </ sup > to C. We consider the Cauchy – Riemann equations at z = 0 assuming ƒ ( z ) = 0, just for notational simplicity – the proof is identical in general case.

Euclid frequently used the method of proof by contradiction, and therefore the traditional presentation of Euclidean geometry assumes classical logic, in which every proposition is either true or false, i. e., for any proposition P, the proposition " P or not P " is automatically true.

Empire magazine praised the film saying " the gaudily gory, virtuoso, hyper-kinetic horror sequel / remake uses every trick in the cinematic book " and confirms that " Bruce Campbell and Raimi are gods " and Caryn James of The New York Times called it " genuine, if bizarre, proof of Sam Raimi's talent and developing skill.

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.

It says that for any first-order theory T with a well-orderable language, and any sentence S in the language of the theory, there is a formal proof of S in T if and only if S is satisfied by every model of T ( S is a semantic consequence of T ).

The cut-elimination theorem for a calculus says that every proof involving Cut can be transformed ( generally, by a constructive method ) into a proof without Cut, and hence that Cut is admissible.

Cromwell arranged for leaflets with the stamps featuring Momotombo to be sent to every Senator, as " proof " of the volcanic activity in Nicaragua.

The four color theorem states that such colorings are possible for any planar map, but every known proof involves a computational search that is too long to check by hand.

These cases demonstrate a paradox not in the sense that they demonstrate a logical contradiction, but in the sense that they demonstrate a counter-intuitive result that is provably true: the situations " there is a guest to every room " and " no more guests can be accommodated " are not equivalent when there are infinitely many rooms ( an analogous situation is presented in Cantor's diagonal proof ).

They proved that the existence of a proof system in which every true formula has a short proof is equivalent to NP = coNP.

We will go over a typical application of Zorn's lemma: the proof that every nontrivial ring R with unity contains a maximal ideal.

A similar proof shows that every Boolean ring is commutative:

In fact, it is possible to give a proof that is a Noetherian ring without appealing to its order structure and this proof applies more generally to principal ideal rings ( i. e., rings in which every ideal is generated by a single element ).

The proof that Homer does not belong to that school — and that his poetry is not in any true sense ballad poetry — is furnished by the higher artistic structure of his poems and, as regards style, by the fourth of the qualities distinguished by Arnold: the quality of nobleness.

The proof of this statement uses the linearity of classical probability, and has exactly the same structure as the proof of the quantum no-cloning theorem.

The definition of theorems as elements of a formal language allows for results in proof theory that study the structure of formal proofs and the structure of provable formulas.

While the proof that is a Noetherian ring uses the order structure of, typical proofs in ring theory in general do not assume such additional structure on the ring.

His most celebrated single result is his proof in 1956 of the existence of 7-dimensional spheres with nonstandard differential structure.

However, the more difficult proof of the converse ( see below ) makes use of the vector space structure: Since both of the factors are vector spaces over Q, the tensor product can be taken over Q.

Gentzen's proof proceeds by assigning to each proof in Peano arithmetic an ordinal number, based on the structure of the proof, with each of these ordinals less than ε < sub > 0 </ sub >.

Secondly, while Ramsey theory results do say that sufficiently large objects must necessarily contain a given structure, often the proof of these results requires these objects to be enormously large – bounds that grow exponentially, or even as fast as the Ackermann function are not uncommon.

The ASI report said there is sufficient proof of existence of a massive and monumental structure having a minimum dimension of 50x30 metres in north-south and east-west directions respectively just below the disputed structure.

Several important logics have come from insights into logical structure arising in structural proof theory.

The concept of dual cell structure, which Henri Poincaré used in his proof of his Poincaré duality theorem, contained the germ of the idea of cohomology, but this was not seen until later.

The structural induction proof is a proof that the proposition holds for all the minimal structures, and that if it holds for the immediate substructures of a certain structure S, then it must hold for S also.

The brittleness of admissibility comes from the way it is proved: since the proof can induct on the structure of the derivations of the premises, extensions to the system add new cases to this proof, which may no longer hold.

Many logicians feel that this symmetric presentation offers a deeper insight in the structure of the logic than other styles of proof system, where the classical duality of negation is not as apparent in the rules.

A credulousness, a distaste for documentation, an uncritical reliance on contemporary accounts, and a proneness to assume a theory as true before adequate proof was provided were all evidences of his failure to comprehend the use of the scientific method or to evaluate the responsibilities of the historian to his reading public.

The first is the strictly scientific, which demands concrete proof and therefore may err on the conservative side by waiting for evidence in the flesh.

If T is a linear operator on an arbitrary vector space and if there is a monic polynomial P such that Af, then parts ( A ) and ( B ) of Theorem 12 are valid for T with the proof which we gave.

Since the change to better nutrition, he feels he can report on improvements in health, though he considers the following statements observations and not scientific proof.

Most of them, the world over, operate on the same principle by which justice is administered in France and some other Latin countries: the customer is to be considered guilty of abysmal ignorance until proven otherwise, with the burden of proof on the customer himself.

At some point, he was alleged to have accompanied Swein on a pilgrimage to the Holy Land, but proof is lacking.

* Natarajan Shankar SRI International, work on decision procedures, little engines of proof, co-developer of PVS.

Following Desargues ' thinking, the sixteen-year-old Pascal produced, as a means of proof, a short treatise on what was called the " Mystic Hexagram ", Essai pour les coniques (" Essay on Conics ") and sent it — his first serious work of mathematics — to Père Mersenne in Paris ; it is known still today as Pascal's theorem.

where the rule is that wherever instances of "" and "" appear on lines of a proof, "" can validly be placed on a subsequent line.

where the rule is that wherever an instance of "" appears on a line of a proof, either "" or "" can be placed on a subsequent line ;

Although there is no proof, the name " Thomas Corbett " does appear on the list of dead and missing.

The burden of proof should be on the people who make these statements, to show where they got their information from, to see if their conclusions and interpretations are valid, and if they have left anything out.

An inscription on a stone built into the wall of a summer house in Lancarffe furnishes proof of a settlement in Bodmin in the early Middle Ages.

He / she must have a valid passport and either have an invitation letter or a bank statement with enough money to survive the length of the stay in Costa Rica, plus proof of onward travel ( ticket to exit Costa Rica & legal ability to travel to the destination stated on the ticket ).

Maimonides argued that executing a defendant on anything less than absolute certainty would lead to a slippery slope of decreasing burdens of proof, until we would be convicting merely " according to the judge's caprice ".

Bolzano's proof relied on the method of bisection: the sequence was placed into an interval that was then divided into two equal parts, and a part containing infinitely many terms of the sequence was selected.

# Every net on X has a convergent subnet ( see the article on nets for a proof ).

The first proof relies on a theorem about products of limits to show that the derivative exists.

In anticipation of its eventual proof, some have proceeded to develop further proofs which are contingent on the truth of this conjecture.