He uses this as a proof of the existence of atoms:

In this case, if a proof uses this statement, researchers will often look for a new proof that doesn't require the hypothesis ( in the same way that it is desirable that statements in Euclidean geometry be proved using only the axioms of neutral geometry, i. e. no parallel postulate.

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.

The proof uses Euclid's lemma ( Elements VII, 30 ): if a prime p divides the product of two natural numbers a and b, then p divides a or p divides b ( or perhaps both ).

The proof uses modal logic, which distinguishes between necessary truths and contingent truths.

The proof of Gödel's completeness theorem given by Kurt Gödel in his doctoral dissertation of 1929 ( and a rewritten version of the dissertation, published as an article in 1930 ) is not easy to read today ; it uses concepts and formalism that are outdated and terminology that is often obscure.

George Boolos has since sketched an alternative proof of the first incompleteness theorem that uses Berry's paradox rather than the liar paradox to construct a true but unprovable formula.

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.

Newman's proof is arguably the simplest known proof of the theorem, although it is non-elementary in the sense that it uses Cauchy's integral theorem from complex analysis.

Perelman's proof uses a modified version of a Ricci flow program developed by Richard Hamilton.

The proof uses the classification of finite simple groups.

Although BB84 does not use entanglement, Ekert's protocol uses the violation of a Bell's inequality as a proof of security.

The proof of Euler's identity uses only the formula for the geometric series and the fundamental theorem of arithmetic.

A sig block is easily forged, whereas a digital signature uses cryptographic techniques to provide verifiable proof of authorship.

* Centipede game, the Nash equilibrium of which uses a similar mechanism as its proof.

The alternative is formal software verification, which uses mathematical proof techniques to show the absence of bugs.

An alternative proof uses Prüfer sequences.

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.

One proof of the impossibility of finding a planar embedding of K < sub > 3, 3 </ sub > uses a case analysis involving the Jordan curve theorem, in which one examines different possibilities for the locations of the vertices with respect to the 4-cycles of the graph and shows that they are all inconsistent with a planar embedding.

The original proof uses hypergeometric functions and innovative tools from the theory of Hilbert spaces of entire functions, largely developed by de Branges.

In logic, a formal proof is not written in a natural language, but instead uses a formal language consisting of certain strings of symbols from a fixed alphabet.

Chaitin prefaces his definition with: " I'll show you can't prove that a program is ' elegant '"— such a proof would solve the Halting problem ( ibid ).

The basic idea of his proof is that a proposition that holds of x if x = n for some natural number n can be called a definition for n, and that the set

One proof of the chain rule begins with the definition of the derivative:

Noting another proof that self-replicating machines are possible is the simple fact that all living organisms are self replicating by definition.

While it is one of the most commonly used concepts in logic it must not be mistaken for a logical law ; rather, it is one of the accepted mechanisms for the construction of deductive proofs that includes the " rule of definition " and the " rule of substitution " Modus ponens allows one to eliminate a conditional statement from a logical proof or argument ( the antecedents ) and thereby not carry these antecedents forward in an ever-lengthening string of symbols ; for this reason modus ponens is sometimes called the rule of detachment.

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.

A definition is also accepted without proof since it simply gives the meaning of a word or phrase in terms of known concepts.

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

Nearly every aspect of SRA was controversial, including its definition, the source of the allegations and proof thereof, testimonials of alleged victims, and court cases involving the allegations and criminal investigations.

This allows the definition of a formal proof to be precisely specified without any ambiguity.

: The only thing that was used in the proof that was not a definition was the sine law.

Early laws simply prohibited driving while intoxicated, requiring proof of a state of intoxication with no specific definition of what level of inebriation qualified.

A formal proof, from the definition of a derivative, is:

The international protection of refugees hardly existed before the First World War and people faced difficulty in providing proof of identity identity but also there was no internationally agreed definition of " refugee ".

The * points to Exercise # 24 ( p. 189 ) wherein he asks for a proof of the following: " In section 53, the definition of the number ' 0 ' was stated by way of an example.

The noun triviality usually refers to a simple technical aspect of some proof or definition.

Probabilistic computation is also critical for the definition of most classes of interactive proof systems, in which the verifier machine depends on randomness to avoid being predicted and tricked by the all-powerful prover machine.

For a detailed definition of free current and bound current, and the proof that the two formulations are equivalent, see the " proof " section below.

A rigorous proof of the product rule can be given using the properties of limits and the definition of the derivative as a limit of Newton's difference quotient.

On the other hand, the simplicity of the algebra in this proof perhaps makes it easier to understand than a proof using the definition of differentiation directly.

IntuiSense's patented technology includes a high definition monitor ( 55 "- 32 "), a custom built CPU, MDB monitor, and the IS Gesture Recognition system all in a vandal proof construction.

( Without an accepted definition of large cardinal property, it is not subject to proof in the ordinary sense ).