From the criterion it also follows that Carmichael numbers are cyclic .< ref > Proof sketch: If is square-free but not cyclic, for two prime factors and of.

#* Proof: Then so Thus However, is prime, so or In the former case, hence ( which is a contradiction, as neither 1 nor 0 is prime ) or In the latter case, or If however, which is not prime.

#* Proof: If q divides 2 < sup > p </ sup > − 1 then 2 < sup > p </ sup > ≡ 1 ( mod q ).

Proof: If there is no possible move, then the lemma is vacuously true ( and the first player loses the normal play game by definition ).

Proof: If n is a prime-power, then a group of order n has a nontrivial center and, therefore, is not simple.

Proof: If is diagonalizable, then A is annihilated by some polynomial, which has no multiple root ( since ) and is divided by the minimal polynomial of A.

: Proof: If we assume, w. l. o. g., Q < sub > A </ sub >∩ Q < sub > B </ sub > is empty then L ( A )∪ L ( B ) is recognized by the Büchi automaton ( Q < sub > A </ sub >∪ Q < sub > B </ sub >, Σ, Δ < sub > A </ sub >∪ Δ < sub > B </ sub >, I < sub > A </ sub >∪ I < sub > B </ sub >, F < sub > A </ sub >∪ F < sub > B </ sub >).

Canibus name dropped Eminem's long time deceased friend Proof, Canibus said " If Proof Was Alive He'd Be Dying Inside ".

Tentative Proof: If the underlying Hilbert space is finite-dimensional, the spectral theorem says that N is of the form

Thus, boundedness of T < sup >∗</ sup > on its domain does not imply boundedness of T. On the other hand, if T < sup >∗</ sup > is defined on the whole space then T is bounded on its domain and therefore can be extended by continuity to a bounded operator on the whole space .< ref > Proof: being closed, the everywhere defined T < sup >∗</ sup > is bounded, which implies boundedness of T < sup >∗∗</ sup >, the latter being the closure of T. See also for the case of everywhere defined T .</ ref > If the domain of T < sup >∗</ sup > is dense, then it has its adjoint T < sup >∗∗</ sup >.

T .< ref > Proof: If T is closed densely defined, then T < sup >∗</ sup > exists and is densely defined.

Proof: If we have a description in, we can convert it into a description in our optimal language by first describing as a computer program ( part 1 ), and then using the original description as input to that program ( part 2 ).

Topics of the book include: “ The Pope: A Scandal and a Mystery ,” “ How does the Pope Pray ?” “ Does God Really Exist ?” “ Proof: Is it Still Valid ?” “ If God Exists, Why is He Hiding ?” “ Is Jesus the Son of God ?” “ Why Is There So Much Evil in the World ?” “ What Does To Save Mean ?” “ Why So Many Religions ?” “ Buddha ?” “ Muhammad ?” “ Judaism ?” “ What Is the New Evangelization ?” “ Is There Really Hope in the Young ?” “ Was God at Work in the Fall of Communism ?” “ Is Only Rome Right ?” “ In Search of Lost Unity ,” “ A Qualitative Renewal ,” “ The Reaction of the World ,” “ Does Eternal Life Exist ?” “ Human Rights ,” “ The Mother of God ,” and “ Be Not Afraid .”

Proof: Since p is a prime number the only possible values of gcd ( p < sup > k </ sup >, m ) are 1, p, p < sup > 2 </ sup >, ..., p < sup > k </ sup >, and the only way for gcd ( p < sup > k </ sup >, m ) to not equal 1 is for m to be a multiple of p. The multiples of p that are less than or equal to p < sup > k </ sup > are p, 2p, 3p, ..., p < sup > k − 1 </ sup > p = p < sup > k </ sup >, and there are p < sup > k − 1 </ sup > of them.

Proof of first property: from 3. we obtain ( a < sup >− 1 </ sup >)< sup >− 1 </ sup >=

Proof of second property: since a * b and b * b < sup >− 1 </ sup > are defined, so is a * b * b < sup >− 1 </ sup >=

Proof: Let d be the position of the leftmost ( most significant ) nonzero bit in the binary representation of s, and choose k such that the dth bit of x < sub > k </ sub > is also nonzero.

< div class =" NavHead " style =" background :# ccf ; text-align: left ; font-size: larger ;"> Proof of ( 1 )</ div >

Proof: It is straightforward to prove that ∇ satisfies the Leibniz identity and is C < sup >∞</ sup >( S < sup > 2 </ sup >) linear in the first variable.

Proof: Using the same sum for R ( p, n ) as in Lemma 2, we see that since p < sup > 2 </ sup > > 2n, in fact only the first term is nonzero ; this term is exactly the right-hand side of the above equation.

# is a polynomial ( of degree ≤ n − 1 ) in .< ref > Proof: When A is normal, use Lagrange's interpolation formula to construct a polynomial P such that, where are the eigenvalues of A .</ ref >

Proof: The exponent of p in n!

Proof: a character on T is of the form z → z < sup > n </ sup > for n an integer.

: Proof: In a size -( n + 1 ) set, choose one distinguished element.

He famously put the point into dramatic relief with his 1939 essay " Proof of an External World ", in which he gave a common sense argument against scepticism by raising his right hand and saying " Here is one hand ," and then raising his left and saying " And here is another ," then concluding that there are at least two external objects in the world, and therefore that he knows ( by this argument ) that an external world exists.

His wife and he were then issued a Certified Proof of Citizenship on 20 July 1965.

In 1899, Henry Gottlieb Eckstein developed the " waxed sealed package " for freshness, known then as the " Eckstein Triple Proof Package ," a dust, germ and moisture-proof paper package.

Proof was then shot by the bouncer Mario Etheridge, Bender's cousin.

Proof of commutativity can be seen by letting one summand shrink until it is very small and then pulling it along the other knot.

Proof of Zocor's efficacy in reducing cardiovascular disease was only presented five years after its original introduction, and then only for secondary prevention.

The Birmingham Gun Barrel Proof House was established in 1813 by an act of Parliament at the request — and expense — of the then prosperous Birmingham Gun Trade.

Birmingham Proof House was built in 1813, then one of only two such proof houses in England, the other being in London.

Since then, Ladd has portrayed supporting as well as lead roles in films, including Cabin Fever ( 2002 ), Club Dread ( 2004 ), and Death Proof ( 2007 ).

Proof of the bidirectional nature came from providing replicating cells with a pulse of tritiated thymidine, quenching rapidly and then autoradiographing.

: Proof: Let S be a set, and let T be the union of the elements of S. Then every element of S is a subset of T, and hence is of cardinality less than or equal to the cardinality of T. Cantor's theorem then implies that every element of S is of cardinality strictly less than the cardinality of 2 < sup > lTl </ sup >.

Lessing outlined the concept of the religious " Proof of Power ": How can miracles continue to be used as a base for Christianity when we have no proof of miracles?

Eminem references Big Daddy Kane in the lyrics to his song ‘ Yellow Brick Road ’ from his Encore album, saying, “ we ( Eminem and Proof ) was on the same shit, that Big Daddy Kane shit, where compound syllables sound combined ” and he quotes the same lines in his book, The Way I Am – this illustrates how Big Daddy Kane had an influence on both Eminem ’ s and Proof ’ s rhyme technique.

Any proof by contrapositive can also be trivially formulated in terms of a Proof by contradiction: To prove the proposition, we consider the opposite,.

#* Proof: suppose that p is composite, hence can be written with a and Then is prime, but and contradicting statement 1.

* ( German ), reprinted in English translation as " Proof that every set can be well-ordered ", van Heijenoort 1976, pp. 139 – 141.

This informal analysis can be formalized to make a rigorous proof of the incompleteness theorem, as described in the section " Proof sketch for the first theorem " below.

Proof that competitors have shared prices can be used as part of the evidence of an illegal price fixing agreement.

Proof that eccentric films can survive in today's off-the-rack Hollywood!

One such free community, Collectors Proof enables manufacturers and users alike to associate unique identification numbers to virtually any item so that each new owner can update its chain of custody.

The book professes to be " A Short and Easy Method with the Deists wherein the certainty of the Christian Religion is Demonstrated by Infallible Proof from Four Rules, which are incompatible to any imposture that ever yet has been, or that can possibly be " ( 1697 ).

Proof of this can be found on platform Five of Sheffield railway station, because the Sheaf also runs under that, and it also flooded at the same time.

Usually a demonstration is less than a Proof of concept, but can come some of the way to showing how a business project may be justified.

Proof of the devastation can be seen from the Domesday survey of 1086 most of the lands in Cheshire were recorded as ' wasta ', or wasteland, as " abandoned or useless lands ".

Proof of Mexican citizenship is required to attend public schools for free, but foreigners can attend public schools by paying a tuition.

Proof of this can be seen at the Parker Flats at ( the former ) Gage School.

Proof may be rendered invalid if the firearm is damaged or modified significantly ; at this point it is described as " out of proof " and must be re-proofed before it can be sold or transferred.

The theme can also be heard in Quentin Tarantino's movie Death Proof as Rosario Dawson's character's ringtone.

The last equation in Proof 1 above can be expressed as