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 m < n, then we can view R < sup > m </ sup > as a subspace of R < sup > n </ sup >, and the non-empty open subsets of R < sup > m </ sup > are not open when considered as subsets of R < sup > n </ sup >.

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.

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 .”

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?

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

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.

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 sheets were the means by which syndicates provided newspapers with black-and-white line art for the reproduction of strips ( which they arranged to have colored in the case of Sunday strips ).

* Demonstratio evangelica ( Proof of the Gospel ) is closely connected to the Praeparatio and comprised originally twenty books of which ten have been completely preserved as well as a fragment of the fifteenth.

Proof for the existence of a common Germanic goddess once known as * Fraujon does not exist, but scholars have commented that this may simply be due to lack of evidence.

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

Proof that conservation does work, some species have returned home for continuation of programmes to reintroduce them to their own environment

The Maskrafters have also done recent plays including Proof, The Fantasticks, Grease, and Shakespeare's The Tempest.

Proof of concept trials have indicated that Ebola-targeted siRNAs may be effective as post-exposure prophylaxis in humans, with 100 % of non-human primates surviving a lethal dose of Zaire Ebolavirus, the most lethal strain.

Many well-known actors have appeared at the theatre, including Nicole Kidman ( The Blue Room ), Gwyneth Paltrow ( Proof ), Ian McKellen ( The Cut ) and Ewan McGregor ( Othello ).

Proofs that Yield Nothing But Their Validity, or All Languages in NP have Zero-Knowledge Proof systems.

A closed densely defined operator T is bounded if and only if T < sup >∗</ sup > is bounded .< ref > Proof: We have:.

When five years was up, the homesteader had to have two or three witnesses sign a document called " Proof Required Under Homestead Acts May 20, 1862.

In Robert Rodriguez and Quentin Tarantino's Grindhouse double feature, both the Planet Terror and Death Proof segments have references to missing reels, used as plot devices.

* A more recent movie featuring car surfing was the Death Proof ( 2007 ), which may have also re-popularized the trend.

The first use of the term " POP " or " Proof of Payment " on a rail line in North America is believed to have been in Edmonton in 1980.

Similarly, Proof, Romper Stomper, Shine, The Piano and Kiss or Kill have also become artistic successes.

#* 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 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.

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

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