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.

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 .”

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: Let S, T be primitive polynomials in R, and assume that their product ST is not primitive, so that some noninvertible element d of R divides all coefficients of ST.

Proof.

Proof.

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Proof.

* Sanjeev Saxena, " A Simple Proof of Bernoulli's Inequality ", viXra: 1205. 0068, May 2012

Generally this is in a very small detail, such as the number of leaves on the ear of corn on the recent US Wisconsin state quarter: File: 2004 WI Proof. png.

: Proof: Let A be infinite RE.

* Gödel's Proof ( 2002 revised edition ) by Ernest Nagel and James R. Newman, edited by Hofstadter ( ISBN 0-8147-5816-9 ).

‡ Proof.

Proof.

Proof.

Proof.

Proof.

Proof: Suppose that and are two identity elements of.

Proof: Suppose that and are two inverses of an element of.

Proof.

Proof.

Proof.

Proof.

A more successful effort was the Standard Proof for Pythagoras ' Theorem, that replaced the more than 100 incompatible existing proofs.

* 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.