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 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.

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 .”

# 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: 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.

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.

Proof: By symmetry, it suffices to prove that there is some constant c such that for all bitstrings s

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.

* 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.

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.

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

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

#* Proof:, so is a square root of 2 modulo.

* In Hal Clement's short story Proof ( 1942 ), neutronium is the only form of solid matter known to Solarians, the inhabitants of the Sun's interior.

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.

( Proof: either 1 is positive or − 1 is positive.

Proof by contradiction is also known as indirect proof, apagogical argument, proof by assuming the opposite, and reductio ad impossibilem.

* Proof that R is uncountable

Proof: Suppose G = ⟨ H | R ⟩ is a finitely presented, residually finite group.

* Proof of Fermat's Last Theorem is discovered by Andrew Wiles.

Proof: We only give the proof in the simplified case ; the general case is similar.

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 >.