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: Suppose that and are two identity elements of.

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

Among Moore's most famous works are his book Principia Ethica, and his essays, " The Refutation of Idealism ", " A Defence of Common Sense ", and " A Proof of the External World ".

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 of payment is often in the form of adhesive postage stamps, but postage meters are also used for bulk mailing.

Note that the relationships above are only valid for the form of the Fourier transform shown in the Proof section below.

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 credentials are verified before the event or on-site.

In the shortest of them ( 43 pages as of 2009 ), which he titles " Apology for the Proof of the Riemann Hypothesis " ( using the word " apology " in the rarely used sense of apologia ), he claims to use his tools on the theory of Hilbert spaces of entire functions to prove the Riemann Hypothesis for Dirichlet L-functions ( thus proving GRH ) and a similar statement for the Euler zeta function, and even to be able to assert that zeros are simple.

* A 2011 episode of the series Body of Proof features character Dana Delany investigating a homicide in a neighborhood in which all neighbors are in the Lifestyle.

# 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 >

Distributions of the Mizar Proof Checker for all major operating systems are freely available for downloaded at the Mizar Project website.

( 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 of her feelings are further cemented when she moves to kiss Kaidou, asking why, if he wanted to marry into the profession, didn't he choose to marry her due to her father's political influence.

Proof methods that are not direct include proof by contradiction, including proof by infinite descent.

The edge lettering does not occur at the same time as the minting of the coins, allowing for the natural occurrence of the lettering in either orientation, except Proof Coins where the date and lettering are all " right-side-up.

Proof by Contradiction: Assume that there is a non-empty set of natural numbers that are not interesting.

Proof mintages are indicated in italics.

Although not suggested by natural language, and in contrast to usage in other areas, Proof of Principle and Proof of Concept are not synonymous in drug development.

# Recording the results in a centralized location with Proof Of Purchase records.

Newman also wrote Gödel's Proof ( 1958 ) with Ernest Nagel, presenting the main results of Gödel's incompleteness theorem and the mathematical work and philosophies leading up to its discovery in a more accessible manner.

* Benson, Donald C., The Moment of Proof: Mathematical Epiphanies, Oxford University Press, USA ; New Ed edition ( December 14, 2000 ).

In 2000, Crow became romantically involved with his co-star Meg Ryan on the set of Proof of Life.

After his release, Hargrove wrote a book about his ordeal which inspired the 2000 film Proof of Life starring Meg Ryan and Russell Crowe.

Proof is a 2000 play by American playwright David Auburn.

Fusion shred guitarist, and session musician Greg Howe has released solo albums such as Introspection ( 1993 ), Uncertain Terms ( 1994 ), Parallax ( 1995 ), Five ( 1996 ), Ascend ( 1999 ), Hyperacuity ( 2000 ), Extraction ( 2003 ) with electric bassist Victor Wooten and drummer Dennis Chambers, and Sound Proof ( 2008 ).

* Proof of Life ( 2000 )

He is best known for his 2000 play Proof, which won the 2001 Tony Award for Best Play, as well as the 2001 Pulitzer Prize for Drama.

* Proof of Life ( 2000 )

In 2000, Proof toured with Eminem, Dr. Dre, and Snoop Dogg in the Up In Smoke Tour, performing as a hype man for Eminem.

* Proof of Life ( 2000 ) .... Janis Goodman

In 2000 Shenkman gained success in the theatre, co-starring with Mary-Louise Parker in the Manhattan Theatre Club production of Proof, for which he received a 2001 Tony nomination.

Other films include Enchanted April ( 1992 ), Fatherland ( 1994 ), The Hanging Gale ( 1995 ), Kidnapped ( 1995 ), Mrs. Dalloway ( 1997 ), The Railway Children ( 1999 ), Proof of Life ( 2000 ) as Ian Havery and My Week with Marilyn ( 2011 ).

* Proof of Life ( 2000 ) as Terry's Driver

* Rabbit Proof Fence ( 2000 )

Proof of Life is a 2000 American film, directed by Taylor Hackford.

* Proof of Life ( 2000 )

Scotland Yard also guarded Crowe while he was promoting Proof of Life in London in February 2001.

* 2001: Proof * – David Auburn

* 2001: Proof by David Auburn

She succeeded Mary-Louise Parker in the lead role in Proof on Broadway in 2001.

Leigh met independent film writer-director Noah Baumbach in 2001, while she was starring on Broadway in Proof.

In 2001, Heche appeared in 7 episodes of Ally McBeal, and starred in the Pulitzer Prize-winning drama Proof on Broadway.

Parker is also the recipient of the 2001 Tony Award for Best Actress for the Broadway play Proof.

Her notable film roles include The Long Riders ( 1980 ), The Right Stuff ( 1983 ), Kindergarten Cop ( 1990 ), Junior ( 1994 ), Bean ( 1997 ), and Proof of Life ( 2001 ).

File: 2001 RI Proof. png | The bridge's representation on the state quarter