Proof: Suppose that and are two identity elements of.

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

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

Proof: Suppose the sequence converges to zero and is monotone decreasing.

Proof: Suppose MS

* Proof sketch of the product representation of ξ ( s )

* Proof of Concept, a test to prove something intended is actually possible, conceptually identical to the intended final product

Proof of purchase is often required for sales promotions and manufacturer rebates as evidence that the customer purchased the product.

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.

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.

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.

As there is no explicit Federal requirement for a two person train crew the Utah Transit Authority's FrontRunner service was originally planned to be operated by an operator only with revenue collection being handled by a Proof of Payment system.

Proof of two way contacts, either in the form of QSL cards or via digital entry into Logbook of The World ( LoTW ), must be submitted to qualify.

Proof of this relationship and of the strong and progressive views Korais held on how the legal system of the new Greek state should be formed, is based on correspondence exchanged between the two men, during a long period of time, beginning before the Greek Revolution.

In 2002, Bracegirdle co-wrote two songs on Cher's album Living Proof, " You Take It All " and " Alive Again ", the latter which was released as a single in Europe only.

It spawned two spin-offs, CSI: Miami and CSI: NY, and inspired shows such as Body of Proof, Bones, Castle, Cold Case, Criminal Minds, Crossing Jordan, NCIS, The Mentalist and Without A Trace.

Nicholas Rowe in 1709 wrote that the play must date from after the death of Elizabeth in 1603, because its " Elogy upon Q. Elizabeth, and her Successor K. James, in the latter end of his Henry VIII, is a Proof of that Play's being written after the Accession of the latter of those two Princes to the Crown of England ".

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.

The single was pressed with two B-sides: " Things I Don't Understand " and " Proof ".

The two rappers started rapping together and two of their songs, " Living Proof " and " Echo ", were leaked.

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

Bishop co-starred in the first two seasons Dana Delany's medical drama television series Body of Proof in which he plays Medicolegal Investigator Peter Dunlop, assistant to Medical Examiner Dr Megan Hunt.

# Proof ( a rough thought-experiment or argument, decomposing the primitive conjecture into subconjectures ).

Proof: Assume that a signal f ( t ) which has finite support in both domains exists.

Proof: Notice that the nim-sum (⊕) obeys the usual associative and commutative laws of addition (+), and also satisfies an additional property, x ⊕ x = 0 ( technically speaking, the nonnegative integers under ⊕ form an Abelian group of exponent 2 ).

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.

* Discovery and Proof of the True Cross ( 356 x 747 cm ).

: 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 >).

The unusual repudiation of torture is reflected not only in our criminal laws, see e. g. §§ 2340-2340A, but also in international agreements, in centuries of Anglo-American law, see e. g., John H. Langbeen, Torture and the Law of Proof: Europe and England in the Ancien Régime ( 1977 ) (“ Torture and the Law of Proof ”), and in the long standing policy of the United States, repeatedly and recently reaffirmed by the President.

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.

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

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

( 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 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.