Most criminal cases within an adversarial system require that the prosecution proves its contentions beyond a reasonable doubt — a doctrine also called the " Burden of Proof ".

* Ghost Hunters: William James and the Search for Scientific Proof of Life After Death by Deborah Blum, Penguin Press, 2006 ; Briefly discusses Home's career and the scientific investigation of his abilities, in the larger context of the 19th century and the scientific inquiry into life beyond death.

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: If n is a prime-power, then a group of order n has a nontrivial center and, therefore, is not simple.

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 >.

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.

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.

* 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: 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.

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 examples from 1936 to 1942 and 1950 to 1967 were struck at the Philadelphia Mint ; in 1968 proof production was shifted to the San Francisco Mint.

 He made another major contribution to the field in his 1979 paper, joint with his student Robert A. Reckhow, " The Relative Efficiency of Propositional Proof Systems ", in which they formalized the notions of p-simulation and efficient propositional proof system, which started an area now called propositional proof complexity.

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 theory is important in philosophical logic, where the primary interest is in the idea of a proof-theoretic semantics, an idea which depends upon technical ideas in structural proof theory to be feasible.

* Metamath Proof explorer a repository of mathematical theorems along with their proofs using the Metamath proof assistant.

* Proof by exhaustion or brute force method, a method of mathematical proof

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

Discharging these proof obligations is performed using the SPADE Simplifier ( an automated theorem prover ) and the SPADE Proof Checker ( a manual theorem prover, used for those VCs too thorny for the Simplifier to automatically discharge ).

:: Sketch of proof: Proof by contradiction.

* Proof by exhaustion, proof by examining all individual cases

Proof by exhaustion, also known as proof by cases, perfect induction, or the brute force method, is a method of mathematical proof in which the statement to be proved is split into a finite number of cases and each case is checked to see if the proposition in question holds.

Any proof by contrapositive can also be trivially formulated in terms of a Proof by contradiction: To prove the proposition, we consider the opposite,.

A popular front-end for proof assistants is the Emacs-based Proof General, developed at the University of Edinburgh.

Some issues sell for significant sums, for example the 1995-W proof ($ 3, 500 as of 2011 ) and the 2006 20th anniversary set containing a special " Reverse Proof " coin along with a regular proof coin and the new " Burnished Uncirculated " coin ( over $ 300 as of 2011 ).

The LeMats which found their way through the Union blockade were stamped with British proof marks from the Birmingham Proof House, leading to the misapprehension that the pistols were actually manufactured in the UK.

Becker's Postulate later played a role in the formalization given, by Charles Hartshorne, the American process theologian, of the Ontological Proof of God's existence, stimulated by conversations with the logical positivist and opponent of the alleged proof, Rudolf Carnap.

* Composer John Massari would go on to work on such films as Killer Klowns From Outer Space, Skeletons, Retro Puppet Master and such television shows as POV Murder and Prison Break: Proof of Innocence.

In fact, Thurston resolved so many outstanding problems in foliation theory in such a short period of time that, according to Thurston, it led to a kind of exodus from the field, where advisors counselled students against going into foliation theory because Thurston was " cleaning out the subject " ( see " On Proof and Progress in Mathematics ", especially section 6 ).

* Proof by verbosity, sometimes colloquially referred to as argumentum verbosum-a rhetorical technique that tries to persuade by overwhelming those considering an argument with such a volume of material that the argument sounds plausible, superficially appears to be well-researched, and it is so laborious to untangle and check supporting facts that the argument might be allowed to slide by unchallenged.

Proof was provided in the form of unconnected bits of information such as pictures drawn by patients, heavy metal album covers, historical folklore about Satan worshippers, and pictures of mutilated animals.

Proof: Fix a first-order language L, and let Σ be a collection of L-sentences such that every finite subcollection of L-sentences, i ⊆ Σ of it has a model.

# 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 of guilt had been established by magical means such as the Ordeal.

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.

After leaving the U. S. Attorney's office, Turow became a novelist, writing legal thrillers such as The Burden of Proof, Presumed Innocent, Pleading Guilty, and Personal Injuries, which Time magazine named as the Best Fiction Novel of 1999.

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 legal acquisition, such as a bill of sale or purchase receipt, is contributory.

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.

Songs such as " Shatter Proof " and the early short films of Parker and Stone were considered to be a part of the new DVD, but these additions were ultimately rejected.

She has appeared in films such as Kids, Men in Black II, 25th Hour, Sin City, Clerks II, Rent, Death Proof, The Rundown, Eagle Eye, Alexander, Seven Pounds, Percy Jackson and the Olympians: The Lightning Thief, and Unstoppable.

Also included are some original pieces such as " Fate to Despair ", " Awake ", and " Proof of Blood ".

On television she has appeared in the 2006 made-for-TV movie Proof of Lies and episodes of such drama series as What About Brian and Private Practice.

Proof of such a property interest would not, of course, entitle him to reinstatement.

He continued to appear in such high-profile television movies as Skokie ( 1981 ), Split Image ( 1982 ), Day One, ( 1989 ), A Killing in a Small Town ( 1990 ) opposite Barbara Hershey, In Broad Daylight ( 1991 ), Scott Turow's The Burden of Proof and the miniseries A Season in Purgatory.

The Birmingham pen trade prospered and areas such as the Gun Quarter and Jewellery Quarter existed with high standards of production set out by institutions such as the Birmingham Proof House and Birmingham Assay Office.