* Metamath-a language for developing strictly formalized mathematical definitions and proofs accompanied by a proof checker for this language and a growing database of thousands of proved theorems ; while the Metamath language is not accompanied with an automated theorem prover, it can be regarded as important because the formal language behind it allows development of such a software ; as of March, 2012, there is no " widely " known such software, so it is not a subject of " automated theorem proving " ( it can become such a subject ), but it is a proof assistant.

* Metamath – a language for developing strictly formalized mathematical definitions and proofs accompanied by a proof checker for this language and a growing database of thousands of proved theorems

* Norman Megill, developer of Metamath, and maintainer of its site at metamath. org, an online database of automatically verified proofs.

* Metamath.

* Proposition ✸ 54. 43 in a more modern notation ( Metamath )

* Proof theory, a branch of mathematical logic that represents proofs as formal mathematical objects

Proof theory is a branch of mathematical logic that represents proofs as formal mathematical objects, facilitating their analysis by mathematical techniques.

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

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.

Proof that QCD confines at low energy is a mathematical problem of great relevance, and an award has been proposed by the Clay Mathematics Institute for whoever is also able to show that the Yang – Mills theory has a mass gap and its existence.

: Proof theory is the mathematical study of formalized arguments.

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.

This book inspired Douglas Hofstadter to take up the study of mathematical logic, write his famous book Gödel, Escher, Bach, and prepare a second edition of Gödel's Proof, published in 2002.

: Proof: By mathematical induction.

: Proof: We use mathematical induction.

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 that is found in the numerous remains of Roman retreats, the so-called villa rustica uncovered all along the coast.

Barwise, along with his former colleague at Stanford John Etchemendy, was the author of the popular logic textbook Language, Proof and Logic.

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 ).

Proof sheets were the means by which syndicates provided newspapers with black-and-white line art for the reproduction of strips ( which they arranged to have colored in the case of Sunday strips ).

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

* The Nature of Judicial Proof: An Inquiry into the Logical, Legal, and Empirical Aspects of the Law of Evidence ( 1931, with Jerome Michael )

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

 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.

Cook co-authored a book with his student Phuong The Nguyen in this area titled " Logical Foundations of Proof Complexity ".

Proof of this must be supplied with one's application for naturalisation.

In the middle, Simon embarked on the successful Born at the Right Time Tour and promoted the album with further singles, including " Proof " – accompanied with a humorous video featuring, again, Chevy Chase.

* 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 of the affront had been captured with a cell phone camera.

After a highly troubled recording process for Extraction ( 2003 )– a collaboration with drummer Dennis Chambers and bassist Victor Wooten — he took an extended hiatus from recording solo material until the release of his eighth studio album, Sound Proof, in 2008.

* 1994: Revlon " Kiss Proof " with Cindy Crawford

Their first single, " Down in Mexico ," was an R & B hit in 1956 and appears ( in a re-recording from 1970 — still with Gardner singing the lead ) on the soundtrack of Quentin Tarantino's Death Proof.

1922: Proof by Abraham Fraenkel that the axiom of choice cannot be proved from the axioms of Zermelo's set theory with urelements.

Proof of claim: Consider a graph on t vertices and colour its edges with c colours.

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.

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 ).

Lanier is currently working on a book Technology and the Future of the Human Soul, and a music album Proof of Consciousness, in collaboration with Mark Deutsch.

Proof of this was that even though Cheoljong ruled the country for 13 years, until his very last days he had not yet learned how to move with dignity or how to wear royal clothes, so that in even the most luxurious of robes he still looked like a fisherman.

Intending to record one more song for the album (" Living Proof "), he ended up with ten new songs, which became Lucky Town.

Although shadowed by Eminem's success, other members Bizarre and Proof managed to prevail with mildly successful solo careers with the releases of Hannicap Circus and Searching for Jerry Garcia in 2005.