Certain government bureaucracies, such as a department of motor vehicles will issue an abstract of a completed transaction or an updated record intended to serve as a proof of compliance with some administrative requirement.

This key result opened the way for a proof of the Weil conjectures, ultimately completed by his student Pierre Deligne.

Major steps in the proof involve showing how manifolds behave when they are deformed by the Ricci flow, examining what sort of singularities develop, determining whether this surgery process can be completed and establishing that the surgery need not be repeated infinitely many times.

( The proof is actually completed in Volume II, 1st edition, page 86, accompanied by the comment, " The above proposition is occasionally useful.

It was not until 1920 that Gauss ' proof was completed.

According to Quebec law, Montreuil could not change her record of sex because this requires proof of a completed sex reassignment surgery, which she has not had.

The proof of the density of topologically stable mappings was completed by John Mather in 1970, based on the ideas developed by Thom in the previous ten years.

We have proven both lemmas and have completed the proof.

This result is variously known as Matiyasevich's theorem ( because he provided the crucial step that completed the proof ) and the MRDP theorem ( for Yuri Matiyasevich, Julia Robinson, Martin Davis, and Hilary Putnam ).

Thus the claim is true and we have completed the proof for 2 colours.

Adrien-Marie Legendre, one of the referees, soon completed the proof for this case ; Dirichlet completed his own proof a short time after Legendre, and a few years later produced a full proof for the case.

Steiner's proof was completed later by several other mathematicians.

This can be done by harvesting late ( late harvest wine ), freezing the grapes to concentrate the sugar ( ice wine ), or adding a substance to kill the remaining yeast before fermentation is completed ; for example, high proof brandy is added when making port wine.

This proof completed a programme initiated and largely developed by Alexander Grothendieck.

A relation of Croppenburg's ; the Dutch engineer Cornelius Vermuyden present in England at the time of the project on a commission to drain the Fens and involved in repairing the seawall at Dagenham has led to speculation that Vermuyden oversaw the project, but proof appears to be vague, nevertheless the work was completed by around 300 Dutch skilled in the construction of dykes and other sea defences.

Georges Gonthier ( of Microsoft Research, in Cambridge, England ) and Benjamin Werner ( of INRIA ) used Coq to create a surveyable proof of the four color theorem, which was completed in September 2004.

* Four color theorem: formal proof using Coq was completed in September 2004.

* Feit – Thompson theorem: formal proof using Coq was completed in September 2012.

In addition to the Sintra Mountains and Sintra-Cascais Nature Park, the parishes of the town of Sintra are dotted by royal retreats, estates, castles and buildings from the 8th-9th century, in addition to many buildings completed between the 15th and 19th century, including the Castelo dos Mouros, the Pena National Palace and the Sintra National Palace, resulting in its classification by UNESCO as a World Heritage Site in 1995.

The classification system was completed by the national censuses of 1765 and 1772.

The classification system was completed by the national censuses of 1765 and 1772.

In addition to the service completed by its student body, Rockhurst has been recognized as a university by the Carnegie Foundation for the Advancement of Teaching with the foundation's community engagement classification for its ongoing engagement with its neighborhood and the Kansas City community.

Vineyard classification began in 1730 and was completed by the national censuses of 1765 and 1772.

Vineyard classification began in 1730 and was completed by the national censuses of 1765 and 1772.

The name " practical number " is due to, who first attempted a classification of these numbers that was completed by and.

For his doctoral work Nelson completed a re-classification of the vegetation communities of Trinidad and Tobago, a massive undertaking which updates John Stanley Beard's 1946 classification.

He was subsequently drafted, but received a 4-F classification on the Selective Service System and was released from military duty after his preliminary physical was completed.

In 1947 CDB-18 completed the development of Technical Design No. 613 ( called the Whiskey class submarine in NATO classification ) – a torpedo diesel-electric submarine of medium displacement which took into account the combat experiences of Soviet and German submarines from the War.

I decided to see no more of the clerk until the processing of my papers was completed.

His next major work, completed in 1892, was a long fantastic epic in prose, entitled Hans Alienus, which Professor Book describes as a monument on the grave of his carefree and indolent youth.

The doctor, he noticed, was attempting a transverse movement towards the stairs, but before the movement could be completed a distinct and audible cluck ruffled the air in the hollow of the stair-well.

A set of tables containing spectral intensities for 39,000 lines of 70 elements, as observed in a copper matrix in a d-c arc, was completed and published.

A project for the Air Force has been completed in which the NAIR infrared detecting device was developed for area monitoring of noxious or dangerous gases.

Erected on the site of pagan temples and three previous St. Sophias, the first of which was begun by Constantine, this fourth church was started by Justinian in 532 and completed twenty years later.

On November 26 of that year the bridge was completed and opened.

He completed his training in pharmacy also, taking his degree with high honors in 1797, and in 1799 was awarded the degree of Doctor of Philosophy along with a prize for an essay in medicine.

I promised to illustrate the lecture, if they so much as breathed till after the call was completed.

Voting preparations began in the fall of 1959, although the actual demarcation and planning for the rural communes was completed in 1958.

This was nearly completed May 23, 1901 with a promise of lights by June 10, but the first light did not go on until September 28.

Volume 1, was completed in 1941 and published in 1944.

The volume was completed in 1950 and published in 1951.

the burn oozed watery fluid for many days, and finally the healing was completed by bathing it with epsom salts.

The report was completed after nearly eighteen months of work on the question of the organization of the United Nations.

With the announcement of a `` special achievement award '' to William A. ( ( Bill ) Shea, the awards list was completed yesterday for Sunday night's thirty-eighth annual dinner and show of the New York Chapter, Baseball Writers' Association of America, at the Waldorf-Astoria Hotel.

There was also the fact that by the time he meets Mr. Khrushchev, the President will have completed conversations with all the other principal Allied leaders.

After playing a splendid first nine holes in 34 -- two strokes under par -- on this fifth and final day of the tournament ( Sunday's fourth round had been washed out by a violent rainstorm when it was only half completed ), Player's game rapidly fell to pieces.

The Pacific Railway Acts of 1862 and 1864 granted federal support for the construction of the United States ' First Transcontinental Railroad, which was completed in 1869.

The formal ratification by all 13 states was completed in early 1781.

The road between Lubango and Namibe, for example, was completed recently with funding from the European Union, and is comparable to many European main routes.

Integration was never completed as UNITA went back to war in 1992.

A bridge was first completed here in 1887, replaced by another structure in 1949, and subsequently replaced with the current bridge which was completed in 2008.