First note that the fundamental vector fields generating the G action on P provide a bundle isomorphism from the vertical bundle V of P ( with V < sub > p </ sub >= T < sub > p </ sub >( P < sub > π ( p )</ sub >) to.

: First, well-order the real numbers ( this is where the axiom of choice enters via the well-ordering theorem ), giving a sequence < math >

First we prove the theorem when n = 1, in which case the ordering on R can be put to good use.

: First recursion theorem.

First we prove the theorem for the 2-colour case, by induction on r + s.

was his proof, using the irreducibility theorem ( or First Seldon Theorem ) that the population of the people of the galaxy fell just short of being a dynamic system which would be impossible to model adequately mathematically, the definition of ' just ' most likely being one order of magnitude, though this is not discussed within the novels.

# REDIRECT Gödel's incompleteness theorems # First incompleteness theorem

First pass from polynomial to continuous functional calculus by using the Stone-Weierstrass theorem.

First of all, by the fundamental theorem of algebra, the complex numbers are an algebraically closed field.

First, the theorem assumes voters can place all election alternatives along a one-dimensional continuum.

First, Grothendieck changed the statement itself: the theorem was, at the time, understood to be a theorem about a variety, whereas after Grothendieck, it was known to essentially be understood as a theorem about a morphism between varieties.

First, the strong completeness theorem and compactness fail.

( First applied to Corporate Finance by Joel Dean in 1951 ; see also Fisher separation theorem, John Burr Williams: Theory.

First century sources, such as the Abhidharmakosa describe the knowledge and use of abacus in India.

Dee's journals did not describe the language as " Enochian ", instead preferring " Angelical ", the " Celestial Speech ", the " Language of Angels ", the " First Language of God-Christ ", the " Holy Language ", or " Adamical " because, according to Dee's Angels, it was used by Adam in Paradise to name all things.

The use of the title First Lady to describe the spouse or hostess of an executive began in the United States.

The epistles " ( First, Second, and Third John ) " describe the fracturing of the Johannine community itself.

It includes some Inuit and First Nations words ( for example tabanask, a kind of sled ), preserved archaic English words no longer found in other English dialects ( for example pook, a mound of hay ), Irish language survivals like sleveen and angishore, compound words created from English words to describe things unique to Newfoundland ( for example stun breeze, a wind of at least 20 knots ( 37 km / h )), English words which have undergone a semantic shift ( for example rind, the bark of a tree ), and unique words whose origins are unknown ( for example diddies, a nightmare ).

First, it is used to describe a high-low split game without a declaration.

The term is sometimes used to describe Ebionitism, which was rejected by the ecumenical councils, especially in the First Council of Nicaea, which was convened to deal directly with the nature of Christ's divinity.

This term was often used to describe the controlled economies of the great powers in the First World War.

The Unua Libro (, First Book ) was the first publication to describe the international language Esperanto ( then called Lingvo Internacia, " international language ").

Chronicling the First and Second Great Awakenings, Wolfe comes to describe the " Me decade as the " Third Great Awakening.

* July 14 – First recorded use of lime-green to describe a color .< ref > The Daily News ( London ).

The term " family values " has been used as a buzzword by right-wing parties such as the Republican Party in the United States, the Family First Party in Australia, the Conservative party in the United Kingdom and the Bharatiya Janata Party in India to describe support for traditional families, and opposition to the changes the modern world has made in how families live.

It also began to borrow words, especially place names such as " Québec ", " Canada " and " Hochelaga ", and words to describe the flora and fauna such as " atoca " and " achigan " from native Indian languages due to contacts with First Nations peoples.

Dee's journals did not describe the language as " Enochian ", instead preferring descriptors like " Angelical ", the " Celestial Speech ", the " Language of Angels ", the " First Language of God-Christ ", the " Holy Language ", or " Adamical " because, according to Dee's Angels, it was used by Adam in Paradise to name all things.

Western Front was a term used during the First and Second World Wars to describe the contested armed frontier between lands controlled by Germany to the east and the Allies to the west.

*" The letters " is most likely a reference to the Letter to the Corinthians, or First Epistle to the Corinthians, which uses the phrase " dark mirror ," also used by Dream to describe him.

The term was first used to describe those Irish parliamentarians who were elected at the 1918 general election, and who, rather than attending the British House of Commons in London, to which they had been elected, assembled instead in Dublin's Mansion House on 21 January 1919 to create a new Irish parliament: the First Dáil Éireann.

The concept of the First World first originated during the Cold War, where it was used to describe countries that the United States was aligned with.

The Isaiah vision of seraphs in an idealised Jerusalem First Temple represents the sole instance in the Hebrew Bible of this word being used to describe celestial beings.

Kevin Bowyer recorded Kaikhosru Sorabji's First Organ Symphony on it in 1988, for which the organ was an " ideal choice "; the notes to the recording describe the church as " acoustically ideal, with a reverberation period of 3½ seconds ", and notes that the organ has " a luxuriousness of tone " and " a range of volume from practically inaudible to fiendishly loud ".

First, it is employed metaphorically to describe the essence of human existence.

First used in a development industry article in Vancouver to describe irrational local opponents ( fruit cakes, fruit loops or just fruits ) of well-planned developments.

The term is avoided by some historians, who either describe the conflicts as merely a series of West Coast campaigns that took place between the Taranaki War ( 1860 – 1861 ) and Titokowaru's War ( 1868 – 69 ), or an extension of the First Taranaki War.

First, we describe how Euler originally discovered the result.

The term ' Nuu-chah-nulth ' is used to describe fifteen separate but related nations, such as the Nuchatlaht First Nation, whose traditional home is in the Pacific Northwest on the west coast of Vancouver Island.

A close relationship developed between Hasan Ali Shah and the British, which coincided with the final years of the First Anglo-Afghan War ( 1838 – 1842 ).

Salieri collaborated with Casti to produce a parody of the relationship between poet and composer in Prima la musica e poi le parole ( First the Music and then the Words ).

* Since Big Brother 2, the UK series always opens with a twist which have included the public being able to choose the final housemate out of three possibilities ( Big Brother 2 ), the public voting for a housemate to leave during the first week and then the housemates choosing between the two housemates with the least number of votes ( Big Brother 3 ), First Night Nominations ( Big Brother 4 ), Suitcase Nominations ( Big Brother 5 ), Unlucky Housemate 13 ( Big Brother 6 ), Big Brother Hood ( Big Brother 7 ), an all-female House and the first inclusion of twins as contestants ( Big Brother 8 ), the first couple to enter as housemates and set a secret task to hide their real relationship ( Big Brother 9 ), all " housemates " really being " non-housemates " who had to earn their housemate status ( Big Brother 10 ), a mole entering the House with an " Impossible Task " ( Big Brother 11 ), Jackie Stallone entering a house containing her son's ex-wife ( Celebrity Big Brother 3 ), the entrance of a non-celebrity in a celebrity edition ( Celebrity Big Brother 4 ) and Jade Goody's family announced to be visiting.

First Council of the Vatican ( 1870 ; officially, 1870 – 1960 ) defined pope's primacy in church governance and his infallibility, repudiated rationalism, materialism and atheism, addressed revelation, interpretation of scripture and the relationship of faith and reason.

First, collective labour law relates to the tripartite relationship between employee, employer and union.

While there is no documentary evidence connecting Oxford ( or any authorial candidate ) to the plays of Shakespeare, Oxfordian researchers, including Mark Anderson and Charlton Ogburn, believe the connection is provided by considerable circumstantial evidence inferred from Oxford's connections to the Elizabethan theatre and poetry scene ; the participation of his family in the printing and publication of the First Folio ; his relationship with the Earl of Southampton ( believed by most Shakespeare scholars to have been Shakespeare's patron ); as well as a number of specific incidents and circumstances of Oxford's life that Oxfordians believe are depicted in the plays themselves.

First, Grant's relationship with his predecessor Johnson had deteriorated badly, culminating with Johnson's appointment of Grant antagonist William Rosecrans as minister to Mexico.

Further, included in Canada's constitution are the various treaties between the Crown and Canadian First Nations, Inuit, and Métis peoples, who view these documents as agreements directly and only between themselves and the reigning monarch, illustrating the relationship between sovereign and aboriginals.

Family tree showing the relationship between the three members of the First Triumvirate, as well as their relationships with other prominent members of the Roman Republic | Republic.

The relationship between the two was so close that rumors spread BCCI was the real owner of First American.

First then does Ovid begin telling the story describing Myrrha, her father and their relationship, which Doll compares to the mating of Cupid and Psyche: here the lovemaking occurs in complete darkness and only the initiator ( Cupid ) knows the identity of the other as well.

Clayborne is shown to prefer solitude during much of the series, and even her relationship with fellow First Hundred settler Simon ( with whom she has a child ) is subject to introspective silence in most cases.

After a four month dating relationship, on 2 August 1975 Stonehill married Sarah Mae Finch ( born 18 February 1953 in Los Angeles, California ), a school teacher, the sister-in-law of Stephen J. Cannell, at the First Congregational Church of Los Angeles, California, In a 1978 interview Stonehill explained how he met Sarah Finch.

At the First Congress of the Communist Party in Shanghai, Chen was elected ( in absentia ) as the party's first General Secretary ; and, with the assistance of Li Dazhao, he developed what would become a crucial cooperative relationship with the international Communist movement, the Comintern.

First, the relationship between Defence policy and military strategy, and – while highlighting the utility of force – emphasizes the importance of addressing security issues through a comprehensive, rather than an exclusively military, approach.

The song, about a relationship that fell apart due to the protagonist's lover's crack addiction, got her notice from First Priority, who signed Lyte in late 1987.

She eventually rejoins the Scooby Gang as Season Seven focuses on the long fight against the First Evil and eventually becomes friends with Xander ; although Xander and Anya still love each other, they never officially resume their relationship.

Held: Both statutes are unconstitutional under the Religion Clauses of the First Amendment, as the cumulative impact of the entire relationship arising under the statutes involves excessive entanglement between government and religion.

Helen Bryan acknowledged Ann Dandridge and her relationship to Martha Dandridge in her 2001 Martha Washington: First Lady of Liberty, relying upon Wiencek's research.

First published in 1935, and continuing its important relationship with Columbia University, the encyclopedia underwent major revisions in 1950 and 1963 ; the current edition is the sixth, printed in 2000.

That Capua and Rome were allied in 343 is less controversial, as such a relationship underpins the whole First Samnite War.

The First Chapter of volume 1 ( of the full 2 volume report ) considers the relationship between taonga works and intellectual property.

It also had a strong religious theme, with John's dealings with the First of the Fallen, and some storylines, such as the relationship between an angel, Tali, and a succubus demon, Ellie, would go on to be used again as a major plot device in Preacher, one of his most popular works.

Ennis ' run ends with " Rake at the Gates of Hell ", a story which finally brings together the racism storyline, with riots in Mile End, echoing the real-life Brixton and Broadwater Farm riots in London, the revenge attempt of the First of the Fallen, started in " Dangerous Habits ", and the end of John's relationship with Kit Ryan.

First, she decidedly established a cordial relationship with Marie Leszczyńska.