: Given any family of nonempty sets, their Cartesian product is a nonempty set.

Given the similarities in the two characters ' names, professions, written works and generally dark subject matter, it is likely that Lovecraft's Alhazred provided the main inspiration for al-Hazir.

Given a field F, the assertion “ F is algebraically closed ” is equivalent to other assertions:

* Given an algebraic number, there is a unique monic polynomial ( with rational coefficients ) of least degree that has the number as a root.

" Given that Thomas Aquinas died six years before Albertus Magnus ' death, this legend as stated is unlikely.

Given Ealdred's known support of Godwin's family, John of Worcester is probably correct.

* Given any Banach space X, the continuous linear operators A: X → X form a unitary associative algebra ( using composition of operators as multiplication ); this is a Banach algebra.

* Given an R-module M, the endomorphism ring of M, denoted End < sub > R </ sub >( M ) is an R-algebra by defining ( r · φ )( x ) = r · φ ( x ).

Given the other ZF axioms, the axiom of regularity is equivalent to the axiom of induction.

After Christians in Ephesus first wrote to their counterparts recommending Apollos to them, he went to Achaia where Paul names him as an apostle ( 1 Cor 4: 6, 9-13 ) Given that Paul only saw himself as an apostle ' untimely born ' ( 1 Cor 15: 8 ) it is certain that Apollos became an apostle in the regular way ( as a witness to the risen Lord and commissioned by Jesus-1 Cor 15: 5-9 ; 1 Cor 9: 1 ).< ref > So the Alexandrian recension ; the text in < sup > 38 </ sup > and Codex Bezae indicate that Apollos went to Corinth.

Given its " blue-water " bias, the navy is even less inclined to become involved in counterdrug operations than the army or air force.

Given any element x of X, there is a function f < sup > x </ sup >, or f ( x ,·), from Y to Z, given by f < sup > x </ sup >( y ) := f ( x, y ).

Given this, the bidding is said to start at the one-level when contracting for a total of seven tricks, at the two-level for eight tricks and so on to the seven-level to contract to take all thirteen tricks.

While tests have been done to support this with fixed pieces of flat metal, the result is inconclusive with respect to the curved armour of the period. Given the following actions of the archers, it seems likely Froissart was correct.

Given a ket of norm 1, the orthogonal projection onto the subspace spanned by is

Given this influence, it is unfortunate that many of the details of his life remain shrouded in mystery, perhaps forever ; even the only known picture of him, shown above, is heavily retouched, with a fake tie painted in by hand.

Given points P < sub > 0 </ sub > and P < sub > 1 </ sub >, a linear Bézier curve is simply a straight line between those two points.

Given the then GDP ($ 7. 095 bln ) of the country, military spending is roughly estimated to be about $ 300 million.

Given some encrypted data (" ciphertext "), the goal of the cryptanalyst is to gain as much information as possible about the original, unencrypted data (" plaintext ").

Given the federal nature of the holiday, celebrating Canada Day can be a cause of friction in the province of Quebec, where the holiday is overshadowed by Quebec's National Holiday, on June 24.

Given that many journeys are for relatively short distances, there is considerable scope to replace car use with walking or cycling, though in many settings this may require some infrastructure modification, particularly to attract the less experienced and confident.

Given such a sweeping definition, it is apparent that cognition is involved in everything a human being might possibly do ; that every psychological phenomenon is a cognitive phenomenon.

To the members of our Advisory Board, and most specially to its members who constitute our committees of selection, the Foundation is indebted for its successes of choice of Fellows.

Dr. Ray is a Fellow of the Foundation -- appointed thrice to assist his studies of William Makepeace Thackeray and of H. G. Wells -- and, before his appointment to the Foundation's executive staff, had been given our highest scholarly accolade, appointment to the Advisory Board.

An example of a more definite class bias is noted in proceedings of the Commission on the Financing of Higher Education sponsored by the Association of American Universities and supported by the Rockefeller Foundation and the Carnegie Corporation.

Similarly, the American Cancer Society ( ACS ), the Arthritis and Rheumatism Foundation, and the BBB have each stated lately that medical quackery is at a new high.

The mediums with whom the Parapsychology Foundation is working in this experiment are in a waking or only slightly dissociated state, so that the sitter can make comments, ask and answer questions, instead of talking with a `` control '' who speaks through an entranced sensitive.

`` We are back with the ' Met ' again now that the ' Met ' is back in Chicago '', bulletins Mrs. Frank S. Sims, president of the women's board of the University of Chicago Cancer Research Foundation.

The series of ballets is sponsored by the Milenoff Ballet Foundation, Inc., a non-profit foundation with headquarters in Coral Gables.

The American Film Institute ( AFI ) is an independent non-profit organization created by the National Endowment for the Arts, which was established in 1967 when President Lyndon B. Johnson signed the National Foundation on the Arts and the Humanities Act.

However, during the January to September 2010 period, the sector experienced a 5. 2 percent year-on-year decrease, which according to the Civilitas Foundation is an indication of the unsustainability of a sector based on an elite market, with few products for the median or low budgets.

The building is made open to the public by the CatalunyaCaixa Foundation, which manages the various exhibitions and activities and visits to the interior and roof.

The Apache Software Foundation ( ASF ) is a non-profit corporation ( classified as 501 ( c )( 3 ) in the United States ) to support Apache software projects, including the Apache HTTP Server.

The ACLU Foundation is a 501 ( c )( 3 ) non-profit corporation, which does not engage in lobbying, and donations to it are tax deductible .< ref name =" nonprofit ">

In Asimov's Empire and Foundation series, the capital planet Trantor of the galactic empire is a completely built-up planet, covered in its entirety with tall buildings and subterranean structures.

* 1840 – Foundation stone for new Palace of Westminster, London, is laid by wife of Sir Charles Barry.

Brian Lara has established the Pearl and Bunty Lara Foundation, which is a charitable organisation in memory of his parents that aims to address health and social care issues.

For instance, JavaBeans library is used for designing enterprise applications, Windows Forms library is used for designing graphical user interface ( GUI ) applications like Microsoft Word, and Windows Communication Foundation is used for designing web services.

Chalmers is an aktiebolag with 100 shares à 1, 000 SEK, all of which is owned by a private foundation ( Chalmers University of Technology Foundation ) which appoints the university board and the president.

The movement is supported by the Masorti Foundation for Conservative Judaism in Israel, an American organization that provides funding to Masorti programs, which are disadvantaged by the Israeli government's practice of funding only Orthodox institutions.

It is said to have also received funding from the Heritage Foundation.

* Portable. NET, part of the dotGNU project, is a Free Software implementation of ECMA-335 by Free Software Foundation.

According to the National Psoriasis Foundation and the FDA, coal tar is a valuable, safe and inexpensive treatment option for millions of people with psoriasis and other scalp or skin conditions.

ClearType is also an integrated component of the Windows Presentation Foundation text-rendering engine.