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Then and element Then our choice function can choose the least element of every set under our unusual ordering. Then is a group whose identity element is The group inverse of an arbitrary group element is the function inverse * Let the index set I of an inverse system ( X < sub > i </ sub >, f < sub > ij </ sub >) have a greatest element m. Then the natural projection π < sub > m </ sub >: X → X < sub > m </ sub > is an isomorphism. Then, after the pump energy stored in the laser medium has approached the maximum possible level, the introduced loss mechanism ( often an electro-or acousto-optical element ) is rapidly removed ( or that occurs by itself in a passive device ), allowing lasing to begin which rapidly obtains the stored energy in the gain medium. For f a real polynomial in x, and for any a in such an algebra define f ( a ) to be the element of the algebra resulting from the obvious substitution of a into f. Then for any two such polynomials f and g, we have that ( fg ) ( a ) Then, for any non-zero element x of M, the cyclic submodule xR must equal M. Fix such an x. Then, in 1846, the element ilmenium was claimed to have been discovered, but later was determined to be impure niobium. Then in 1908, the Japanese chemist Masataka Ogawa found evidence in the mineral thorianite, which he thought indicated the presence of element 43. Then each element in G is represented in some way by a product Suppose a partially ordered set P has the property that every chain ( i. e. totally ordered subset ) has an upper bound in P. Then the set P contains at least one maximal element. Suppose a non-empty partially ordered set P has the property that every non-empty chain has an upper bound in P. Then the set P contains at least one maximal element. Then p + q = ( p < sub > 1 </ sub >+ q < sub > 1 </ sub >, p < sub > 2 </ sub >+ q < sub > 2 </ sub >); since p < sub > 1 </ sub > = p < sub > 2 </ sub > and q < sub > 1 </ sub > = q < sub > 2 </ sub >, then p < sub > 1 </ sub > + q < sub > 1 </ sub > = p < sub > 2 </ sub > + q < sub > 2 </ sub >, so p + q is an element of W. # Let p = ( p < sub > 1 </ sub >, p < sub > 2 </ sub >) be an element of W, that is, a point in the plane such that p < sub > 1 </ sub > = p < sub > 2 </ sub >, and let c be a scalar in R. Then cp = ( cp < sub > 1 </ sub >, cp < sub > 2 </ sub >); since p < sub > 1 </ sub > = p < sub > 2 </ sub >, then cp < sub > 1 </ sub > = cp < sub > 2 </ sub >, so cp is an element of W. Then for each x in I, there is a base element B < sub > 3 </ sub > containing x and contained in I. Then, for example, (−∞, 1 ) and ( 0, ∞) would be in the topology generated by S, being unions of a single base element, and so their intersection ( 0, 1 ) would be as well. Then if we intersect each element of B with Y, the resulting collection of sets is a base for the subspace Y. " Then with her right hand she would sprinkle water at the head of the child and say, " Behold this element without whose assistance no mortal being can survive. Then the union ( written as +) and the concatenation ( written as ·) of two elements of A again belong to A, and so does the Kleene star operation applied to any element of A. Let B be a complex Banach algebra containing a unit e. Then we define the spectrum σ ( x ) ( or more explicitly σ < sub > B </ sub >( x )) of an element x of B to be the set of those complex numbers λ for which λe − x is not invertible in B. Then the film was rewound, and everything except the foreground element matted out so that the foreground element would now photograph in the previously blacked out area. Then the element c = ba is nilpotent ( if non-zero ) as c < sup > 2 </ sup > = ( ba )< sup > 2 </ sup > = b ( ab ) a = 0.

Then and e Then, removing the catalyst would also result in reaction, producing energy ; i. e. the addition and its reverse process, removal, would both produce energy. Then X is compact if and only if X is a complete lattice ( i. e. all subsets have suprema and infima ). Then is a derivation and is linear, i. e., and, and a Lie algebra homomorphism, i. e.,, but it is not always an algebra homomorphism, i. e. the identity does not hold in general. Then, once this claim ( expressed in the previous sentence ) is proved, it will suffice to prove " φ is either refutable or satisfiable " only for φ's belonging to the class C. Note also that if φ is provably equivalent to ψ ( i. e., ( φ ≡ ψ ) is provable ), then it is indeed the case that " ψ is either refutable or satisfiable " → " φ is either refutable or satisfiable " ( the soundness theorem is needed to show this ). Then, ~ 150, 000 years later ( i. e. around 50, 000 years ago ), sub-groups of this population began to expand our species ' range to regions outside of, and ( later ) within, this continent ( Tishkoff, 1996 ). Then a general definition of isomorphism that covers the previous and many other cases is: an isomorphism is a morphism that has an inverse, i. e. there exists a morphism with and. Then the data are passed through high-performance restartable dithering engine which is used regardless of monitor bit depth, i. e. also for 24 bits per pixel colour. Consider some set P and a binary relation ≤ on P. Thenis a preorder, or quasiorder, if it is reflexive and transitive, i. e., for all a, b and c in P, we have that: Then, in 1905, to explain the photoelectric effect ( 1839 ), i. e., that shining light on certain materials can function to eject electrons from the material, Albert Einstein postulated, based on Planck ’ s quantum hypothesis, that light itself consists of individual quantum particles, which later came to be called photons ( 1926 ). Then it was pointed out that " seeing the most submarines " depended not only on the number of submarines present, but also on the number of eyes looking ; i. e., patrol density. Then any vector in R < sup > 3 </ sup > is a linear combination of e < sub > 1 </ sub >, e < sub > 2 </ sub > and e < sub > 3 </ sub >. Then these dummies could be selected with mouse, and next character from the password " e " is typed, which replaces the dummies " asdfsd ". Then that researcher's Bradford multiplier b < sub > m </ sub > is 2 ( i. e. 10 / 5 ). Then it is used as: General Strike of a city, i. e., " General Strike in Florence ", or a General Strike in a whole country or province, for the purpose of gaining political rights, i. e., the right to vote ; as in Belgium, or Sweden. Then, the markedness of the ruler comes into play: it is " anchored " at point A, and slided and rotated until one mark is at point C, and one at point D, i. e., CD = AB.

Then and S Then Spec ( A < sub > S </ sub >) is homeomorphic to Then came an even more shocking confession: according to the CIA document, al-Faruq said two senior al-Qaeda officials, Abu Zubaydah and Ibn al-Shaykh al-Libi, had ordered him to ' plan large-scale attacks against U. S. interests in Indonesia, Malaysia, ( the ) Philippines, Singapore, Thailand, Taiwan, Vietnam and Cambodia. Then, when the U. S. Army Air Forces on the Marianas Islands ran out of conventional thermite incendiary bombs for its B-29 Superfortresses to drop on Japanese cities, its top commanders, such as General Curtis E. LeMay turned to napalm bombs to continue its fire raids on the large Japanese cities. Then N < sub > x </ sub > is a directed set, where the direction is given by reverse inclusion, so that S ≥ T if and only if S is contained in T. For S in N < sub > x </ sub >, let x < sub > S </ sub > be a point in S. Then ( x < sub > S </ sub >) is a net. Then according to the second isomorphism theorem S ∩ T is normal in T and ST / S ≅ T /( S ∩ T ). Then, it is undecidable to determine whether the language decided by an arbitrary Turing machine lies in S. In 1986, a screening of the entire Monkees television series by MTV led to renewed interest in the group, followed by a single (" That Was Then, This Is Now " reached number 20 on the Billboard Hot 100 in the U. S .), a 20th Anniversary Tour, a greatest hits album and a brand new LP, Pool It! Then, in 1966, the U. S. Congress created a National Council for Marine Resources and Engineering Development. Let S be a subgroup of G, and let N be a normal subgroup of G. Then:

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