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Then, there exists a constant satisfying: for every ground term in such that, there exists a context, a nontrivial context and a ground term such that and, for all.
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Then and there
Then the darkness thinned, and there was light again, and then bright sunlight.
Then there was no saying how many times the marine had blown his nose on the handkerchief.
Then, on July 2, there occurred another incident which set tongues to wagging at a furious clip.
Then suddenly there was a tremendous revulsion of popular feeling.
Then there were the distinguished foreign volunteers.
Then there was Mark Howe and there was Henry Dwight Sedgwick, an accomplished man of letters who wrote in the spirit of Montaigne and produced in the end a formidable body of work.
Then, all but blind, he said there was nothing in Back to Methuselah --, -- `` G.B.S. ought to have known that '', -- and `` I look at my bookshelves despairingly, knowing that I can have nothing more to do with them ''.
Then there had been the auction itself.
Then, of course, there are those of us who either do not want or need or cannot afford another car.
Then there was the caterer's ad which read: `` are you getting married or having an affair??
Then there was the General kissing his wife.
Then there is a diagonalizable operator D on V and a nilpotent operator N in V such that ( A ) Af, ( b ) Af.
Then there are the trustees and officers of the great educational foundations, who inevitably exert an influence on educational decisions by their support or refusal to support various educational programs, experiments, and demonstrations.
Then we drove on, until there was no more road and we traversed dry rice fields, bouncing across their squat earth walls.
Then, with the new affluence, there is actually a sallying forth into the wide, wide world beyond the precincts of New York.
Then there was a chorus of wild barking and baying.
Then there was another sound.
Then on Monday morning -- or it might have to be Tuesday -- get up and leave just the usual time, and last thing, put the money in an envelope under the old woman's purse there in the drawer.
Then Dick Hyde, submarine-ball hurler, entered the contest and only five batters needed to face him before there existed a 3-to-3 deadlock.
Then there is a matchmaker, one Mikeen Flynn, a role for which Eddie Foy was happily selected.
Then there are a pair of old biddies played by Grace Carney and Sibly Bowan who may be right off the shelf of stock Irish characters, but they put such a combination of good will and malevolence into their parts that they're quite entertaining.
Then there was a bad delay in getting Mort Lindsey's 30-piece orchestra wedged into its chairs.
Then there was exercise, boating and hiking, which was not only good for you but also made you more virile: the thought of strenuous activity left him exhausted.
Then there was a spat over something, as there had been lovers' spats before ; ;
Then and exists
Then, he said, `` Unfortunately, only one lamechian linguist exists, and he is too old for this expedition.
Then, for any given sequence of integers a < sub > 1 </ sub >, a < sub > 2 </ sub >, …, a < sub > k </ sub >, there exists an integer x solving the following system of simultaneous congruences.
Then is a totally ordered subset of A, hence there exists a maximal totally ordered subset containing, in particular A contains a maximal totally ordered subset.
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 ƒ is invertible if there exists a function g with domain Y and range X, with the property:
Then there exists a finite set of integer coefficients satisfying the equation:
Then a fuzzy subset s: S of a set S is recursively enumerable if a recursive map h: S × N Ü exists such that, for every x in S, the function h ( x, n ) is increasing with respect to n and s ( x ) = lim h ( x, n ).
Then x is an interior point of S if there exists a neighbourhood of x which is contained in S. Note that this definition does not depend upon whether neighbourhoods are required to be open.
Let us suppose that L is a complete lattice and let f be a monotonic function from L into L. Then, any x ′ such that f ′( x ′) ≤ x ′ is an abstraction of the least fixed-point of f, which exists, according to the Knaster – Tarski theorem.
Then there exists a factorization of the form
Then there exists a real number C such that for every real number x.
Then there exists a real-valued Borel measurable function w on X such that for every Lebesgue integrable function f: Y → R, the function ( f ° φ ) w is Lebesgue integrable on X, and
Then is a proper colouring of if and only if, for all there exists such that.
Let X be a g-dimensional torus given as X = V / L where V is a complex vector space of dimension g and L is a lattice in V. Then X is an abelian variety if and only if there exists a positive definite hermitian form on V whose imaginary part takes integral values on L × L.
Then there exists a lift of f ( that is, a continuous map for which and ) if and only if the induced homomorphisms and at the level of fundamental groups satisfy
Then there exists an open neighbourhood V of F ( 0 ) in Y and a continuously differentiable map G: V → X such that F ( G ( y )) = y for all y in V. Moreover, G ( y ) is the only sufficiently small solution x of the equation F ( x ) = y.
Then there exists a geodesic of maximal length connecting and.
Then, L < sub > 0 </ sub > is a Lie algebra, L < sub > 1 </ sub > is a linear representation of L < sub > 0 </ sub >, and there exists a symmetric L < sub > 0 </ sub >- equivariant linear map such that for all x, y and z in L < sub > 1 </ sub >,
Then, there exists a unique isomorphism g from onto such that g composed with the natural homomorphism induced by equals h.
) Then a S-scheme X is projective if and only if it is proper and there exists a very ample sheaf on X relative to S. Indeed, if X is proper, then an immersion corresponding to the very ample line bundle is necessarily closed.
Then, since span ker T, there exists a set of scalars such that:
Then the main line of Midland Railway was built in 1868 with a station near the main village which still exists today.
Then the fusion of all objects satisfying φ exists.
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