Then, I remembered that the girls had had a banana for dessert every day for the last week.

Then every linear operator T in V can be written as the sum of a diagonalizable operator D and a nilpotent operator N which commute.

Then every component of the graph of F must be defined over a bounded sub-interval.

Then our choice function can choose the least element of every set under our unusual ordering.

After this, the senior priest ( or bishop ) pours pure olive oil and a small amount of wine into the shrine lamp, and says the " Prayer of the Oil ", which calls upon God to "... sanctify this Oil, that it may be effectual for those who shall be anointed therewith, unto healing, and unto relief from every passion, every malady of the flesh and of the spirit, and every ill ..." Then follow seven series of epistles, gospels, long prayers, Ektenias ( litanies ) and anointings.

Then, since every real is the limit of some Cauchy sequence of rationals, the completeness of the norm extends the linearity to the whole real line.

Indeed, following, suppose ƒ is a complex function defined in an open set Ω ⊂ C. Then, writing for every z ∈ Ω, one can also regard Ω as an open subset of R < sup > 2 </ sup >, and ƒ as a function of two real variables x and y, which maps Ω ⊂ R < sup > 2 </ sup > to C. We consider the Cauchy – Riemann equations at z = 0 assuming ƒ ( z ) = 0, just for notational simplicity – the proof is identical in general case.

Then almost every point of A returns to A infinitely often.

Then I got the script and my name was on every page because it was about my car.

In every medium, spy thrillers introduce children and adolescents to deception and espionage at earlier ages, as in the Agent Cody Banks film, the Alex Rider adventure novels by Anthony Horowitz, chick lit novels such as I'd Tell You I Love You, But Then I'd Have to Kill You and the CHERUB series, by Robert Muchamore.

Then the equivalence class of the identity is the symmetry group of the figure, and every equivalence class corresponds to one isomorphic version of the figure.

Those who believe and those who are Jews and Christians, and Sabians, whoever believes in Allah and the Last Day and do righteous good deeds shall have their reward with their Lord, on them shall be no fear, nor shall they grieve ... let there be no compulsion in religion "-that all monotheistic religions or people of the book have a chance of salvation, to the most exclusive teaching common amongst Salafis and Wahhabis, and supported by several works of medieval Islamic theology and by traditions ( hadith ) which are considered correct ( sahih ) by Sunni Muslims, for the most part are summed up in Surah 9: 5, 29: " Then, when the sacred months have passed, slay the idolaters wherever ye find them, and take them, and besiege them, and lay in wait in every stratagem of war.

It looked to me as if the whole town would break out into a riot ... Then suddenly I heard a heavy thump, and as if by magic the whole crowd dispersed in every direction ... One of the regular patrons had felled one of the noisiest rioters .... And it was the effect of this which had scattered everybody so suddenly.

Then I shall crucify you every one.

Then he ordered 2 golden pieces to be given to every householder in Constantinople and 200 pounds of gold ( including 200 silver pieces annually ) to be given to the Byzantine Church.

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 ).

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, when the clock reaches midnight, there will be firework at Taipei 101 and every year since, the firework has last longer than the year before.

Then, still black with smoke and with his uniform in shreds, Nelson went on board Victory where he was received on the quarter-deck by Admiral Jervis – " the Admiral embraced me, said he could not sufficiently thank me, and used every kind expression which could not fail to make me happy.

Then is an open cover of S, but any finite subcollection of has the form of C discussed previously, and thus cannot be an open subcover of S. This contradicts the compactness of S. Hence, every accumulation point of S is in S, so S is closed.

Then every hom-set has an addition, endowing it with the structure of an abelian monoid, and such that the composition of morphisms is bilinear.

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.

is partially ordered by set inclusion, therefore it contains a maximal totally ordered subset P. Then the set satisfies the desired properties.

Then a subset is called a σ-algebra if it satisfies the following three properties:

* Then extend the definition of truth to include sentences that predicate truth or falsity of one of the original subset of sentences.

For f ∈ R, define D < sub > f </ sub > to be the set of ideals of R not containing f. Then each D < sub > f </ sub > is an open subset of Spec ( R ), and is a basis for the Zariski topology.

Then, given any point x and neighbourhood G of x, there is a closed neighbourhood E of x that is a subset of G.

Let K be a closed subset of a compact set T in R < sup > n </ sup > and let C < sub > K </ sub > be an open cover of K. Then

Pick a fixed subset L of U. Then the maps F and G, where F ( M ) is the intersection of L and M, and G ( N ) is the union of N and ( U

Then, a tail event is an event whose occurrence or failure is determined by the values of these random variables but which is probabilistically independent of each finite subset of these random variables.

Then the subset of X

A local trivialization of a principal bundle P is given by a section s of P over an open subset U of M. Then the pullback s < sup >*</ sup > ω of a principal connection is a 1-form on U with values in.

Let S a multiplicatively closed subset of R, i. e. for any s and t ∈ S, the product st is also in S. Then the localization of M with respect to S, denoted S < sup >− 1 </ sup > M, is defined to be the following module: as a set, it consists of equivalence classes of pairs ( m, s ), where m ∈ M and s ∈ S. Two such pairs ( m, s ) and ( n, t ) are considered equivalent if there is a third element u of S such that

Furthermore, let S be a subset of P that has a least upper bound s. Then f preserves the supremum of S if the set f ( S ) =

Given a compact subset K of X and an open subset U of Y, let V ( K, U ) denote the set of all functions such that Then the collection of all such V ( K, U ) is a subbase for the compact-open topology on C ( X, Y ).

If E is a Borel subset of R, and 1 < sub > E </ sub > is the indicator function of E, then 1 < sub > E </ sub >( T ) is a self-adjoint projection on H. Then mapping

Let S be an ( a, b )- separator, that is, a vertex subset that separates two nonadjacent vertices a and b. Then S is a minimal ( a, b )- separator if no proper subset of S separates a and b. More generally, S is called a minimal separator if it is a minimal separator for some pair ( a, b ) of nonadjacent vertices.

Then a subset B ⊂ X is bounded if for each open neighborhood U of 0 in X there exists a number m > 0 such that

) Then X < sub > i </ sub > is the value ( or realization ) produced by a given run of the process at time i. Suppose that the process is further known to have defined values for mean μ < sub > i </ sub > and variance σ < sub > i </ sub >< sup > 2 </ sup > for all times i. Then the definition of the autocorrelation between times s and t is

Then X is reflexive if and only if each X < sub > j </ sub > is reflexive.

Then X is separable if and only if X ′ is separable.

Then X is compact if and only if X is a complete lattice ( i. e. all subsets have suprema and infima ).

Then all elements of X equivalent to each other are also elements of the same equivalence class.

Then the quotient space X /~ can be naturally identified with a torus: take a square piece of paper, bend and glue together the upper and lower edge to form a cylinder, then bend the resulting cylinder so as to glue together its two open ends, resulting in a torus.

Then the expectation of this random variable X is defined as

Then a presheaf on X is a contravariant functor from O ( X ) to the category of sets, and a sheaf is a presheaf which satisfies the gluing axiom.

Then the joint distribution of X and Y is completely determined by our channel and by our choice of, the marginal distribution of messages we choose to send over the channel.

Then ƒ is invertible if there exists a function g with domain Y and range X, with the property:

* 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 for a specific value x of X, the function L ( θ | x )

Then the observation that X

Then the emf, E < sub > X </ sub >, of the same cell containing the solution of unknown pH is measured.

Then X has cardinality at most and cardinality at most if it is first countable.

Then A is dense in C ( X, R ) if and only if it separates points.

Then ρ will be the finest completely regular topology on X which is coarser than τ.

Then the Zariski tangent space at a point p ∈ X is the collection of K-derivations D: O < sub > X, p </ sub >→ K, where K is the ground field and O < sub > X, p </ sub > is the stalk of O < sub > X </ sub > at p.