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Then and p
Then each point p of the line can be specified by its distance from O, taken with a + or − sign depending on which half-line contains p.
Then b < sub > 0 </ sub > is the value of p ( x < sub > 0 </ sub >).
#* Proof: suppose that p is composite, hence can be written with a and Then is prime, but and contradicting statement 1.
Then p < sub > 1 </ sub >= MU < sub > 1 </ sub >, p < sub > 2 </ sub >= MU < sub > 2 </ sub >.
Then the first case of Fermat's Last Theorem holds true for p.
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.
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 and <
Then the energy of the vacuum is exactly E < sub > 0 </ sub >.
) 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 the cotangent space at x is defined as the dual space of T < sub > x </ sub > M:
Then I < sub > x </ sub > and I < sub > x </ sub >< sup > 2 </ sup > are real vector spaces and the cotangent space is defined as the quotient space T < sub > x </ sub >< sup >*</ sup > M = I < sub > x </ sub > / I < sub > x </ sub >< sup > 2 </ sup >.
Then the complex derivative of ƒ at a point z < sub > 0 </ sub > is defined by
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, 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 the overall runtime is O ( n < sup > 2 </ sup >).
Then the Cartesian product set D < sub > 1 </ sub > D < sub > 2 </ sub > can be made into a directed set by defining ( n < sub > 1 </ sub >, n < sub > 2 </ sub >) ≤ ( m < sub > 1 </ sub >, m < sub > 2 </ sub >) if and only if n < sub > 1 </ sub > ≤ m < sub > 1 </ sub > and n < sub > 2 </ sub > ≤ m < sub > 2 </ sub >.

Then and sup
Then, dividing the units of energy ( such as eV ) by a fundamental constant that has units of velocity ( M < sup > 0 </ sup > L < sup > 1 </ sup > T < sup >-1 </ sup >), facilitates the required conversion of using energy units to describe momentum.

Then and 2
Then, on July 2, there occurred another incident which set tongues to wagging at a furious clip.
Then in 2 we show that any line involution with the properties that ( A ) It has no complex of invariant lines, and ( B ) Its singular lines form a complex consisting exclusively of the lines which meet a twisted curve, is necessarily of the type discussed in 1.
Then Chris Broad scored three hundreds in successive Tests and bowling successes from Graham Dilley and Gladstone Small meant England won the series 2 – 1.
Then, in 396 BC, Agesilaus crossed into Asia with a force of 2, 000 neodamodes ( freed helots ) and 6, 000 allies ( including 30 spartiates ) to liberate Greek cities from Persian dominion.
Then Crapsey decided to make the criterion a stanza of five lines of accentual-syllabic verse, in which the lines comprise, in order, 1, 2, 3, 4, and 1 stresses and 2, 4, 6, 8, and 2 syllables.
Then NTPF became the Afloat Prepositioning Force ( AFP ) and eventually Maritime Prepositioning ship Squadron Two ( MPSRON 2 ) consisting of 20 deep-water pre-positioned logistics ships anchored in the lagoon.
Let t and s ( t > s ) be the sides of the two inscribed squares in a right triangle with hypotenuse c. Then s < sup > 2 </ sup > equals half the harmonic mean of c < sup > 2 </ sup > and t < sup > 2 </ sup >.

Then and is
Then he would get to his feet, as though rising in honor of his own remarkable powers, and say almost invariably, `` Gentlemen, this is an amazing story!!
Then, Jesus indicated that God's forgiveness is unlimited.
Certainly, the meaning is clearer to one who is not familiar with Biblical teachings, in the New English Bible which reads: `` Then Jesus arrived at Jordan from Galilee, and he came to John to be baptized by him.
Then it added: `` It is not possible to determine how extensive these ill effects will be -- nor how many people will be affected ''.
Then the words fell into a pattern: `` Mollie the Mutton is scratching her nose, Scratching her nose in the rain.
Then he thought of Aaron Blaustein standing in his rich house saying: `` God is tired of taking the blame.
Then it is marked on the inside where it comes in contact with the transom, frames, keelson and all the battens.
Then it is replaced and fastened.
Then the chines are rounded off and the bottom is rough-sanded in preparation.
Then, a group of eggs is deposited in a cavity in the beebread loaf and the egg compartment is closed.
Then there is a diagonalizable operator D on V and a nilpotent operator N in V such that ( A ) Af, ( b ) Af.
Then, too, the utmost clinical flexibility is necessary in judiciously combining carefully timed family-oriented home visits, single and group office interviews, and appropriate telephone follow-up calls, if the worker is to be genuinely accessible and if the predicted unhealthy outcome is to be actually averted in accordance with the principles of preventive intervention.
Then the editorial added prophetically: `` how far they may reach in Asia is yet undetermined, but they fall far short of our dreams of the war conferences ''.
Then she catapults into `` everything and everybody '', putting particular violence on `` everybody '', indicating to the linguist that this is a spot to flag -- that is, it is not congruent to the patient's general style of speech up to this point.
Then comes the time when the last wire is removed and Susie walks out a healthier and more attractive girl than when she first went to the orthodontist.
Then, with the new affluence, there is actually a sallying forth into the wide, wide world beyond the precincts of New York.
Then, if the middle number is activated to its greatest potential in terms of this square, through multiplying it by the highest number, 9 ( which is the square of the base number ), the result is 45 ; ;

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