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.

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 it would be that by recursion ( using the axiom of countable choice ) that a sequence of polynomials could be found so that, letting of minimal degree.

Then, when similar diagram is plotted for a cluster whose distance is not known, the position of the main sequence can be compared to that of the first cluster and the distance estimated.

Then there does not exist a strictly increasing sequence of open sets ( equivalently

Then there is an exact sequence relating the kernels and cokernels of a, b, and c:

Then it does whatever operation ( or sequence of operations ) it was going to do.

Then its negation ¬ φ, together with the field axioms and the infinite sequence of sentences 1 + 1 ≠ 0, 1 + 1 + 1 ≠ 0, …, is not satisfiable ( because there is no field of characteristic 0 in which ¬ φ holds, and the infinite sequence of sentences ensures any model would be a field of characteristic 0 ).

Then the sheaf axioms can be expressed as the exactness of the sequence

Then there is a long exact sequence of homotopy groups

Then, in one afternoon, they wrote his entire body of work: 17 poems, none longer than a page, and all intended to be read in sequence under the title The Darkening Ecliptic.

Then A, B and their intersection A ∩ B are homotopy equivalent to circles, so the nontrivial part of the sequence yields

Then divide the sequence into an initial segment, and a tail of small terms: given any ε > 0 we can take M large enough to make the initial segment of terms up to c < sub > N </ sub > average to at most ε / 2, while each term in the tail is bounded by ε / 2 so that the average is also.

Then continue the sequence, where each subsequent term is the sum of the previous n terms.

Then this polynomial sequence is of binomial type, i. e. it satisfies the binomial identity

Then there is a constant C depending only on b < sub > 1 </ sub >, ..., b < sub > s </ sub >, such that sequence

Then once a relationship is established contexts can be removed from site in the reverse order they arrived in accordance with the stratigraphic excavation and the creation of a Harris matrix for the sequence being investigated.

Then, in August, Cuellar accomplished a sequence of 35 batters gotten out in a row.

Then, in the novel's most controversial sequence, she opens her dress and offers her breast to Lecter ; he accepts her offer and the two became lovers.

" Hughes went on to comment positively on the episode's debut of the high-definition opening sequence, and its introduction of Joyce Kinney as a replacement for news anchor Diane Simmons, who was killed off in " And Then There Were Fewer ".

Then the sequence of Y < sub > i </ sub > has exactly the probability distribution of tosses made with the second coin.

Then the sequence is pure exact if and only if C is flat.

Then she wakes up, showing that she dreamed this sequence at least.

Then follows in sequence: the dance of the hours of dawn, the hours of day, the hours of the night and the morning.

Then, helpfully, as she merely stared at him in weary silence, `` Maybe you could write it down for me, huh??

Then he would realize they were really things that only he himself could think.

Then he could tell them to go home, while the administration continued to wage the battle with the $28 million in extra revenues the sales tax measure would bring in over an eight months period.

Then, a little later, Shilkret discovered there was no one to play the brief celesta solo during the slow section, so he hastily asked Gershwin if he might play the solo ; Gershwin said he could and so he briefly participated in the actual recording.

Then the accuracy of the response could be measured.

Then, in the late 1960s and early 1970s, Vera Rubin, a young astronomer at the Department of Terrestrial Magnetism at the Carnegie Institution of Washington, presented findings based on a new sensitive spectrograph that could measure the velocity curve of edge-on spiral galaxies to a greater degree of accuracy than had ever before been achieved.

Then we could have written a formula of degree k which is equivalent to φ, namely.

Then the Lord let me see why there was none upon the earth that could speak to my condition, namely, that I might give Him all the glory ; for all are concluded under sin, and shut up in unbelief as I had been, that Jesus Christ might have the pre-eminence who enlightens, and gives grace, and faith, and power.

For instance, suppose that each input is an integer z in the range 0 to N − 1, and the output must be an integer h in the range 0 to n − 1, where N is much larger than n. Then the hash function could be h

Then a monk appeared saying that the jaundice could be cured by making a medicine out of the arm and eye of one without anger.

Then it mushroomed to nearly 800, 000 members in 1886, but its frail organizational structure could not cope and it was battered by charges of failure and violence.

A rhetorical question such as " Then please explain, how could I possibly have beaten a wife that I've never had?

Then Aguilar could interpret from Mayan to Spanish, until Marina learned Spanish and could be the sole interpreter.

Then a different musician could change the song / beat / lyrics to their preference without getting into legal trouble.

Then the mean and standard deviation of heights of American adults could be calculated as:

Then, when personal computers became available, gamers could simply " sit down and play " without learning masses of rules, clearing physical space, and finding and coordinating schedules with opponents.

Then, by reinserting the end of one whisker into the flame, he could create a very small, high-quality glass sphere.

Then suddenly we found that chimpanzees could be brutal — that they, like us, had a darker side to their nature.

Then I read David Copperfield and realized I could never write at that level, and therefore I should find something else to do.

Then the brothers spotted something really intriguing — what could have been a natural shelter.

Then, anticipating the motorcycle boom and intending to have a stake in it, the Aermacchi chiefs looked for a designer who could produce an attractive lightweight.

Then on 20th and 21st Boedromion, the initiates entered a great hall called Telesterion ; in the center stood the Anaktoron (" palace "), which only the hierophants could enter, where sacred objects were stored.

Then these dummies could be selected with mouse, and next character from the password " e " is typed, which replaces the dummies " asdfsd ".