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 K < sub > X </ sub > is the sheaf associated to the presheaf K < sub > X </ sub >< sup > pre </ sup >.

Then he was on his way at a gallop.

Then he went on to the Cheyennes and told them that the Sioux was goin' to move up.

Then he hunkered down on the heels of his handmade boots, peered into the orderly chaos of axle, shock absorber, and spring.

Then, with a glory that almost wiped out the deep, downward sags in her careworn face, Matilda leaned over the wheel and shouted to Hez, who was stumbling along in the heat and the dust on the opposite side of the wagon `` Pa!!

Then, and only then, with the Jacksons and Dan as their true guests of honor, did the Harrows take time to catch up on the news.

Then maybe next time he won't be so quick on the trigger ''.

Then he calmly and carefully slugged the remaining five shots into the venomous head -- caught in the wicker back of the chair, the eyes dead on him as the life finally went out of the brute.

Then there was no saying how many times the marine had blown his nose on the handkerchief.

Then when Miss Langford was on the end of the line of girls, Jack, in the middle of the line, gave an extra hard pull and the young teacher sprawled backwards, sitting down hard, her dress flying over her head.

Then, on July 2, there occurred another incident which set tongues to wagging at a furious clip.

Then she rounded on Weston and cried, `` You always did Wright's dirty work!!

Then Miriam varied the senseless psychological warfare by suddenly withdrawing a suit for separate maintenance that had been pending, and asking for divorce on the grounds of cruelty, with the understanding that Wright would not contest it.

Then he launches into an attack on the Tory ministers, whom he calls the `` New Converts '' ; ;

Then he called on the governor to explain why.

Then I spoke at the ninetieth birthday party of W. E. Burghardt Du Bois, who embarked on a fictional trilogy at eighty-nine and who, with The Crisis, had created a Negro intelligentsia that had never existed in America before him.

Then followed a period in which he wrote reviews for The New York Times Book Review, The Commonweal, Commentary, had a small piece in Partisan Review, and moved on to Hudson, The Village Voice, and Exodus.

Then let the whole lot be hanged in a public mass execution on July 4, 1963.

Then, to conclude on an indeterminate note: `` Nevertheless, if fallout increased substantially, or remained high for a long time, it would become far more important as a potential health hazard in this country and throughout the world ''.

Then he slipped and went down on his hands and knees in the melting snow.

Then the figure moved on.

Then came their bathroom, and then a bedroom that, judging by the photographs on the walls, must belong to Mme Cestre.

Then Rector, attired in his best blue serge suit, sat in a chair out on the lawn, in the shade of a tree, smoking a cigarette and waiting.

Then she went back to the wicker chair and resolutely adjusted her eyes to the glare on the water.

Then he spread out the last list on the counter.

Then he stood back to look at Mr. Jack, who was pulling on his pigskin gloves.

) 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 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.