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 he was appointed Assistant Chief of Staff in charge of the new Operations Division ( which replaced WPD ) under Chief of Staff General George C. Marshall, who spotted talent and promoted accordingly.

Then, once this claim ( expressed in the previous sentence ) is proved, it will suffice to prove " φ is either refutable or satisfiable " only for φ's belonging to the class C. Note also that if φ is provably equivalent to ψ ( i. e., ( φ ≡ ψ ) is provable ), then it is indeed the case that " ψ is either refutable or satisfiable " → " φ is either refutable or satisfiable " ( the soundness theorem is needed to show this ).

Then suppose definition B contains 32 instances of A ; C contains 32 instances of B ; D contains 32 instances of C ; and E contains 32 instances of D. The design now contains 5 definitions ( A through E ) and 128 instances.

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

Let U be a functor from D to C, and let X be an object of C. Then the following statements are equivalent:

Then we have a universal morphism for each object in C and D:

* Douglas K. Smith, Robert C. Alexander, Fumbling the Future: How Xerox Invented, Then Ignored, the First Personal Computer ( William Morrow and Company, New York, 1988 ) ISBN 1-58348-266-0

Then in 1952 the Barbados Advocate newspaper polled several prominent Barbadian politicians, lawyers, businessmen, the Speaker of the Barbados House of Assembly and later as first President of the Senate, Sir Theodore Branker, Q. C.

Then, after the medical team has made sure they are on the right track, the tip of the electrode is advanced to the plane of the cingulate where it is heated to 75-90 C. Once the first lesion is created it serves as a center around which several other lesions are created.

Then FF. C followed with Simeia ton kairon, and TXC ( now Artemis / Efthimis ) with their single Terror X Crew in 1995.

Then F and G can be restricted to D < sub > 1 </ sub > and C < sub > 1 </ sub > and yield inverse equivalences of these subcategories.

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.

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

Introduction " The One, The Only R. C., Then Now and Forever " by Peter Straub.

Then that formula must be copied into all of the cells in column C, making sure to change the reference to A1 to a new reference for A2, etc.

Then, the markedness of the ruler comes into play: it is " anchored " at point A, and slided and rotated until one mark is at point C, and one at point D, i. e., CD = AB.

Then around four hundred million B. C., a small continent that was long and thin, collided with proto North America.

Then, in January 1849, Michael C. Nye and William Foster Nye, brothers-in-law to Covillaud's wife, Mary Murphy, bought the other half of the Cordua ranch.

Then in 1943 he received orders transferring him to the Secret Intelligence Branch of the Office of Strategic Services ( OSS ) in Washington, D. C.

Then an enriched category C ( alternatively, in situations where the choice of monoidal category needs to be explicit, a category enriched over M, or M-category ), consists of

Then Wollheim changed everything when he brought out an unauthorized paperback edition of J. R. R. Tolkien's The Lord of the Rings in three volumes — the first mass-market paperback edition of Tolkien's epic.

Let R be a domain and f a Euclidean function on R. Then:

Then the probability of the measurement outcome lying in an interval B of R is | E < sub > A </ sub >( B ) ψ |< sup > 2 </ sup >.

Line QP can be extended beyond P to some point T, and line GP can be extended beyond P to some point R. Then and are vertical, so they are equal ( congruent ).

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, thus by T, which means w R w using the definition of.

Then R / I is a ring with unity, ( respectively, R / A is a finitely generated module ), and so the above theorems can be applied to the quotient to conclude that there is a maximal ideal ( respectively maximal right ideal ) of R containing I ( respectively, A ).

Then they were treated with protease enzymes, which removed the proteins from the cells before the remainder was placed with R strain bacteria.

Then the remnants of the R strain bacteria were treated with a deoxyribonuclease enzyme which removed the DNA.

Then for arbitrary ε > 0 there is an embedding ( or immersion ) ƒ < sub > ε </ sub >: M < sup > m </ sup > → R < sup > n </ sup > which is

Then any vector in R < sup > 3 </ sup > is a linear combination of e < sub > 1 </ sub >, e < sub > 2 </ sub > and e < sub > 3 </ sub >.

Let R be a Noetherian ring and let I be an ideal of R. Then I may be written as the intersection of finitely many primary ideals with distinct radicals ; that is:

Then W is a subspace of R < sup > 2 </ sup >.

# 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 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 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, 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 ; ;