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 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 ).

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

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 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 C ( R ) is a subspace of R < sup > R </ sup >.

Then, he tied for first with Alexander Alekhine at St. Petersburg 1913 / 14 ( the eighth All-Russian Masters ' Tournament ).

Then it was covered with logs and poles and then sealed by drit / mud / soil and ice and moss.

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 Xorn / Magneto assaults New York, where Cyclops, Fantomex and a few students confront him.

Then on July 20, 2007 guitarist Jack Owen ( ex-Cannibal Corpse ) announced that Deicide is " on hiatus " and he has joined Ohio based death / thrash combo Estuary for touring purposes.

Then, in 405 / 6, an enormous barbarian horde, composed of Ostrogoths, Alans, Vandals and Quadi, crossed the frozen Rhine and invaded Gaul.

Then, not long before the finish, the Automobile Club de l ' Ouest ( ACO ), organizers of the Le Mans event, informed Ford that the geographical difference in starting positions would be taken into account at a close finish – meaning that the McLaren / Amon vehicle, which had started perhaps behind the Hulme-Miles car, would have covered slightly more ground over the 24 hours and would therefore be the winner.

Then, he would introduce a song he / she wrote and was made popular by another artist.

Then, the Parseval / GEIG project compared phrase-structure grammars ( Black 1991 ).

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

Then according to the second isomorphism theorem S ∩ T is normal in T and ST / S ≅ T /( S ∩ T ).

In the early ballads Robin's men usually kneel before him in strict obedience: in A Gest of Robyn Hode the king even observes that " His men are more at his byddynge / Then my men be at myn.

Then the ensuing field goal attempt failed when punter / holder Mike Horan fumbled the snap.

Then I and I < sup > 2 </ sup > are real vector spaces, and T < sub > x </ sub > M may be defined as the dual space of the quotient space I / I < sup > 2 </ sup >.

Some of these songs, along with others performed but not included on the album, were included on Colin Newman's post-Wire solo albums ( 5 / 10, We Meet Under Tables ), while others were released by Gilbert and Lewis ' primary post-Wire outlet Dome ( And Then ..., Ritual View ).

* In the 1939 song Strange Fruit, written to condemn the practice of lynching, the Magnolia flower was referenced as being associated with the Southern United States, where most lynchings took place: " Pastoral scene of the gallant south / The bulging eyes and the twisted mouth / Scent of magnolias, sweet and fresh ,/ Then the sudden smell of burning flesh.

Then, the chance that the first letter typed is ' b ' is 1 / 50, and the chance that the second letter typed is a is also 1 / 50, and so on.

Then, as and when each party communicates a bit to the other, the number of choices for the answer reduces as this eliminates a set of rows / columns resulting in a submatrix of A.

* "( Now and Then There's ) A Fool Such as I " / " Where'd You Go Last Night " ( Decca 30873 )

Then I saw the father's head slightly turn ; ;

`` Then I return to the United States for engagements at the Hollywood Bowl and in Philadelphia '', he added.

Then, after overtures to accept a settlement and go through with a divorce, Miriam gave a ghastly echo of Mrs. Micawber by suddenly stating, `` I will never leave Mr. Wright ''.

Then I spent the next two days at the baseball park and at Jack Doyle's pool parlors.

My own stern hand has rent the ancient bond, And thereof shall the ending not have end: But not for me, that loved her, to be fond Lightly to please me with a newer friend Then hold it more than bravest-feathered song, That I affirm to thee, with heart of pride, I knew not what did to a friend belong Till I stood up, true friend, by thy true side ; ;

Then I asked, `` What does that teach you ''??

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, all but blind, he said there was nothing in Back to Methuselah --, -- `` G.B.S. ought to have known that '', -- and `` I look at my bookshelves despairingly, knowing that I can have nothing more to do with them ''.

Then we pull out under our mortar and artillery cover, but nobody pulls out until I say so.

Then he was asking himself the usual early morning questions: What the Hell am I doin here??

Then epistolatory me was a foreign correspondent dispatching exciting cables and communiques, full of dash and wit and glamor, quoting from the books I read, imitating the grand styles of the authors recommended by a teacher in whose special, after-school class I was enrolled.

Then I have seen the pride of country well in the eyes of these young people.

Then I cover the sowing with a board.

Then I plan my attack: the parts I will finish first, the range of values, the accenting of minor details -- all in all, mechanics of producing the finished job with a maximum of crispness.

Then I dry the sheet under mild pressure so that it will lie flat as a board.

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

Then I realized that she had been deliberately showing me, this time, what Granny was like ; ;

Then I replied, coldly, `` If you despise me, why don't you say so, directly ''??

Then later I had quit my safe, secure five-a-week spot on a network soap opera to take a part in this play.

Then I shut off the device again.