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 used resolution to attempt to obtain a proof by contradiction by adding the clausal form of the negation of the theorem to be proved.

Then, mathematicians started worrying that they were assuming the existence of a continuum of real numbers without proof.

Then the existence of an eigenvalue is equivalent to the ideal generated by ( the relations satisfied by ) being non-empty, which exactly generalizes the usual proof of existence of an eigenvalue existing for a single matrix over an algebraically closed field by showing that the characteristic polynomial has a zero.

Then, according to Michael Rapport, " He pointed to Napoleon's bloody, pallid face as proof.

Then, given the description of A ( n ) as the advice, the machine will correctly decide the problem on all inputs of length n. The other direction uses a simulation of a polynomial-time Turing machine by a polynomial-size circuit as in one proof of Cook's Theorem.

Then in late September he began reading " for amusement " the 6th edition of Malthus's An Essay on the Principle of Population which reminded him of Malthus's statistical proof that human populations breed beyond their means and compete to survive, at a time when he was primed to apply these ideas to animal species.

Then they insisted on its proof.

Then Janos Pintz joined the team and they completed the proof in 2005.

Then the dried paper wound around a bar to make a roll of water proof paper.

Then there had been the auction itself.

Then, in chronological order, Mr. White covers the primary campaigns, the conventions and the Presidential campaign itself.

Then, after many more billions of years, when all the galaxies are whistling toward a common center, this movement will slow down and reverse itself again.

Then a title appears, saying " Reversed ", and the action is repeated in reverse, so that the castle re-erects itself under his blows.

Then, after the pump energy stored in the laser medium has approached the maximum possible level, the introduced loss mechanism ( often an electro-or acousto-optical element ) is rapidly removed ( or that occurs by itself in a passive device ), allowing lasing to begin which rapidly obtains the stored energy in the gain medium.

Then, in 1905, to explain the photoelectric effect ( 1839 ), i. e., that shining light on certain materials can function to eject electrons from the material, Albert Einstein postulated, based on Planck ’ s quantum hypothesis, that light itself consists of individual quantum particles, which later came to be called photons ( 1926 ).

Then material objects and the Earth itself begin to dissolve into transparency.

Then, if the total flux is known, the field itself can be deduced at every point.

Then, two centuries later in 330 BCE the Macedonian emperor Alexander the Great overthrew the Persians and took control of Mesopotamia itself, bringing Hellenic influence to the region with the Seleucid Empire.

Then followed relegation in 1981 and promotion in 1982, before the club finally managed to establish itself in the 2nd division.

Then, providing a localization package for a given language is in itself a non-trivial undertaking, requiring specialized technical writers to construct a culturally appropriate syntax for potentially complicated concepts, coupled with engineering resources to deploy and test the localization elements.

Then we can apply Gauss – Bonnet to the surface T formed by the inside of that triangle and the piecewise boundary given by the triangle itself.

Theorem: Let R be a Dedekind domain with fraction field K. Let L be a finite degree field extension of K and denote by S the integral closure of R in L. Then S is itself a Dedekind domain.

Then, around 5700 BC, Mazama collapsed into itself during a tremendous volcanic eruption, losing in height.

Then, in the form of plates rivetted together and made watertight, it was used to form the hull itself.

Then, on 12th May 1725, the society took it upon itself to " pass " Brother Cotton and Brother Papillion Ball as Master Masons.

Then again pull factors are issues that can arise from a location itself and therefore ‘ push ’ an individual to choose to experience it.

Then, lord Krishna smilingly declared that as Earth, Moon and Sun are together in one place, this in itself was an eclipse.

Then the just will reign with Christ for ever, glorified in body and soul, and the material universe itself will be transformed.

Then, before the insect can extricate itself, the spider rushes over to bite it and wrap it in silk.

Then, when an alien mecha attacks the city, Poemy encounters a mysterious and silent man playing the shamisen, who is apparently her real father, and he gives her a fish, which when gutted and used as a magic wand, allows her to transform into Puni Puni Poemy, a magical girl embodying the will of Earth itself.

As to the origins of the name itself, " Heller admitted in later years that the name ' Yossarian ' was derived from the name of one of his Air Force buddies, Francis Yohannan, but that the character of Yossarian himself was ' the incarnation of a wish ' ( Now and Then 175-6 ).

Then, because any node dominates itself and node A dominates itself, node B is in the dominance frontier of node A .).

In " A Village Sketch ," author Miss Mitford wrote: " Then comes a sun-burnt gipsy of six, beginning to grow tall and thin and to find the cares of the world gathering about her ; with a pitcher in one hand, a mop in the other, an old straw bonnet of ambiguous shape, half hiding her tangled hair ; a tattered stuff petticoat once green, hanging below an equally tattered cotton frock, once purple ; her longing eyes fixed on a game of baseball at the corner of the green till she reaches the cottage door, flings down the mop and pitcher and darts off to her companions quite regardless of the storm of scolding with which the mother follows her runaway steps.

Then it follows that

Then follows a seven-year period of suffering in which the Antichrist will conquer the world and persecute those who refuse to worship him.

Then is a compact topological space ; this follows from the Tychonoff theorem.

Let G be a set and let "~" denote an equivalence relation over G. Then we can form a groupoid representing this equivalence relation as follows.

Then follows a site investigation of soil, rock, fault distribution and bedrock properties on and below an area of interest to determine their engineering properties including how they will interact with, on or in a proposed construction.

Then the inverse of A can be found as follows.

Then follows an account of the storming of the castle.

Then, the kinetic equation for relative populations can be written as follows:

Then follows chills and shivers, and a little fever ...

* Then, to return a new sample given the most recent sample, we proceed as follows:

" Then follows the Collect prayer, and the reading or chanting of,,, and the Passion account from the Gospel of John, traditionally divided between three deacons, yet often divided between the celebrant and more than one singer or reader.

Then follows the sound of screeching tires and an explosion.

Then the area of the parallelogram with vertices at a, b and c is equivalent to the absolute value of the determinant of a matrix built using a, b and c as rows with the last column padded using ones as follows:

Then follows an explanation of chords with table of chords ; observations of the obliquity of the ecliptic ( the apparent path of the Sun through the stars ); and an introduction to spherical trigonometry.

Then the set of rational points on C may be determined as follows:

Then the dual basis vectors are given as follows:

Then seeing that a fraction is less than one, if I multiply by a fraction, it follows that I do take it less than once.

Then is the length of the path from source to v that follows the path from source to u and then goes to v.

Then follows the name of the original title in its original publication.

Then, using the formula, the asymptotic behavior can be determined as follows:

Then the construction or machinery follows.

Then can be broken down to its column vectors as follows:

Then the equations can be solved as follows, using the same process for both: