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 is a totally ordered subset of A, hence there exists a maximal totally ordered subset containing, in particular A contains a maximal totally ordered subset.

is partially ordered by set inclusion, therefore it contains a maximal totally ordered subset P. Then the set satisfies the desired properties.

* Then extend the definition of truth to include sentences that predicate truth or falsity of one of the original subset of sentences.

Then every subset of X is either considered " almost everything " ( has measure 1 ) or " almost nothing " ( has measure 0 ).

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 a fuzzy subset s: S of a set S is recursively enumerable if a recursive map h: S × N Ü exists such that, for every x in S, the function h ( x, n ) is increasing with respect to n and s ( x ) = lim h ( x, n ).

Suppose a partially ordered set P has the property that every chain ( i. e. totally ordered subset ) has an upper bound in P. Then the set P contains at least one maximal element.

Then, given any point x and neighbourhood G of x, there is a closed neighbourhood E of x that is a subset of G.

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

Pick a fixed subset L of U. Then the maps F and G, where F ( M ) is the intersection of L and M, and G ( N ) is the union of N and ( U

Then, a tail event is an event whose occurrence or failure is determined by the values of these random variables but which is probabilistically independent of each finite subset of these random variables.

Then the subset of X

A local trivialization of a principal bundle P is given by a section s of P over an open subset U of M. Then the pullback s < sup >*</ sup > ω of a principal connection is a 1-form on U with values in.

Let S a multiplicatively closed subset of R, i. e. for any s and t ∈ S, the product st is also in S. Then the localization of M with respect to S, denoted S < sup >− 1 </ sup > M, is defined to be the following module: as a set, it consists of equivalence classes of pairs ( m, s ), where m ∈ M and s ∈ S. Two such pairs ( m, s ) and ( n, t ) are considered equivalent if there is a third element u of S such that

Furthermore, let S be a subset of P that has a least upper bound s. Then f preserves the supremum of S if the set f ( S ) =

Given a compact subset K of X and an open subset U of Y, let V ( K, U ) denote the set of all functions such that Then the collection of all such V ( K, U ) is a subbase for the compact-open topology on C ( X, Y ).

If E is a Borel subset of R, and 1 < sub > E </ sub > is the indicator function of E, then 1 < sub > E </ sub >( T ) is a self-adjoint projection on H. Then mapping

Let S be an ( a, b )- separator, that is, a vertex subset that separates two nonadjacent vertices a and b. Then S is a minimal ( a, b )- separator if no proper subset of S separates a and b. More generally, S is called a minimal separator if it is a minimal separator for some pair ( a, b ) of nonadjacent vertices.

Then a subset B ⊂ X is bounded if for each open neighborhood U of 0 in X there exists a number m > 0 such that

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 ; ;

Then he called on the governor to explain why.

Then he called in his friend Walton and turned over the problem to him, with instructions to work out what was best -- provided it didn't pile unnecessary burdens on the President.

Then he asked to use the phone and called Brian Thayer, who said that he was just leaving to keep a lunch date but would be home by two o'clock.

Then Via called to say they had decided to cremate her -- as they had Ellen, the thought leaped to my mind -- and did I want to meet her at the funeral home the next morning.

Then the manager called for a doctor.

Then, as their grokking made them ever closer and Mike felt himself almost ready to discorporate her voice called him back: `` Oh!!

Then and throughout the war, Lincoln came under heavy, often vituperative attack from antiwar Democrats, called Copperheads.

:" Then there is also in that place the abode called Breidablik, and there is not in heaven a fairer dwelling.

Then g ( E ) is called the density of states if the energy spectrum is continuous.

Then the reservists have to serve up to three weeks a year and can be called up to serve two weeks during a non-military crisis.

Then he moved to the Humboldt University of Berlin ( then called the Friedrich William University ) in 1878 where he continued his study of mathematics under Leopold Kronecker and the renowned Karl Weierstrass.

Then the preparer grinds the coffee using a traditional tool called a mokecha.

Then, on 1 July 1948, Tirana called on all Yugoslav technical advisors to leave the country and unilaterally declared all treaties and agreements between the two countries null and void.

Then an element e of S is called a left identity if e * a = a for all a in S, and a right identity if a * e = a for all a in S. If e is both a left identity and a right identity, then it is called a two-sided identity, or simply an identity.

Then the pair (( A < sub > i </ sub >)< sub > i ∈ I </ sub >, ( f < sub > ij </ sub >)< sub > i ≤ j ∈ I </ sub >) is called an inverse system of groups and morphisms over I, and the morphisms f < sub > ij </ sub > are called the transition morphisms of the system.

Then Israel called all of his sons in and prophesied their blessings or curses to all twelve of them in order of their ages.

“ And that Christ being Lord, and God the Son of God, and appearing formerly in power as Man, and Angel, and in the glory of fire as at the bush, so also was manifested at the judgment executed on Sodom, has been demonstrated fully by what has been said .” Then I repeated once more all that I had previously quoted from Exodus, about the vision in the bush, and the naming of Joshua ( Jesus ), and continued: “ And do not suppose, sirs, that I am speaking superfluously when I repeat these words frequently: but it is because I know that some wish to anticipate these remarks, and to say that the power sent from the Father of all which appeared to Moses, or to Abraham, or to Jacob, is called an Angel because He came to men ( for by Him the commands of the Father have been proclaimed to men ); is called Glory, because He appears in a vision sometimes that cannot be borne ; is called a Man, and a human being, because He appears arrayed in such forms as the Father pleases ; and they call Him the Word, because He carries tidings from the Father to men: but maintain that this power is indivisible and inseparable from the Father, just as they say that the light of the sun on earth is indivisible and inseparable from the sun in the heavens ; as when it sinks, the light sinks along with it ; so the Father, when He chooses, say they, causes His power to spring forth, and when He chooses, He makes it return to Himself.

The character and themes from the comic strip Little Nemo were used in a song " Scenes from a Night's Dream " written by Phil Collins and Tony Banks of the progressive rock group Genesis on their 1978 recording, ... And Then There Were Three ... Another progressive rock group, from Germany, called Scara Brae also recorded a musical impression of the comic on their rare self titled disc from 1981 ( the track was actually recorded 2 years earlier ).

Then find a root of the plant called bryony.

Then you ’ d better watch what you say .” He noted that Ann Coulter and Bill O ' Reilly called the remarks in question " treasonous ".

Finally, everything comes to what is a plain flat plain at the lowest elevation possible ( called " baseline ") This plain was called by Davis ' " peneplain " meaning " almost plain " Then the rejuvenation occurs and there is another mountain lift and the cycle continues.