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 Goursat's theorem asserts that ƒ is analytic in an open complex domain Ω if and only if it satisfies the Cauchy – Riemann equation in the domain.

Then the action of H on Ω extends in a natural way to an action of H on the group K by

Then the unrestricted wreath product A Wr < sub > Ω </ sub > H of A by H is the semidirect product K ⋊ H. The subgroup K of A Wr < sub > Ω </ sub > H is called the base of the wreath product.

Then ( Ω, Σ, μ ) is a measure space.

Then the module of Kähler differentials of " S " can be equivalently defined by Ω < sup > 1 </ sup >< sub > S / R </ sub >

Then an upper bound of the numbers that the ground state can take is estimated as Ω < 2 < sup > 2N </ sup >( 6 / 16 )< sup > N </ sup >.

Then every component of the graph of F must be defined over a bounded sub-interval.

Then codenamed " Stella ," the machine was also set to utilize cartridges ; after seeing the Channel F, Atari realized they needed to release it before the market was flooded with cartridge-based machines.

Then Monroe travelled to New York to appear at a birthday celebration for John F. Kennedy at Madison Square Garden, where she famously serenaded the President.

Let the line of symmetry intersect the parabola at point Q, and denote the focus as point F and its distance from point Q as f. Let the perpendicular to the line of symmetry, through the focus, intersect the parabola at a point T. Then ( 1 ) the distance from F to T is 2f, and ( 2 ) a tangent to the parabola at point T intersects the line of symmetry at a 45 ° angle.

Then by definition, torque τ = r × F.

Let X be a normed topological vector space over F, compatible with the absolute value in F. Then in X *, the topological dual space X of continuous F-valued linear functionals on X, all norm-closed balls are compact in the weak -* topology.

Then F is one-sided Lipschitz if

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

In particular, let p define the coordinates of points in a reference frame M coincident with a fixed frame F. Then, when the origin of M is displaced by the translation vector d relative to the origin of F and rotated by the angle φ relative to the x-axis of F, the new coordinates in F of points in M are given by

Then F < sub > 43 </ sub >

Then one finds a free module F < sub > 2 </ sub > and a surjective homomorphism p < sub > 2 </ sub >: F < sub > 2 </ sub > → ker ( p < sub > 1 </ sub >).

To accommodate Cibber's voice range the recitative " Then shall the eyes of the blind " and the aria " He shall feed his flock " were transposed down to F major.

Then the total number of intersection points of X and Y with coordinates in an algebraically closed field E which contains F, counted with their multiplicities, is equal to the product of the degrees of X and Y.

Then by the theorem, the equation also holds for D, E and F ′.

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 on March 26, 1916 after being there six months, Stroud was reprimanded by cafeteria guard, Andrew F. Turner, for a minor rule violation which would have annulled Stroud's visitation privilege to meet his younger brother, whom he had not seen in eight years.

Then the periodic Bernoulli functions P < sub > n </ sub > are defined as

Then, in terms of P < sub > n </ sub >( x ), the remainder

Then, using the periodic Bernoulli function P < sub > n </ sub > defined above and repeating the argument on the interval, one can obtain an expression of ƒ ( 1 ).

Then, Gödel defined essences: if x is an object in some world, then the property P is said to be an essence of x if P ( x ) is true in that world and if P entails all other properties that x has in that world.

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

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 ).

Consider some set P and a binary relation ≤ on P. Then ≤ is a preorder, or quasiorder, if it is reflexive and transitive, i. e., for all a, b and c in P, we have that:

Then, since implies a contradiction, conclude < big > P </ big >.

Then P is a justified true belief.

Then transfinite induction tells us that P is true for all ordinals.

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.

Suppose a non-empty partially ordered set P has the property that every non-empty chain has an upper bound in P. Then the set P contains at least one maximal element.

That puts the sequential problem in P. Then, it will be in NC if and only if it is parallelizable.

Then, 2002 and 2003 also saw the release of the new material from Front 242 in a decade: the E. P.

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 ; ;