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 we could have written a formula of degree k which is equivalent to φ, namely.

Then Φ ( x ) = φ.

Then γ < sub > 1 </ sub > and γ < sub > 2 </ sub > are called equivalent at 0 if the ordinary derivatives of φ ∘ γ < sub > 1 </ sub > and φ ∘ γ < sub > 2 </ sub > at 0 coincide.

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 the subspace of consisting of functions φ such that all functional derivatives of at φ are zero, that is:

Then its negation ¬ φ, together with the field axioms and the infinite sequence of sentences 1 + 1 ≠ 0, 1 + 1 + 1 ≠ 0, …, is not satisfiable ( because there is no field of characteristic 0 in which ¬ φ holds, and the infinite sequence of sentences ensures any model would be a field of characteristic 0 ).

Then any model of B is a field of characteristic greater than k, and ¬ φ together with B is not satisfiable.

Let U be an open set in R < sup > n </ sup > and φ: U → R < sup > n </ sup > an injective differentiable function with continuous partial derivatives, the Jacobian of which is nonzero for every x in U. Then for any real-valued, compactly supported, continuous function f, with support contained in φ ( U ),

Then φ ( U ) is measurable, and for any real-valued function f defined on φ ( U ),

Then there exists a real-valued Borel measurable function w on X such that for every Lebesgue integrable function f: Y → R, the function ( f ° φ ) w is Lebesgue integrable on X, and

Then replace the ( r, φ ) plane with a surface dimpled in the w direction according to the equation ( Flamm's paraboloid )

Then φ < sub > 0 </ sub > and φ < sub > 1 </ sub > have inverses

Then the final formula in (*) must be true, and φ is also true.

The frame bundle F ( E ) can be given a natural topology and bundle structure determined by that of E. Let ( U < sub > i </ sub >, φ < sub > i </ sub >) be a local trivialization of E. Then for each x ∈ U < sub > i </ sub > one has a linear isomorphism φ < sub > i, x </ sub >: E < sub > x </ sub > → R < sup > k </ sup >.

Then, when heated to 980 ° C ( calcined ), the aluminium hydroxide decomposes to aluminium oxide, giving off water vapor in the process:

Then he hunkered down on the heels of his handmade boots, peered into the orderly chaos of axle, shock absorber, and spring.

Then he took off his wet boots and dropped down into the water to talk with the beasts, needing their comfort more than they needed his.

Then he calmly and carefully slugged the remaining five shots into the venomous head -- caught in the wicker back of the chair, the eyes dead on him as the life finally went out of the brute.

Then they were tumbling again, and the big man reached into the same pocket he had gone for earlier, and came up with a vicious switchblade.

Then he launches into an attack on the Tory ministers, whom he calls the `` New Converts '' ; ;

Then, in some way, this lack of faith in the cavalry became mixed up in his mind with the dragging effect of wagon trains and was hardened into a prejudice.

Then they moved offices of the East German puppet government into East Berlin and began illegally to treat it as the capital of East Germany.

Then they disappeared and Scotty got up and went into his own room and got into bed.

Then it disappeared into the wings.

Then the words fell into a pattern: `` Mollie the Mutton is scratching her nose, Scratching her nose in the rain.

Then, getting back into bed, he thought unashamedly of Mrs. Flannagan, planning where they would next meet and what they would do.

Then pour water or light oil from a graduated beaker into the chamber to fill the chamber to its gasket surface.

Then spread a generous amount of glue on the four pieces and bend them into place on the jig.

Then, at last, his day would fall into an ordered pattern and he would be free to read, or garden or just wander through the woods in the late afternoon, accompanied by his dogs.

Then they had always been romping around him on these walks, yelping with delight, dashing off into the bushes on fruitless hunting expeditions, returning to jump up on him triumphantly with muddy paws.

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, with the new affluence, there is actually a sallying forth into the wide, wide world beyond the precincts of New York.

Then he turned the telephone over to Rourke, and went into the bedroom to change his slippers for dry socks and shoes.

Then, six weeks after the day Kitti first came into the office, Stanley announced he and Kitti were married.

Then the man he saved turned and looked squarely into the truck driver's face, without saying a word.

Then Throneberry rapped into a fast double play.

Then it will be a `` fearful thing to fall into the hands of the living God '' if you have abused Him in your hands.

Then there are a pair of old biddies played by Grace Carney and Sibly Bowan who may be right off the shelf of stock Irish characters, but they put such a combination of good will and malevolence into their parts that they're quite entertaining.

Then there was a bad delay in getting Mort Lindsey's 30-piece orchestra wedged into its chairs.

Then every linear operator T in V can be written as the sum of a diagonalizable operator D and a nilpotent operator N which commute.

Then we can express the variables as the sum of the ( time averaged ) mean field () that varies in space and a small fluctuating field () that varies in space and time.

Then the sum total of all the maintenance actions are bundled into maintenance intervals so that maintenance is not occurring around the clock, but rather, at regular intervals.

Then, calculate the flow rate for each and sum the flows.

Since they are even, they can be written as x = 2a and y = 2b respectively for integers a and b. Then the sum.

Then the resultant perturbation can be written as a linear sum of the individual types of perturbations,

Then it will be spontaneously re-emitted, either in the same frequency as the original or in a cascade, where the sum of the energies of the photons emitted will be equal to the energy of the one absorbed ( assuming the system returns to its original state ).

Then the zero vector of this space can be expressed as a linear combination of no elements, which again is an empty sum.

Then any linear functional can be written in these coordinates as a sum of the form:

Since they are even, they can be written as x = 2a and y = 2b respectively for integers a and b. Then the sum.

Then negotiations began over how much money U. S. businesses owed Iran — Iran believing the sum to be $ 20 to $ 60 billion and the United States estimating it at " closer to $ 20 to 60 million ".— and how much Iran owed U. S. businesses.

* Let ρ be a unitary representation of a compact group G on a complex Hilbert space H. Then H splits into an orthogonal direct sum of irreducible finite-dimensional unitary representations of G.

Then the sum of the signed distances from the circumcenter D to the sides of triangle ABC is

If we take E to be the sum of the even exterior powers of the cotangent bundle, and F to be the sum of the odd powers, define D = d + d *, considered as a map from E to F. Then the topological index of D is the Euler characteristic of the Hodge cohomology of M, and the analytical index is the Euler class of the manifold.

Then came chivalry to Rome, and the sum of learning, which now is come to France.

Then, luminance is calculated as a weighted sum of the three linear-intensity values.

Then continue the sequence, where each subsequent term is the sum of the previous n terms.

Then χ ( M, N ), defined as the alternating sum of the lengths of the modules Tor < sub > i </ sub >< sup > R </ sup >( M, N ) is 0 if dim M + dim N < d, and positive if the sum is equal to d. ( N. B.

Then the statement is that the sum of delta-functions at each point of Λ, and at each point of Λ ′, are again Fourier transforms as distributions, subject to correct normalization.

# Then decide the input connections of OR matrix to generate the sum terms.

Then the Gauss sum mod p, g ( a ; p ), is the following sum of the pth roots of unity:

Then A ( G ) splits as a sum over characters λ of T of the subrepresentations A ( G )< sup > λ </ sup > of elements transforming according to λ.