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 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 came an even more shocking confession: according to the CIA document, al-Faruq said two senior al-Qaeda officials, Abu Zubaydah and Ibn al-Shaykh al-Libi, had ordered him to ' plan large-scale attacks against U. S. interests in Indonesia, Malaysia, ( the ) Philippines, Singapore, Thailand, Taiwan, Vietnam and Cambodia.

Then, when the U. S. Army Air Forces on the Marianas Islands ran out of conventional thermite incendiary bombs for its B-29 Superfortresses to drop on Japanese cities, its top commanders, such as General Curtis E. LeMay turned to napalm bombs to continue its fire raids on the large Japanese cities.

Then U acts as a copier provided that

Let U be a functor from D to C, and let X be an object of C. Then the following statements are equivalent:

In 1986, a screening of the entire Monkees television series by MTV led to renewed interest in the group, followed by a single (" That Was Then, This Is Now " reached number 20 on the Billboard Hot 100 in the U. S .), a 20th Anniversary Tour, a greatest hits album and a brand new LP, Pool It!

Then, in 1966, the U. S. Congress created a National Council for Marine Resources and Engineering Development.

Twelve Deadly Cyns ... and Then Some, was released worldwide in 1994 ( except in the U. S., where it was held back until the summer of 1995 ).

Then French Prime Minister Dominique de Villepin sent an emergency aid package worth 36 million Euro ($ 57. 6M U. S. dollars ) and deployed approximately five hundred French troops in an effort to eradicate mosquitoes.

Then in 1885, the U. S. Postal Service rejected the use of two words for a post office name, so Rufus submitted just the name Helix and also became the first postmaster in Spring Valley ; the Helix Post Office was operated out of his home.

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, in December 2001, Krispy Kreme opened its first store outside the U. S. in Mississauga, Ontario, Canada, just outside Toronto.

Then a subfunctor of U < sup > N </ sup > is called an N-ary predicate and a

In 1970, these recordings were compiled onto an LP released via their fan club called From Then to You ( U. S. title: The Beatles ' Christmas Album ).

In 2008, U. K. label Stage Door Records released the retrospective collection Shirley Jones — Then & Now featuring twenty-four songs from Jones's musical career, including songs from the films Oklahoma !, Carousel and April Love.

boundary of U. Then

Then whenever F ( X ) would appear in a statement, you can replace it with a new symbol Y of type U and include another statement P ( X, Y ).

" Then, Fuentes reminded Cárdenas how the U. S. launched a massive manhunt and investigation after the kidnap, torture, and assassination of the DEA agent Enrique Camarena in 1985 in Mexico.

Then it was broadcast, with English-dubbed voices, in the United States and other English-speaking markets, beginning in September, 1966 ( it was first commissioned for U. S. development by NBC Enterprises adapted by Fred Ladd, for syndicated broadcast ).

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 ; ;