Then the sheet is automatically graded by a scanning machine.

Then, passing to the associated graded, one gets a canonical morphism T ( L ) → grU ( L ), which kills the elements vw-wv for v, w ∈ L, and hence descends to a canonical morphism S ( L ) → grU ( L ).

Then the tangent cone to X at x is the spectrum of the associated graded ring of O < sub > X, x </ sub > with respect to the m-adic filtration:

Then if M is a graded module over R for which for i sufficiently negative ( in particular, if M is finitely generated and R does not contain elements of negative degree ) such that, then.

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

Then by Arzelà – Ascoli theorem the space K is compact.

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, 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 according to the second isomorphism theorem S ∩ T is normal in T and ST / S ≅ T /( S ∩ T ).

# 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.

Let us suppose that L is a complete lattice and let f be a monotonic function from L into L. Then, any x ′ such that f ′( x ′) ≤ x ′ is an abstraction of the least fixed-point of f, which exists, according to the Knaster – Tarski theorem.

Then Cauchy's theorem can be stated as the integral of a function holomorphic in an open set taken around any cycle in the open set is zero.

Then use the SAS congruence theorem for triangles OPA ' and OPB ' to conclude that angles POA and POB are equal.

Then the group action is by classification of G-orbits ( also known as the orbit-stabilizer theorem ).

Abstractly, we can say that D is a linear transformation from some vector space V to another one, W. We know that D ( c ) = 0 for any constant function c. We can by general theory ( mean value theorem ) identify the subspace C of V, consisting of all constant functions as the whole kernel of D. Then by linear algebra we can establish that D < sup >− 1 </ sup > is a well-defined linear transformation that is bijective on Im D and takes values in V / C.

Then Noether's theorem states that the following quantity is conserved,

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

Then they are independent, but not necessarily identically distributed, and their joint probability distribution is given by the Bapat – Beg theorem.

Then one proves that if the theorem is true for pieces resulting from a cutting of a Haken manifold, that it is true for that Haken manifold.

Then there is a cyclic cubic field inside the cyclotomic field of pth roots of unity, and a normal integral basis of periods for the integers of this field ( an instance of the Hilbert – Speiser theorem ).

Then it is a theorem that

Then, the Riemann-Roch theorem states that

Then the integral in Mercer's theorem reduces to a simple summation

Then the Riemann – Roch theorem states: if g is a genus of X,

Then the theorem states that for analytic functions f, if

Then the Radon – Nikodym theorem provides the function g, equal to the density of μ with respect to Q.

Then v ( p ) is the displacement vector of this projected point relative to p. According to the hairy ball theorem, there is a p such that v ( p )

Then the Pythagorean theorem for two curves intersecting orthogonally at is:

Then Wilson's theorem says that

Then, all but blind, he said there was nothing in Back to Methuselah --, -- `` G.B.S. ought to have known that '', -- and `` I look at my bookshelves despairingly, knowing that I can have nothing more to do with them ''.

Then you can do the finishing touches at your leisure.

Then during washing, the greasy soil rolls back at the edges so that emulsified droplets can disengage themselves from the sorbed oil mass, with the aid of mechanical action, and enter the aqueous phase.

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 the enthusiasm and energy of all elements can be channeled to produce cumulative progress toward a common objective.

Then, after I'm back, another fifty so you can put some mileage on yourself and have a solid alibi somewhere while I take care of your seat cover boy ''.

Then, with the hymn writer of old, you can say:

Then our choice function can choose the least element of every set under our unusual ordering.

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 momentum equation can be expressed as

Then the mass balance equation can be written as

Then the mass balance equation can be expressed as

Then one can rewrite the field in the form

Then nodules of blue earth have to be removed and an opaque crust must be cleaned off, which can be done in revolving barrels containing sand and water.

Then, in 1828, Friedrich Wöhler published a paper on the synthesis of urea, proving that organic compounds can be created artificially.

Then the bra can be computed by normal matrix multiplication.

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 we can solve for f ′.

Then it can be shown that one can write a thermodynamic version of the above calorimetric rules:

Then, according to Adkins ( 1975 ), it can be shown that one can write a further thermodynamic version of the above calorimetric rules:

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 each point p of the line can be specified by its distance from O, taken with a + or − sign depending on which half-line contains p.

Then the Cartesian product set D < sub > 1 </ sub > D < sub > 2 </ sub > can be made into a directed set by defining ( n < sub > 1 </ sub >, n < sub > 2 </ sub >) ≤ ( m < sub > 1 </ sub >, m < sub > 2 </ sub >) if and only if n < sub > 1 </ sub > ≤ m < sub > 1 </ sub > and n < sub > 2 </ sub > ≤ m < sub > 2 </ sub >.