Then is a group whose identity element is The group inverse of an arbitrary group element is the function inverse

Let be an inverse of Then:

Then a general definition of isomorphism that covers the previous and many other cases is: an isomorphism is a morphism that has an inverse, i. e. there exists a morphism with and.

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.

* Let the index set I of an inverse system ( X < sub > i </ sub >, f < sub > ij </ sub >) have a greatest element m. Then the natural projection π < sub > m </ sub >: X → X < sub > m </ sub > is an isomorphism.

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

Then, the inverse of this matrix is S, the lower triangular matrix of Stirling numbers of second kind.

Then the convolution of f with itself is proportional to the inverse discrete Fourier transform of the pointwise product of Y with itself.

Then, hit the inverse tangent key ( or tan < sup >- 1 </ sup >) and the answer just calculated.

Then each double coset is of the form HxK, and G is partitioned into its ( H, K ) double cosets ; each of them is a union of ordinary right cosets Hy of H in G and left cosets zK of K in G. In another aspect, these are in fact orbits for the group action of H × K on G with H acting by left multiplication and K by inverse right multiplication.

Let f < sup >− 1 </ sup > denote the inverse function of f. Then, for all a ∈ and b ∈,

Then is an inverse semigroup under this multiplication, with

Then is an E-unitary inverse semigroup.

Then for all the mapping is a bijection with inverse.

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 >.

Then maybe next time he won't be so quick on the trigger ''.

Then it would be a choice between starvation and suicide.

Then, with staring eyes and lips drawn thin, Miriam said to the young woman, `` You are ugly -- uglier than you used to be, and you were always very ugly.

My own stern hand has rent the ancient bond, And thereof shall the ending not have end: But not for me, that loved her, to be fond Lightly to please me with a newer friend Then hold it more than bravest-feathered song, That I affirm to thee, with heart of pride, I knew not what did to a friend belong Till I stood up, true friend, by thy true side ; ;

Then a full-time planning office will be established in Rome to work with a five-member Georgia Tech research staff for development of an area planning and industrial development program.

Then off to a remote place in an underdeveloped country where the diet, culture, language and living conditions will be different.

`` Then cometh Jesus from Galilee to Jordan unto John, to be baptized of him.

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 let the whole lot be hanged in a public mass execution on July 4, 1963.

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 Chinese hostler, who rode with Vernon on the box, would break open a hamper and produce filets of smoked bass or sturgeon, sandwiches, pickled eggs, and a rum sangaree to be heated over a spirit lamp.

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 to get up and be General Burnside.

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

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 Miss Lillian Colman came from Vermont to be kitchen manager.

Then the boycott would not be secondary, but a primary one.

Then on Monday morning -- or it might have to be Tuesday -- get up and leave just the usual time, and last thing, put the money in an envelope under the old woman's purse there in the drawer.

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 Heywood Sullivan, Kansas City catcher, singled up the middle and Throneberry was across with what proved to be the winning run.

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