Thus if E is sufficiently small, there can be only one intersection of C and Af near Q, for if there were more than one intersection for every E then the difference between C and Af near Q would not be a monotone function.

Thus every part and plane of the picture keeps changing place in relative depth with every other part and plane ; ;

For every group G there is a natural group homomorphism G → Aut ( G ) whose image is the group Inn ( G ) of inner automorphisms and whose kernel is the center of G. Thus, if G has trivial center it can be embedded into its own automorphism group.

Thus, if every historian were to claim that there was a solar eclipse in the year 1600, then though we might at first naively regard that as in violation of natural laws, we'd come to accept it as a fact.

Thus, if a publicly traded company specialises in the acquisition of privately held companies, from a per-share perspective there is a gain with every acquisition that falls within these guidelines.

Thus, a single " rule ," like mapping every real number x to x < sup > 2 </ sup >, can lead to distinct functions and, depending on whether the images under that rule are understood to be reals or, more restrictively, non-negative reals.

By a theorem of Gelfand and Naimark, given a B * algebra A there exists a Hilbert space H and an isometric *- homomorphism from A into the algebra B ( H ) of all bounded linear operators on H. Thus every B * algebra is isometrically *- isomorphic to a C *- algebra.

Thus in Austinian terms a moral code can objectively determine what people ought to do, the law can embody whatever norms the legislature decrees to achieve social utility, but every individual remains free to choose what to do.

Thus, every chlorine is in the middle of a cube formed by caesium atoms and every caesium is in the middle of a cube formed by chlorine.

Thus, body forces are specified by vector fields which are assumed to be continuous over the entire volume of the body, i. e. acting on every point in it.

Thus, the molecular formula of ethanol may be written in a paper in conformational, three-dimensional, full two-dimensional ( indicating every bond with no three-dimensional directions ), compressed two-dimensional ( CH < sub > 3 </ sub >– CH < sub > 2 </ sub >– OH ), separating the functional group from another part of the molecule ( C < sub > 2 </ sub > H < sub > 5 </ sub > OH ), or by its atomic constituents ( C < sub > 2 </ sub > H < sub > 6 </ sub > O ), according to what is discussed.

Thus, every non-Han dynasty saw itself as the legitimate holder of the " Mandate of Heaven ", which legitimized the dynastic cycle regardless of social or ethnic background as it was moral integrity and benevolent leadership that had been validating the " Mandate of Heaven.

Thus every proposition has three components: the two terms, and the " copula " that connects or separates them.

Thus every module direct summand of R is generated by an idempotent.

Thus we have in the first five centuries such epithets applied to her as " in every respect holy ", " in all things unstained ", " super-innocent " and " singularly holy "; she is compared to Eve before the fall, as ancestress of a redeemed people ; she is " the earth before it was accursed.

Thus it is true in every case above except case 2, because this is the only case when is true but is not.

Thus, every MUMPS command or function name could be abbreviated from one to three letters in length, e. g. Quit ( exit program ) as Q, $ P

Thus a point y in V is a cluster point of the net if and only if every neighborhood Y of y contains infinitely many elements of the sequence.

Thus the number of preorders is the sum of the number of partial orders on every partition.

( Thus we consider something like the winding number of the orientation of the sides, where at every vertex the contribution is between − and winding.

Thus every even perfect number is a pernicious number.

Thus DA means that a cocking action separate from the trigger pull is unnecessary ; and every trigger pull will result in a complete cycle.

Thus, only a small number of computers ( the servers ) need to be rigorously secured, rather than securing every single client computer.

Thus, many users of a given BBS usually lived in the same area, and activities such as BBS Meets or Get Togethers were common, where users of the board would gather at a local restaurant, the SysOp ’ s home or similar venue and meet face to face.

Thus a player could in theory simultaneously have as many as ten knights, ten bishops, ten rooks or nine queens on the board.

Thus, to grow the local monetary base an equivalent amount of foreign currency must be held in reserves with the currency board.

Thus, SA Brewing had a seat on the Cooper's board of Directors from 1962 to 1995, but in 1995 the Cooper family once again became sole owners of the company.

Thus Zarkov's rocket is the only means to escape certain death and they board willingly.

Thus anthropologists Henri Beuchat and Diamond Jenness, both designated for the Southern Party, found themselves sailing with Karluk, while their equipment was on board Alaska.

Thus, in the early days of panel board systems, alarms were regulated by both real estate, and cost.

Thus, while the story Baret concocted for Bougainville's benefit to explain her presence on board ship was carefully designed to shield Commerson from involvement, there is clear documentary evidence of their previous relationship, and it is highly improbable that Commerson was not complicit in the plan himself.

Thus, the President's security detail decided to take him out of the Casa Rosada on board an Air Force helicopter.

Thus, the party attacking a board decision as uninformed must rebut the presumption that its business judgment was an informed one.

Thus the board used in the card game became the tiled map in the video game.

Thus came Tawfeek Al Kheshen who took the presidency of the board and the honorary presidency was given to Mohammad Hassan Helmy.

Thus Riker dreams of fighting the tar creature Armus ( see " Skin of Evil "), the fake Admiral Gregory Quinn ( see " Conspiracy "), or the Klingon officer Klag on board the warship Pagh ( see " A Matter of Honor ").

Thus killing himself and the many Nazi officials on board.

Thus, the club board made the decision to move to Tele2 Arena for the 2013 season.

Thus users often need to bend down and set the wheel, or come down from the board to set the wheel.

Thus it would be much better if the gearing were on the base so that the wheel could be pushed with the foot, but tradition ( consistency from board to board ) dictates maintaining a " backwards " convention.

Thus the rugby union had no representation on the board.

Thus though Sikander was approved by the Bombay censor board, it was later banned from some of the theatres serving British Indian Army cantonments.