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 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 the multiplicity of Af for a given T must be an even number.

Thus when Premier Khrushchev intimated even before inauguration that he hoped for an early meeting with the new President, Mr. Kennedy was confronted with a delicate problem.

This latter construal is sometimes expressed by saying " there is no fact of the matter as to whether or not P ." Thus, we may speak of anti-realism with respect to other minds, the past, the future, universals, mathematical entities ( such as natural numbers ), moral categories, the material world, or even thought.

Thus, in Steiner's view, we can overcome the subject-object divide through inner activity, even though all human experience begins by being conditioned by it.

Thus even if the macro "" is not defined to signify non-ANSI compliance, "" will work as shown.

Thus, Cobb's totals may be even greater than is recorded.

Thus, even when family and friends learn to recognize the mood swings, the individual often will deny that anything is wrong.

" Thus, even when reception was effected by a constitution, the common law was still subject to alteration by a legislature's statute.

Thus, the universe would have been even more special before the thermalization than after .” The problem of specific or “ fine-tuned ” initial conditions would not have been solved ; it would have gotten worse.

Thus, for example, the English form is is an apparent cognate of German ist, Latin est and Russian < nowiki > jest '</ nowiki >, even though the Germanic, Italic, and Slavic language groups split at least three thousand years ago.

Thus, all carbon isotopes have nearly identical chemical properties because they all have six protons and six electrons, even though carbon atoms may differ in number of neutrons.

In particular, one should think of the condition number as being ( very roughly ) the rate at which the solution, x, will change with respect to a change in b. Thus, if the condition number is large, even a small error in b may cause a large error in x.

Thus, the action of digitalis ( even on IV injection ) is delayed.

Thus, for a period, economics worked in favour of cable cars even in relatively flat cities.

Thus, even systems which didn't use the EISA bus gained the advantage of having the ISA standardized, which contributed to its longevity.

Thus, civilization was opposed in a dualist manner to barbary, a classic opposition constitutive of the even more commonly-shared ethnocentrism.

Thus, most security researchers set maximum times ( such as 14 days or 30 days ) before fully revealing a vulnerability to the public, since otherwise many vendors would never fix even critical vulnerabilities in their products.

Thus, the overall findings of the study are clear and require that even we skeptics, who have long doubted parental claims of the effects of various foods on the behavior of their children, admit we might have been wrong.

Thus in 941, when the Byzantines were facing the vastly more numerous Rus ' fleet, siphons were placed also amidships and even astern.

Thus, numerical mathematical methods are required, even to solve the system of one nucleus and two electrons.

Thus Microsoft sought to ensure some degree of compatibility between the Chicago design and Windows NT, even though the two systems had radically different internal architectures.

Thus Heracles's name means " the glory of Hera ", even though he was tormented all his life by Hera, the Queen of the Gods.

Thus, even when the Electors of Saxony were Catholics during the eighteenth century, they continued to preside over the corpus evangelicorum, since the state of Saxony was officially Protestant.

Thus for the movement category there are powers that can be used for Running, Swimming, Climbing, Leaping, Gliding, Flying, Tunneling through solid surfaces, and even Teleportation.

Thus, even when tin became available again, iron was cheaper, stronger, and lighter, and forged iron implements superseded cast bronze tools permanently.