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 the chief magistrate of the republic at Genoa was called.

Thus, most of the electors were Plebeians, and yet any magistrate elected by a joint assembly had jurisdiction over both Plebeians and Patricians.

Thus `` America '', the most widely sung of the patriotic songs, was written by a New England Baptist clergyman, Samuel Francis Smith ( 1808-1895 ), while a student in Andover Theological Seminary.

Thus the only member churches of the present Anglican Communion existing by the mid-18th century were the Church of England, its closely linked sister church, the Church of Ireland ( which also separated from Roman Catholicism under Henry VIII ) and the Scottish Episcopal Church which for parts of the 17th and 18th centuries was partially underground ( it was suspected of Jacobite sympathies ).

Thus, in England and Wales or in Hong Kong, a person born on February 29, 1996, will have legally reached 18 years old on March 1, 2014.

Thus 59 " counties in themselves ", or ' county boroughs ', were created to administer the urban centres of England.

Thus a foal conceived in England but foaled in the United States is regarded as being bred in the US.

Thus he succeeded to the throne of England, guaranteeing his succession in defiance of William and Robert's earlier agreement.

Thus, Peter Wimsey remained forever fixed on the background of inter-war England, and the books are nowadays often read for their evocation of that period as much as for the intrinsic detective mysteries.

Thus if Portugal specialized in wine and England in cloth, both states would end up better off if they traded.

Thus the Investiture Contest strengthened local power in the Holy Roman Empire-in contrast to the trend in France and England, where centralized royal power grew.

Thus by 1057 the Godwin brothers controlled all of England subordinately apart from Mercia.

" Thus, Magna Carta established the rule of law in England by not only requiring the monarchy to obey the law of the land, but also limiting how the monarchy could change the law of the land.

Thus the Act's enfranchising clauses created 65 new county seats and 65 new borough seats in England and Wales with the total number of members representing England falling by seventeen and the number representing Wales rising by four.

Thus an ex facie absolute disposition, qualified by a back-bond expressing the limited nature of the right actually held by the person to whom the disposition is made, would constitute what in England is termed a deed of trust.

Thus by the opening of the thirteenth century, it was already significant in trade with the continent of Europe and ranked as a port of the Hanseatic League It was one of the official " staple towns " of England, authorized to carry on the import and export trade.

Thus Æthelfrith's death in battle has been seen as causing " a near total revolution in the politics of what is now northern England ".

Thus Rupert was the nephew of King Charles I of England, who created him Duke of Cumberland and Earl of Holderness, and the first cousin of King Charles II of England.

Thus, while in England " Maundy Thursday " is the normal term, this term is rarely used in Ireland or Scotland in religious contexts.

Thus Matthew of Paris, in recording the burial of Henry II of England in 1189, mentions that he was buried in his coronation robes with a golden crown on his head and gloves on his hands.

Thus the English Jews, by an act of omission, as it were, became full citizens, subject to no more disabilities than those inherent in their own unwillingness, like Catholics and Nonconformists, to belong to the Church of England or, in their particular case, to swear Christian oaths.

Thus it seems possible that Shakespeare, in conforming to the growing " Tudor Myth " of the day, as well as taking into account new theologies of divine action and human will becoming popular in the wake of the Protestant Reformation, sought to paint Richard as the final curse of God on England in punishment for the deposition of Richard II in 1399.

Thus, when Tyndall returned to live in England in summer 1851, he probably had as good an education in experimental science as anyone in England.

Thus, the town would hold New England style town meetings that covered business for the greater town, followed by a village corporation meeting to deal with urban needs, including police service, fire department service and town engineer service ( public works ).