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 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 French crown passed from the House of Capet after the death of Charles IV to Philip VI of France of the House of Valois, a cadet branch of the Capetian dynasty,

Thus, Luxembourg returned to the heirs of the old Luxembourg dynasty – at least those of the line of Elisabeth.

Thus, Lenin's practical application of Marxism and working-class urban revolution to the social, political, and economic conditions of the agrarian peasant society that was Tsarist Russia sparked the “ revolutionary nationalism of the poor ” to depose the absolute monarchy of the three-hundred-year Romanov dynasty ( 1613 – 1917 ).

Thus, without holding any land directly under the Holy Roman Emperors, the Liechtenstein dynasty was unable to meet the primary requirement to qualify for a seat in the Imperial diet, the Reichstag, although its head was elevated to princely rank in the late 17th century.

Thus head of the House of Bourbon became the now eldest male heir of the dynasty Juan, Count of Montizón of the Spanish line of the house who was also Carlist claimant to the throne of Spain.

Thus, Rudolph surpassed the two heirs of the Hohenstaufen dynasty that he had earlier served so loyally.

Thus the effective border of dynasty was pushed further south and east.

Thus, for the shorter dynasty which is the subject of this article, " Liu Song " has become the term preferred in most contexts.

Thus, by 1900, the Qing dynasty, which had ruled China for more than two centuries, was crumbling and Chinese culture was under assault by powerful and unfamiliar religions and secular cultures.

Thus Neustria formed the western part of the kingdom of the Franks under the rule of the Merovingian dynasty during the sixth to eighth centuries.

Thus, a dynasty of dual Lancastrian and Yorkist descent, the House of Tudor, was founded.

Thus began the Ptolemaic dynasty, during which began the city's gradual decline.

Thus, Rama Varma Kulasekara, the last emperor of Chera dynasty, is probably the founder of the Venad royal house, and the the title of Chera kings, Kulasekara, was thenceforth kept by the rulers of Venad.

Thus the end of Second Chera dynasty in the 12th century marks the independence of the Venad.

Thus, although modern Chinese read " Confucius " () as Kǒng Fūzǐ, the same title would have been Khuwng Pjutsi in the Tang dynasty and was probably something like an atonal Khongʔ Patsəʔ during his lifetime.

Thus he had access to the early Han dynasty archives, edicts, and records.

Thus, and without any territory held directly under the Imperial throne, the Liechtenstein dynasty was unable to meet a primary requirement to qualify for a seat in the Imperial diet, the Reichstag.

Thus, the two ancient names as Jambhiyagram and Jambubani prove that this district was important as a religious place for Jains and it was also a place of Gupta dynasty in the 19th century.

Thus, the heirs of the five regent houses ( go-seike ) of the Fujiwara dynasty ( Konoe, Takatsukasa, Kujō, Ichijō, and Nijō ) all became princes.

Thus the Carolingian dynasty ended and the Capetian began.

Thus the structure displays a marked variation in architectural styles from Aibak to that of Tughlaq dynasty.

Thus, if Lars Porsena did indeed capture Rome, he may have done so with the intent of controlling it himself, not restoring the former dynasty.

Thus, the history is divided into the Dhanyawadi, Waithali, Laymro and Mrauk U. Mrauk U was conquered by the Konbaung dynasty of Burma in 1784-85, after which Rakhine became part of the Konbaung kingdom of Burma.

Thus, through Burgundy and Anjou, Louis ensured the continuation of the senior Bourbon line on the throne of France and the establishment of the cadet Spanish Bourbon dynasty respectively.