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 a third style, the " curved lip " style was produced, and all official and third-party cartridges during the console's lifespan were released ( or re-released ) using this style.

Thus, the third phase of the war ( roughly August 1920 – July 1921 ) involved the IRA taking on a greatly expanded British force, moving away from attacking well defended barracks and instead using ambush tactics.

Thus, he said, it was necessary to find " a third way ".

Thus, the locomotive is loaded or unloaded in about a third of the time.

* " Then opened he their understanding, that they might understand the scriptures, And said unto them, Thus it is written, and thus it behooved Christ to suffer, and to rise from the dead the third day: And that repentance and remission of sins should be preached in his name among all nations, beginning at Jerusalem.

Thus a rocket would need about 6x more energy ; and allowing for the specific energy of rocket propellant being around one third that of conventional air fuel, roughly 18x more mass of propellant would need to be carried for the same journey.

Thus the area contains about one third of the population of the Kuyavia-Pomerania region ( which has about 2. 1 million inhabitants ).

Thus the length of the edges is one third of that of the original edges.

Thus, if each person votes to get as much cake as possible, choice 4 would be third from the top in everyone's list, and would in any direct choice lose 2 to 1 against an unequal distribution.

Thus the sail second up the mizzen-mast is the " mizzen topsail ", and the third sail up the fore-mast is the " fore topgallant sail ".

Thus, like the Grantville Gazettes the e-publication date antedates the print copy by about two months — the interval before the release of the last third and the hardcover print edition is simultaneously released.

Thus it is only our inability to observe the third group telescopically that makes the difference.

Thus the Elohist describes Balaam constructing giving two blessings, making sacrifices on seven altars, at the high places of Baal, before each, then deciding not to seek enchantments after the third set of sacrifices, but to set his face upon the wilderness, which Balak views as a third blessing, and so Balaam then gives the three final predictions of fate.

Thus, the reunification was not a merger that created a third state out of the two, but an incorporation, by which West Germany absorbed East Germany.

Thus, if a censor uses this method to intercept messages, he or she may then let the letter be sent to the intended recipient who will be unaware that the secret message has already been intercepted by a third party.

Thus, variation 3 is a canon at the unison, variation 6 is a canon at the second ( the second entry begins the interval of a second above the first ), variation 9 is a canon at the third, and so on until variation 27, which is a canon at the ninth.

Thus, 1 indicates the first black square from the left ( a1 in algebraic notation ), 2 indicates the second black square from the left ( c1 ), 3 indicates the third ( e1 ), and 4 indicates the fourth ( g1 ).

Thus " the climax " may refer to the point of climax or to the third phase of the drama.

Thus a sanctuary to Archilochus ( the Archilocheion ) was established on his home island Paros sometime in the third century BC, where his admirers offered him sacrifices, as well as to gods such as Apollo, Dionysus and the Muses.

Thus, the mizzen topgallant staysail can be found dangling from the stay leading from above the mizzen ( third ) mast's topgallant sail to some place ( usually two sails down ) on the second ( main ) mast.

Thus ( Irenaeus, i. 30, p. 109 ) the first archon sent Moses, Joshua, Amos, and Habakkuk ; the second Samuel, Nathan, Jonah, and Micah ; the third Elijah, Joel, and Zechariah ; the fourth Isaiah, Ezekiel, Jeremiah, and Daniel ; the fifth Book of Tobit and Haggai ; the sixth Micah ( qu.

Thus the elements of second row of matrix are identical to those of the first row and hence the third and fourth rows etc.

Thus the elements of second row of matrix are not identical to that of the first row, neither are the third and fourth rows, etc.

Thus spoke Brahma to him “ O Angirasa, you are my third “ manasa putra ” ( the one born out of divine intellect ).