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 pencils of tangents to Q are transformed into pencils of tangents.

Thus, intuitionists are ready to accept a statement of the form " P or Q " as true only if we can prove P or if we can prove Q:

Thus Q is implied by the premises.

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

Two computers P and Q are called Turing equivalent if P can simulate Q and Q can simulate P. Thus, a Turing-complete system is one that can simulate a Turing machine ; and, per the Church-Turing thesis, that any real-world computer can be simulated by a Turing machine, it is Turing equivalent to a Turing machine.

Thus, a high Q tuned circuit in a radio receiver would be more difficult to tune, but would have greater selectivity, it would do a better job of filtering out signals from other stations that lie nearby on the spectrum.

Thus the queen's file is named " Q " and the king's file is named " K ".

Thus, the two substrates of this enzyme are dihydroquinone ( QH2 ) and ferri-( Fe < sup > 3 +</ sup >) cytochrome c, whereas its 3 products are quinone ( Q ), ferro-( Fe < sup > 2 +</ sup >) cytochrome c, and H < sup >+</ sup >.

Q containing these elements contains all the roots of p. Thus

Thus the intuitive reading of a Hoare triple is: Whenever P holds of the state before the execution of C, then Q will hold afterwards, or C does not terminate.

Thus, a high Q tuned circuit in a radio receiver would be more difficult to tune, but would have more selectivity ; it would do a better job of filtering out signals from other stations that lie nearby on the spectrum.

Thus, an animal can swim ( Q ), or it is not a bear (¬ P ).

Thus g preserves labels if g lies in SL < sub > n </ sub >( Q < sub > p </ sub >).

Thus, " Q: 2. 7073 " next to the Labor Party list indicates that the Labor candidates between them polled 2. 7073 quotas.

Thus, the ring of integers of Q (√ d ) is a principal ideal domain ( and hence a unique factorization domain ) if and only if the class number of Q (√ d ) is equal to 1.

Thus, it is a vector space over Q and so there exists a set I such that

Thus the columns of Q are the eigenvectors.

The sum of A and B is obtained by first finding the intersection point of line AB with MP, which is M. Next A and B add up to the second intersection point of the cone with line EM, which is D. Thus if Q is the second intersection point of the cone with line EN, then

Thus λ is a root of the characteristic polynomial for Q,

Thus, to cite but one example, the Pax Britannica of the nineteenth century, whether with the British navy ruling the seas or with the City of London ruling world finance, was strictly national in motivation, however much other nations ( e.g., the United States ) may have incidentally benefited.

Thus we are compelled to face the urbanization of the South -- an urbanization which, despite its dramatic and overwhelming effects upon the Southern culture, has been utterly ignored by the bulk of Southern writers.

Thus Faulkner reminds us, and wisely, that the `` new '' South has gradually evolved out of the Old South, and consequently its agrarian roots persist.

Thus, the avant-garde choreographers have extended the scope of materials available for dance composition.

Thus, there is freshness not only in the individual movements of the dance but in the shape of their continuity as well.

Thus, paradoxically, the beat writers resort to `` religious '' metaphors: they are in search of mana, the spiritual, the numinous, but not anything connected with formal religion.

Thus jazz is transmuted into something holy, the sacred road to integration of being.

Thus, on the highroad, a troop of soldiers `` marched in their own dust and sang '', while on the footpath one man walks alone.

Thus the image of man has suffered complete fragmentation in personal and spiritual qualities, and complete objectification in sub-human and quasi-mechanistic powers.

Thus the fictional detective is much more than a simple businessman.

Thus the transformation of Adam Smith's ideal entrepreneur into a mythological detective coincides closely with the decline of the real entrepreneur in economic life.

Thus the cocktail party would appear to be the ideal system, but there is one weakness.

Thus, if corporations are not to run away with us, they must become quasi-governmental institutions, subject to public control and needs.

Thus in both types attention is focused on the community itself, and its phenomenological life.

Thus, it is no mystical intuition, but an analyzable conception to say that man and his tradition can `` fall out of existence ''.

Thus human perception and human volition is the immanent cause of all social change and this most truly when the change reaches the civilizational level.

Thus with regard to the loss of tradition, in the change from order to disorder the metaphysics of change works itself out as a disruption of the individual soul, a change in which man continues as an objective ontological existent, but no longer as a man.

Thus, circular motion is itself one of the essential characteristics of completely perfect celestial existence.

Thus earth has fallen to the center of the universe.

Thus, Margenau remarks: `` A large number of unrelated epicycles was needed to explain the observations, but otherwise the ( Ptolemaic ) system served well and with quantitative precision.

Thus, in no ordinary sense of ' simplicity ' is the Ptolemaic theory simpler than the Copernican.

Thus he complains, with considerable justice, that the Tory writers have resorted to libel instead of answering his arguments.

Thus Burns's `` My love is like a red, red rose '' and Hopkins' `` The thunder-purple sea-beach, plumed purple of Thunder '' although clearly intelligible in content, hardly present ideas of the sort with which we are here concerned.

Thus the student of literature may sometimes find it helpful to classify a poem or an essay as being in idea or in ideal content or subject matter typical or atypical of its period.

Thus ideas like `` grace '', `` salvation '', and `` providence '' cluster together in traditional Christianity.