Thus there are polynomials Af such that Af.

Thus the Af are projections which correspond to some direct-sum decomposition of the space V.

Thus Af is divisible by the minimal polynomial P of T, i.e., Af divides Af.

Thus we see that Af and Af.

Thus, the study of the solutions to the equation Af is reduced to the study of the space of solutions of a differential equation of the form Af.

Thus Af if and only if Af.

Thus F satisfies Af if and only if F has the form Af.

Thus in the three-dice example Af, Af, and the independence assumption imply that the probability that the three dice fall ace, not-ace, ace in that order is Af.

Thus in independent voting, Af or 0.52 of the couples would agree.

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

Thus, the efficiency **yt couplers is given by -- Af or approximately 50 percent each.

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

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

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

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

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

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

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 science is the savior of mankind, and in this respect Childhood's End only blueprints in greater detail the vision of the future which, though not always so directly stated, has nevertheless been present in the minds of most science-fiction writers.

Thus the copywriter in the world of the space merchants is the person who in earlier ages might have been a lyric poet, the person `` capable of putting together words that stir and move and sing ''.

Thus there is a clearer division of authority, administrative and legislative.

Thus, as a development program is being launched, commitments and obligations must be entered into in a given year which may exceed by twofold or threefold the expenditures to be made in that year.

Thus, the need for the B-70 as a strategic weapon system is doubtful.

Thus technical efficiency is achieved at the expense of actual experience.

Thus, there is an added incentive to stay on the job.

Thus, the Span of its ossification was shortened and the center's ability to `` catch up '' in ossification is demonstrated.

Thus T is not diagonalizable.

Thus we do not score the number of bull's-eyes, and the random variable is not the number of successes.

Thus in it's incredible what that boy can eat dominant stress is likely to be on incredible, and eat will have strong stress also.

Thus, if one accelerometer is properly aligned, the other is also.

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 ).

“ Acts, then is a continuation of the Lucan Gospel, not in the sense that it relates what Jesus continued to do, but how his followers carried out his commission under the guidance of his Spirit .” Thus, part of the answer to the purpose of Acts is that Luke is writing to Theophilus, who is also mentioned in Luke 1: 3, in order to explain to him the occurrences that take place in the church that fulfill Jesus ’ promise to his disciples that “ you will be baptized with, the Holy Spirit not many days from now ” ( Acts 1: 5 ).

Thus, Abulafia in the 13th century assumed that the language spoken in Paradise had been different from Hebrew, and rejected the claim then current also among Christian authors, that a child left unexposed to linguistic stimulus would automatically begin to speak in Hebrew.

Thus the Bauhaus style, also known as the International Style, was marked by the absence of ornamentation and by harmony between the function of an object or a building and its design.

Thus, any error in technology or judgment may have far-reaching consequences ( though the same can also happen through natural reproduction ).

In his fictional historical essay " The Hyborian Age ", Howard describes how the people of Atlantis — the land where his character King Kull originated — had to move east after a great cataclysm changed the face of the world and sank their island, settling where Ireland and Scotland would eventually be located, Thus they are ( in Howard's work ) the ancestors of the Irish and Scottish ( the Celtic Gaels ) and not the Picts, the other ancestor of modern Scots who also appear in Howard's work.

Thus Post in his 1936 was also discounting Kurt Gödel's suggestion to Church in 1934 – 5 that the thesis might be expressed as an axiom or set of axioms.

Thus, Judaism has also been characterized as a culture or as a civilization.

Thus, at UV frequencies and higher ( and probably somewhat also in the visible range ), electromagnetic radiation does far more damage to biological systems than simple heating predicts.

Thus, Elizabeth was as at ease among the upper classes as she was among the fishing folk of the area also enjoyed good health, which she maintained throughout her life.

Thus, if rainfall amounts and intensities increase in many parts of the world as expected, erosion will also increase, unless amelioration measures are taken.

Thus, the electromagnetic field may be viewed as a dynamic entity that causes other charges and currents to move, and which is also affected by them.

Thus the Eastern First Council of Constantinople became ecumenical only when its decrees were accepted in the West also.

Thus, the search for extrasolar planets also includes the study of planetary habitability, which considers a wide range of factors in determining an extrasolar planet's suitability for hosting life.

Thus, entropy is also a measure of the tendency of a process, such as a chemical reaction, to be entropically favored, or to proceed in a particular direction.

Thus, any other number c that divides both a and b must also divide g. The greatest common divisor g of a and b is the unique ( positive ) common divisor of a and b that is divisible by any other common divisor c.

Thus, the nation had now also taken on the responsibility of the Church, which included paying the clergy and caring for the poor, the sick and the orphaned.

Thus the DTFT of the s sequence is also the Fourier transform of the modulated Dirac comb function .< ref group =" note "> We may also note that:

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

Thus they were involved in the struggle for power within France as princes but they also had a religious authority over Roman Catholicism in France as King.

Thus, powerful but refined breeds such as the Andalusian or the Lusitano developed in the Iberian peninsula as riding horses that also had a great aptitude for dressage, while heavy draft horses such as the Clydesdale and the Shire developed out of a need to perform demanding farm work and pull heavy wagons.