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 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 Af is also continuous at Af, and in a neighborhood of Af which does not contain a tangent point.

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

Thus, if public pressure sets the effective limit to the price that the industry may charge, this pressure is itself a function of the wage rate.

Thus, during any given year ( A ) if the revenues from the residential service are $7,000,000, ( B ) if the operating expenses imputed to this class of service come to $5,000,000, and ( C ) if the net investment in ( or value of ) the plant and equipment deemed devoted to this service amounts to $30,000,000, the cost analyst will report that residential service, in the aggregate, has yielded a return of $2,000,000 or 6-2/3 per cent.

Thus if the gyro and platform-controller combination maintains the platform with zero angular deviation about the **yf axis, the system can be rotated with an angular velocity Af if a torque is supplied to the gyro output axis Aj.

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

Thus, if we are to observe American folklore in the twentieth century, we will do well to establish the relationships between folklore, nationalism and imperialism at the outset.

Thus, red wine must, if possible, never be disturbed or shaken ; ;

Thus, if what is at issue is whether `` All S is P '', it is indifferent whether `` Some S is not P '' or `` No S is P '', since in either case the judgment in question is false.

Thus most Affidavits now require to be Apostilled if used for cross border issues.

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 throwing a rock at someone for the purpose of hitting him is a battery if the rock in fact strikes the person and is an assault if the rock misses.

Thus, a court officer taking possession of goods under a court order may use force if reasonably necessary.

Thus, there is no assault if an ox gores a man.

Thus, if P is a program in L < sub > 2 </ sub > which is a minimal description of s, then InterpretLanguage ( P ) returns the string s. The length of this description of s is the sum of

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

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, for non-negative changes in the basic wage rate, the industry becomes the active wage-setter, since any increase in the basic wage rate can occur only by reason of industry acquiescence.

Thus far, the cases which have come before the courts have involved only the issue of referral where the job is vacant due to a strike -- condition ( 1 ) in the Regulation of the Secretary.

Thus, giving grades and promotions based only on an individual's performance relative to a small local group, as is common, may reduce cooperative behaviors in the group.

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 far the only people who have been attested with a high level of genetic, historical, linguistic and cultural research to be the descendants of the ancient Mesopotamians are the Assyrian Christians of Iraq and its surrounding areas in north west Iran, north east Syria and south eastern Turkey.

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 they add two equivalents of bromine whereas an alkene adds only one equivalent.

Thus, negotiation generally features only in games for three or more people.

Thus, if the score was 3 – 2 with the bases loaded in the bottom of the ninth, and the batter smacked an " over the fence home run ", the game would end at 4 – 3, with the batter only allowed a double, and the runners officially stopped on 2nd and 3rd ( since they were not needed to win the game ).

Thus, a PICMG backplane can provide any number and any mix of ISA, PCI, PCI-X, and PCI-e slots, limited only by the ability of the SBC to interface to and drive those slots.

Thus only the target might be affected adversely.

Thus, a player should use Blackwood only when he can ascertain that the partnership holds at least second-round controls in all suits ( kings or, if a suit fit is found, singletons ).

Thus, only the immediate purchaser could recover for a product defect, and if a part was built up out of parts from parts manufacturers, the ultimate buyer could not recover for injury caused by a defect in the part.

Thus the castrati came to supplant both boys ( whose voices broke after only a few years ) and falsettists ( whose voices were weaker and less reliable ) from the top line in such choirs.

Thus, covalent bonding does not necessarily require the two atoms be of the same elements, only that they be of comparable electronegativity.

Thus, only Division I wins ( whether FBS, FCS, or FCS non scholarship ) are considered for playoff selection.

Thus, it would be biochemically possible to sustain life in environments that are only periodically consistent with life as we know it.

Thus the era was indeed a golden age but only for a small but powerful and highly visible elite.

Thus, only the Logos incarnate suffered and died on the Cross and therefore the Son was able to suffer without suffering.

Thus e. g. an O atom with only one Si bond ( a dangling bond ) can be considered a defect in silica.