Thus, " where a court of equity has all the parties before it, it will adjudicate upon all of the rights of the parties connected with the subject matter of the action, so as to avoid a multiplicity of suits.

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

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

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

Thus, many a creativity-oriented aspirant for a career in architecture, drama, or journalism, resigns himself to a real estate business ; ;

Thus a new pattern of days began to develop, for Granny Albright did not die.

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

Thus, you avoid headaches when an employee wants off for his fourth cousin's funeral.

Thus it was that Barco, apprehended for mere larceny, now began to suspect that one or another of his murders had been uncovered.

Thus, the dotted line shown in figure 4 was taken as typical for the temperature distribution for all blowing rates.

Thus, for aqueous media, we can think of the idealized organic active as an oleophilic or hydrophobic surface-active agent, and of an idealized builder as a oleophobic or hydrophilic surface-active agent.

Thus, an enemy would probably use this weapon for attack on static population centers such as large cities.

Thus, the alignment of the `` dots '' and `` tips '', respectively, indicate individual variability of the 21 growth centers of each child with respect to the mean values for these boys and girls.

Thus, although the agenda of external assistance in the economic sphere are cumulative, and many of the policies suggested for nations in the earlier stages remain relevant, the basic purpose of American economic policy during the later stages of development should be to assure that movement into a stage of self-sustaining growth is not prevented by lack of foreign exchange.

Thus the lack of effective recognition of the responsibilities involved in caring for two babies showed signs of becoming a disabling problem.

Thus, the writer decided to hold one experimental section of the functional preparation for marriage course in the spring semester of 1960 exclusively for persons already married -- that is, prerequisite: `` marriage ''.

Thus the films seen as they came in ( coordinated for the regular sections ), were often out of context.

Thus, when more than one distinct form leads to a particular cell in the X-region, a chain of information cells must be created to accommodate the forms, one cell in the chain for each form.

Thus in his friends are stranger than his sisters' strong stresses are normal for his and sisters', but in his friends are stranger than his sisters strong stresses are normal for friends and sisters.

Thus in Mary wrote an account of the trip first strong stress on Mary marks Mary as the first in a series of people who wrote accounts of the trip, strong stress on wrote marks the writing as the first of a series of actions of Mary's concerned with an account of her trip ( about which she may later have made speeches, for example ), and strong stress on trip makes the trip the first of a series of subjects about which Mary wrote accounts.

Thus his persistence values for some stem frequencies per meaning are: stem identical in 8 languages, 100% ; ;

Thus while his theory or technique may not be oversubscribed, it is commonplace for bullish and bearish positions to become temporarily over-subscribed.

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 public-limit price is raised further by a given wage increase the longer it has been since the previous price increase.

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, when specifically permitted, the operand of a given line on the Autocoder coding sheet may be continued in the operand of from one to four additional lines which immediately follow.

Thus, the planetary model of the atom was discarded in favor of one that described atomic orbital zones around the nucleus where a given electron is most likely to be observed.

Thus for an ion which traverses a 1 m flight path, across a time of 2000 ns, given an initial accelerating voltage of 5000 V and noting that one amu is 1 × 10 < sup >− 27 </ sup > kg, the mass-to-charge ratio ( more correctly the mass-to-ionisation value ratio ) becomes Z amu / charge.

Thus, many users of a given BBS usually lived in the same area, and activities such as BBS Meets or Get Togethers were common, where users of the board would gather at a local restaurant, the SysOp ’ s home or similar venue and meet face to face.

Thus the discovery report by the Berkeley group reads: " It is suggested that element 97 be given the name berkelium ( symbol Bk ) after the city of Berkeley in a manner similar to that used in naming its chemical homologue terbium ( atomic number 65 ) whose name was derived from the town of Ytterby, Sweden, where the rare earth minerals were first found.

Thus, the vector space B ( X, Y ) can be given the operator norm

Thus, in time, the bishop changed from being the leader of a single church confined to an urban area to being the leader of the churches of a given geographical area.

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, at a given angular rate of rotation, a concave surface represents the stable situation, and the more rapid the rotation, the more concave this surface.

Thus, by monitoring eye movements, we can study what information is being processed at a given time.

Thus, the total applied torque about the origin is given by

Thus, the sum of all applied forces and torques ( with respect to the origin of the coordinate system ) in the body can be given by

Thus a person who owns a quarter of the shares of a joint-stock company owns a quarter of the company, is entitled to a quarter of the profit ( or at least a quarter of the profit given to shareholders as dividends ) and has a quarter of the votes capable of being cast at general meetings.

Thus, the law of " interpenetrating opposites " records the inextricable interdependence of components: the " transformation of quantity to quality " defends a systems-based view of change that translates incremental inputs into alterations of state ; and the " negation of negation " describes the direction given to history because complex systems cannot revert exactly to previous states.

Thus, churches actually had money-changing tables at which each coin would be examined separately and a token of actual worth given to the layperson so that he or she could be seen by the other parishioners as putting money in the basket during that part of the service.

Thus a lion, when presented in a given context in medieval literature, could only be interpreted in a single, approved way.

This identifies the tangent space T < sub > e </ sub > at the identity with the space of left invariant vector fields, and therefore makes the tangent space at the identity into a Lie algebra, called the Lie algebra of G, usually denoted by a Fraktur Thus the Lie bracket on is given explicitly by = < sub > e </ sub >.

Thus, contrary to the first impression its name might convey, and as realized in specific approaches and disciplines ( e. g. Fuzzy Sets and Systems ), intuitionist mathematics is more rigorous than conventionally founded mathematics, where, ironically, the foundational elements which Intuitionism attempts to construct / refute / refound are taken as intuitively given.

Thus, " the Law " is understood to be the religious teachings and rules given by God.

Thus, we should amend the expression given above into: Metallic bonding is an extremely delocalized communal form of electron deficient covalent bonding.