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

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

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.

Thus, the goal of a conditional proof is to demonstrate that if the CPA were true, then the desired conclusion necessarily follows.

Thus this code number of a proof of contradition of T must be a non-standard number.

If a statement P is provable, then it is certainly impossible to prove that there is no proof of P. But even if it can be shown that no disproof of P is possible, we cannot conclude from this absence that there is a proof of P. Thus P is a stronger statement than not-not-P.

Thus the proof of the existence of a mathematical object is tied to the possibility of its construction.

Thus the only thing we don't have is a formal proof of consistency of whatever version of set theory we may prefer, such as ZF.

Thus the claim is true and we have completed the proof for 2 colours.

Thus, assuming the Axiom of Choice, we can infer that κ is a strong limit cardinal, which completes the proof of its inaccessibility.

Thus, though hard proof is lacking but given the extent of possible habitat and the precautionary principle, it is believed to be extant for the time being.

Thus, the proof shown above is not entirely rigorous when.

Thus the conjecture of Pólya and Hilbert now has a more solid basis, though it has not yet led to a proof of the Riemann hypothesis.

Thus, the perfect graph theorem can be used to prove Dilworth's theorem from the ( much easier ) proof of Mirsky's theorem, or vice versa.

Thus, there is a lower burden of proof, making it easier for the DOJ to pursue infringers.

Thus, the proof is completed.

Thus shall be done for all grievances or thievery, if the accuser cannot provide proof, and the grievance is for less than half of gold grivna, then give him a trial by iron in captivity ; if the grievance is for less than that, but more than two silver grivnas, then trial by water ; if it is even less, then he must make an oath.

Thus, the complementation property of perfect graphs can provide an alternative proof of Dilworth's theorem.

Thus, like the typical small claims court, the plaintiff has the burden of proof and must argue his / her case before the Judge and answer any questions directed to him / her.

Thus, loosely speaking, a proof calculus is a template or design pattern, characterized by a certain style of formal inference, that may be specialized to produce specific formal systems, namely by specifying the actual inference rules for such a system.

Thus, we have the condition that ƒ is a ( Hurwitz ) stable polynomial if and only if p − q = n ( the proof is given below ).

Thus, an act is illegal without extrinsic proof of any surrounding circumstances such as lack of scienter ( knowledge ) or other defenses.

Thus they are easier to prove in some circumstances, but conversely the burden of proof is on the defendant.

Thus the proof might, after all, beg the question against a kind of scepticism radical enough to put in doubt the rule of non-contradiction.

He is callous to supplicants in need .” Steblin ( 2007, p. 169 ) also discovered a document headed " Table of Rules " and dated 5 – 11 July with a list of ethical categories in the hand-writing of Christoph von Stackelberg: " Thus this whole document, dated at the time when ... he ... was deliberating about his future, is surely further proof that Josephine was left alone ... in June and July 1812.

Thus, the scheme serves mainly as a proof of concept, and more efficient provably-secure schemes such as Elgamal have been developed since.