Thus, when criminals are cheated or abused ( e. g., a cocaine dealer is swindled by his supplier ) they cannot turn to the police and instead rely on powerful gangsters to protect the gangsters ' income.

Thus, France also has a mixed delivery system with complete patient freedom of supplier choice.

Thus, the frictional loss between user, acquirer, supplier and developer is reduced.

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, in `` The Story Of Ruth '' we have Ruth, Naomi, and Boaz and sets that are meticulously authentic.

Thus the negation of the axiom of choice states that there exists a set of nonempty sets which has no choice function.

Thus all the sets comprise a general directed graph ( ownership defines a direction ), or network construct.

Thus intuitionists absolutely disallow the blanket assertion: " For all propositions P concerning infinite sets D: P or ~ P " ( Kleene 1952: 48 ).

Thus, null sets may be interpreted as negligible sets, defining a notion of almost everywhere.

Thus, 10 % of possible sets differ in one feature, 30 % in two features, 40 % in three features, and 20 % in all four features.

( Thus, the nowhere dense sets need not form a sigma-ideal.

Thus, sets of complementary wavelets are useful in wavelet based compression / decompression algorithms where it is desirable to recover the original information with minimal loss.

Thus, what the axiom is really saying is that two sets are equal if and only if they have precisely the same members.

Thus the Third Age of Middle-earth begins, with Isildur's fateful decision to keep the Ring of Power, which sets in motions the events that will lead to the War of the Ring.

Thus the axioms of ZFC refer only to sets, not to urelements ( elements of sets which are not themselves sets ) or classes ( collections of mathematical objects defined by a property shared by their members ).

Thus an oracle machine with a noncomputable oracle will be able to compute sets that are not computable without an oracle.

Thus the relatively small amount of network color programming, combined with the high cost of color television sets, meant that as late as 1964 only 3. 1 percent of television households in the U. S. had a color set.

Thus, the cardinality of the Cartesian product of no sets is 1.

Thus crystal sets produce rather weak sound and must be listened to with sensitive earphones, and can only pick up stations within a limited range.

Thus, unlike copy editing, proofreading's defining procedure is to work directly with two sets of information at the same time.

Thus a sheaf can often be defined by giving its values on the open sets of a basis, and verifying the sheaf axioms relative to the basis.

Thus, one could instead of sequences also consider sets.

Thus there might be several such sets.

Thus, players are more vulnerable than in other action games, and the hostility of the environment sets up a narrative where the odds are weighed decisively against the avatar.

Thus, a child's Skeletal Age `` dots '' may be classified as `` advanced '' when they appear above the middle curve, `` moderate '' when they appear immediately above or below the middle curve, and `` delayed '' when they appear below the lower curve.

Thus, the white middle and lower classes are arrayed against the Negro middle and lower classes.

Thus while Persian-speakers were concentrated on the lower rungs of the work force, managers tended to be brought in from some distance.

Thus Lewis and Clark had first to travel the lower Missouri to the Mandan country in North Dakota.

Thus lower cost liquid lubricants may be used.

Thus enhanced electrotonic coupling results in lower tonic activity of the coupled cells while increasing rhythmicity.

Thus, for example, MOVEI A, 3 ( C ) adds 3 to the 18 lower bits of register C and puts the result in register A, without touching memory.

Thus, when these stars expand and cool, they do not brighten as much as lower mass stars ; however, they were much brighter than lower mass stars to begin with, and are thus still brighter than the red giants formed from less massive stars.

" Thus, a lesion at the T8 level will produce pain radiating from the spine laterally along the lower costal margins.

Thus in strong alkali at high pH, it forms the dually charged full urate ion, but at biological pH or in the presence of carbonic acid or carbonate ions, it forms the singly charged hydrogen or acid urate ion as its pKa < sub > 1 </ sub > is lower than the pKa < sub > 1 </ sub > of carbonic acid.

Thus, reducing the current by a factor of 2 will lower the energy lost to conductor resistance by a factor of 4.

Thus, if in this example the composer actually wanted the note a semitone lower than G-natural, he might put first a natural sign to cancel the previous G-sharp, then the flat.

Thus, any exciting frequency lower than the cutoff frequency will attenuate, rather than propagate.

Thus it was the preferred choice of that era for a high-speed backbone, but FDDI has since been effectively obsolesced by fast Ethernet which offered the same 100 Mbit / s speeds, but at a much lower cost and, since 1998, by Gigabit Ethernet due to its speed, and even lower cost, and ubiquity.

Thus range encoding can achieve greater compression ratios than the one-bit-per-symbol lower bound on Huffman encoding and it does not suffer the inefficiencies that Huffman does when dealing with probabilities that are not exact powers of two.

Thus the two justices would have adjudicated the case and upheld the lower court opinion striking down the ban on corporate and union spending.

Thus, the Maasai, according to their own oral history, migrated south from the lower Nile valley north of Lake Turkana in about the 15th century, arriving in a long trunk of land stretching from what is now northern Kenya to what is now central Tanzania during the 17th and 18th centuries.

Thus when there is a wide temperature differential between the hot and cold reservoirs, the COP is lower ( worse ).

Thus, Barbadian manufacturers must compete with those from other regional economies, whose wage costs and other overheads are usually much lower.

Thus, it was explained by the early 1950s that Kryptonians were powerless on their own planet, and would gain super powers only within a lower gravity environment.

Thus, according to the standard quantum-theoretic viewpoint, none of this energy can be withdrawn without altering the system to a different form in which the system has a lower zero-point energy.

Thus while defenders argue that the three-prong test embodies the maxim that " opportunity drives interest ", critics argue that the three-prong test goes beyond Title IX original purpose of preventing discrimination, and instead amounts to an exercise in which athletic opportunities are taken away from male students and given to female students, despite the comparatively lower interest levels of those female students.