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

Thus twenty `` black souls '' were to remain ignorant of their imminent journey to the land of free men.

Thus such great American documentaries as The River and The Plow That Broke The Plains were composed as visual stories rather than as illustrated lectures.

Thus all differences were leveled, and all contrasts erased, in a realm of no distinction, and the harmonious balance of the Lo Shu square could effectively symbolize the world in balanced harmony around a powerful central axis.

Thus #, $ and % were placed to correspond to 3, 4, and 5 in the adjacent column.

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, before the modern assembly line took shape, there were prototypical forms in various industries, as outlined below.

Thus Myspace and YouTube were not liable for a dyslexic man's inability to navigate the site regardless of how impressive the " online theater " is.

Thus we were spared the affliction of his absence and the triumph of suffering for his principles.

Thus we have the story of his riding across a stick ( horse made of stick ) with his children and upon being discovered by a friend desiring that he not mention till he himself were the father of children ; and because of the affection of his son Archidamus ' for Cleonymus, he saved Sphodrias, Cleonymus ' father, from execution for his incursion into the Piraeus, and dishonorable retreat, in 378 BC.

Thus for example two separate quotes by Athenaeus were united by Theodor Bergk to form fr. 362.

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

Thus, on January 16, 1997, the Diamondbacks were officially voted into the National League while their expansion counterparts in Tampa Bay were voted into the American League.

Thus a third style, the " curved lip " style was produced, and all official and third-party cartridges during the console's lifespan were released ( or re-released ) using this style.

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 farmers were placed in a disadvantageous economic position.

Thus, dark blue jerseys, were introduced with white shorts and the red socks of 1908.

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 in 1880 and 1882, Benedictine teaching monks were effectively exiled ; this was not completed until 1901.

Thus, in 250-1, the Bastarnae were probably involved in the Gothic and Sarmatian invasions which culminated in the Roman defeat at the Battle of Abrittus and the slaying of the emperor Decius ( 251 ).

Thus parliamentary constituencies were derived from the ancient boroughs.

Thus, the Greeks most often associated Apollo's name with the Greek verb ἀπόλλυμι ( apollymi ), " to destroy ".

Thus, boundaries between CAM and mainstream medicine, as well as among different CAM systems, are often blurred and are constantly changing.

Thus, even when family and friends learn to recognize the mood swings, the individual often will deny that anything is wrong.

Thus, it has been known for many years that, due to repulsive Coulombic interactions, electrically charged macromolecules in an aqueous environment can exhibit long-range crystal-like correlations with interparticle separation distances, often being considerably greater than the individual particle diameter.

Thus it is often described as the " bank of banks ".

Thus the limbs of the castrati often grew unusually long, as did the bones of their ribs.

Thus, this leads to the often critiqued narrative of the " white man " saving the " brown woman " from the " brown man.

Thus, though often de facto the case, Conservative Judaism's halakhic system does not inherently see Orthodox halakhic practice as acceptable and legitimate halakhic practice for a Conservative Jew.

Thus, texts like the Annals will often state whether the calendar they use ( the calendar of Lu ) is in phase with the Royal calendar ( used by the Zhou kings ).

Thus, interlaced video captures samples the scene motion twice as often as progressive video does, for the same number of frames per second.

Thus, students are forced to think in the mindset of the larger culture, and individual creativity is often squelched.

Thus, online conversations often involve a variety of assumptions and motives unique to each individual user.

Thus began what is now often called " The Golden Age of Hollywood ", which refers roughly to the period beginning with the introduction of sound until the late 1940s.

Thus, in Chile, Argentina and the Canary Islands, godo was an ethnic slur used against European Spaniards, who in the early colony period often felt superior to the people born locally ( criollos ).

Thus in diplomatic affairs, heads of state are often the first person to greet an important foreign visitor.

Thus, a patient suffering from hyperthyroidism can often obtain immediate temporary relief until the hyperthyroidism can be characterized with the Radioiodine test noted above and more permanent treatment take place.

Thus, Heracles ' very existence proved at least one of Zeus ' many illicit affairs, and Hera often conspired against Zeus ' mortal offspring as revenge for her husband's infidelities.

Thus, because of the O ( n log n ) upper bound on heapsort's running time and constant upper bound on its auxiliary storage, embedded systems with real-time constraints or systems concerned with security often use heapsort.

Thus his teaching methods often seem paradoxical – Johnstone is often famous for asking his students to ‘ be boring ’, ‘ be obvious ’.

Thus Joseph ’ s half brothers were jealous of him and they ridiculed him often.

The book takes the term from Friedrich Nietzsche's Thus Spoke Zarathustra, where it is used in the context of the often nauseating quality of existence.

Thus, Peter Wimsey remained forever fixed on the background of inter-war England, and the books are nowadays often read for their evocation of that period as much as for the intrinsic detective mysteries.

Thus, liquid crystal sheets are often used in industry to look for hot spots, map heat flow, measure stress distribution patterns, and so on.