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

Thus Af if and only if Af.

Thus F satisfies Af if and only if F has the form Af.

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, 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.

In mathematics, an associative algebra A is an associative ring that has a compatible structure of a vector space over a certain field K or, more generally, of a module over a commutative ring R. Thus A is endowed with binary operations of addition and multiplication satisfying a number of axioms, including associativity of multiplication and distributivity, as well as compatible multiplication by the elements of the field K or the ring R.

) Thus, for example, a 2x6 rectangle and a 3x4 rectangle are equal but not congruent, and the letter R is congruent to its mirror image.

Thus every module direct summand of R is generated by an idempotent.

Thus, Hillel called attention to seven commonly used in the interpretation of laws ( baraita at the beginning of Sifra ); R. Ishmael, thirteen ( baraita at the beginning of Sifra ; this collection is largely an amplification of that of Hillel ).

Thus Euclidean geometry corresponds to the choice of the group E ( 3 ) of distance-preserving transformations of the Euclidean space R < sup > 3 </ sup >, conformal geometry corresponds to enlarging the group to the conformal group, whereas in projective geometry one is interested in the properties invariant under the projective group.

Thus, on 10 February of that year, a Junker 52 of the Réseau des Liaisons Aériennes Francaises ( R. L. A. F ), later known as Air France, landed in Plaisance.

Thus, as from 1945, the R. L. A. F operated the Paris-Mauritius line.

Thus the set of all polynomials with coefficients in the ring R forms itself a ring, the ring of polynomials over R, which is denoted by R. The map from R to R sending r to rX < sup > 0 </ sup > is an injective homomorphism of rings, by which R is viewed as a subring of R. If R is commutative, then R is an algebra over R.

Thus the ( real < ref > The complex spinors are obtained as the representations of the tensor product H ⊗< sub > R </ sub > C = Mat < sub > 2 </ sub >( C ).

CSMA / CA can optionally be supplemented by the exchange of a Request to Send ( RTS ) packet sent by the sender S, and a Clear to Send ( CTS ) packet sent by the intended receiver R. Thus alerting all nodes within range of the sender, receiver or both, to not transmit for the duration of the main transmission.

Thus R ⊆ A < sub > 0 </ sub > and the A < sub > i </ sub > are R modules.

Thus these are precisely the class of domains for which Frac ( R )/ Prin ( R ) forms a group, the ideal class group Cl ( R ) of R. This group is trivial if and only if R is a PID, so can be viewed as quantifying the obstruction to a general Dedekind domain being a PID.

Thus, r < sub > k </ sub > is smaller than its predecessor r < sub > k − 1 </ sub > for all k ≥ 0.

Thus I < sub >(| X | ≥ a )</ sub > = 1 if the event | X | ≥ a occurs, and I < sub >(| X | ≥ a )</ sub > = 0 if | X | < a.

Thus I ( X ; X ) ≥ I ( X ; Y ), and one can formulate the basic principle that a variable contains at least as much information about itself as any other variable can provide.

Thus for n ≥ 3, the special linear group SL < sub > n </ sub >( Z ) has property ( T ).

Thus in determining whether to shut down a firm should compare total revenue to total variable costs ( VC ) rather than total costs ( FC + VC ).

Thus, the VC dimension of this particular classifier is 3.

Thus Vice-county ( or VC ) H3 is " West Cork ".

Thus, in the example cited above Fromm rests his whole case on the premise that the workers are being deprived unconsciously, unknowingly, of fulfillment, and then supports this with survey data reporting conscious, experienced frustrations.

Thus the possibility of re-incorporating Portugal ( up to then Southern Galicia ) into a Kingdom of Portugal and Galicia as before was eliminated and Afonso became sole ruler ( Duke of Portugal ) after demands for independence from the county's church and nobles.

Thus, the Turun Sanomat Building first heralded Aalto's move towards modernism, and this was then carried forward both in the Paimio Sanatorium and in the on-going design for the library.

Thus, if a two-dimensional array has rows and columns indexed from 1 to 10 and 1 to 20, respectively, then replacing B by B + c < sub > 1 </ sub >-− 3 c < sub > 1 </ sub > will cause them to be renumbered from 0 through 9 and 4 through 23, respectively.

“ Acts, then is a continuation of the Lucan Gospel, not in the sense that it relates what Jesus continued to do, but how his followers carried out his commission under the guidance of his Spirit .” Thus, part of the answer to the purpose of Acts is that Luke is writing to Theophilus, who is also mentioned in Luke 1: 3, in order to explain to him the occurrences that take place in the church that fulfill Jesus ’ promise to his disciples that “ you will be baptized with, the Holy Spirit not many days from now ” ( Acts 1: 5 ).

Thus, Abulafia in the 13th century assumed that the language spoken in Paradise had been different from Hebrew, and rejected the claim then current also among Christian authors, that a child left unexposed to linguistic stimulus would automatically begin to speak in Hebrew.

Thus if the instrument depends on the pressure or suction effect alone, and this pressure or suction is measured against the air pressure in an ordinary room, in which the doors and windows are carefully closed and a newspaper is then burnt up the chimney, an effect may be produced equal to a wind of 10 mi / h ( 16 km / h ); and the opening of a window in rough weather, or the opening of a door, may entirely alter the registration.

The city will be taken, the houses looted, and the women raped … Then the Lord will go forth and fight against those nations … Thus the Lord my God will come, and all the saints with you .” 14: 2-5 In Matthew, Jesus says, " The sign of the Son of Man will appear in heaven, and then all the tribes of the earth will mourn, and they will see the Son of Man coming on the clouds of heaven with power and great glory.

Thus, if ( M, d ) and ( N, d ) are two metric spaces, and, then there is a constant k such that

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

Thus, if n < sub > r </ sub > is the average number of times a new civilization reappears on the same planet where a previous civilization once has appeared and ended, then the total number of civilizations on such a planet would be ( 1 + n < sub > r </ sub >), which is the actual reappearance factor added to the equation.

Thus, man then directs his path to that One, and the love for, and of, Christ guides man's very nature to become centered on the One, and on his neighbor as well.

Thus, the only logical possibilities are to accept non-Euclidean geometry as physically real, or to reject the entire notion of physical tests of the axioms of geometry, which can then be imagined as a formal system without any intrinsic real-world meaning.

Thus, when either field is changing in time, then a field of the other is necessarily induced.

Thus, if the Cl < sup >–</ sup > ion is favored for reduction, then the water reaction is favored for oxidation producing gaseous oxygen, however experiments show gaseous chlorine is produced and not oxygen.

Thus if the universe is hot enough ( approximately 10 < sup > 15 </ sup > K, a temperature exceeded until shortly after the Big Bang ) then the electromagnetic force and weak force will merge into a combined electroweak force.

Thus if a represents Socrates then Phil ( a ) asserts the first proposition, p ; if a instead represents Plato then Phil ( a ) asserts the second proposition, q.

Thus the loop f ∗ g first follows the loop f with " twice the speed " and then follows g with twice the speed.

Thus, despite Gemma's intimate connection with a powerful American politician, who was then director of the United Nations Relief and Rehabilitation Administration ( UNRRA ), it took two years to be cleared and sent to the United States.

Thus the first galaxies were then formed.

Thus, if there is no alternate rationale for prosecuting some people more harshly for the same crime based on who the victim is, then different defendants are treated unequally under the law, which violates the United States Constitution.