If F is algebraically closed and p ( x ) is an irreducible polynomial of F, then it has some root a and therefore p ( x ) is a multiple of x − a.

If one quantizes a real scalar field, then one finds that there is only one kind of annihilation operator ; therefore, real scalar fields describe neutral bosons.

If a player bets 1 unit on red, his chance of winning 1 unit is therefore 18 / 38 and his chance of losing 1 unit ( or winning-1 ) is 20 / 38.

If f is also surjective and therefore bijective ( since f is already defined to be injective ), then S is called countably infinite.

If such an eigenvalue is not stable, the dynamics of this eigenvalue will be present in the closed-loop system which therefore will be unstable.

If the level of an attractant remains high, the level of phosphorylation of CheA ( and therefore CheY and CheB ) will remain low, the cell will swim smoothly, and the level of methylation of the MCPs will increase ( because CheB-P is not present to demethylate ).

If a person is free to anger you at will, you have no control over your internal world, and therefore no freedom.

If the passer drops the ball before this moment it is a fumble and therefore a loose ball.

If on the other hand Theorem 2 holds and φ is valid in all structures, then ¬ φ is not satisfiable in any structure and therefore refutable ; then ¬¬ φ is provable and then so is φ, thus Theorem 1 holds.

If the founders were to contact the Secret Chiefs, therefore, it had to be done on their own.

If hammering upwards gravity will reduce the acceleration during the hammer stroke and therefore reduce the energy delivered with each blow.

If the domain is the real numbers, then each element in Y would correspond to two different elements in X (± x ), and therefore ƒ would not be invertible.

The Council of Trent decreed: " If anyone shall say that a man once justified can sin no more, nor lose grace, and that therefore he who falls and sins was never truly justified ; or, on the contrary, that throughout his whole life he can avoid all sins even venial sins, except by a special privilege of God, as the Church holds in regard to the Blessed Virgin: let him be anathema.

If, on the other hand, a currency is under-valued, its exports will become cheaper and therefore more competitive internationally.

If μ is an unknown constant and W is random noise with probability density, then has probability density and is therefore a location family ..

If a commodity's value is not the same as its price, and therefore the magnitudes of each likely differ, then what is the relation between the two, if any?

Cafiero explains in Anarchy and Communism ( 1880 ) that private property in the product of labor will lead to unequal accumulation of capital and, therefore, the reappearance of social classes and their antagonisms, and thus the resurrection of the state: " If we preserve the individual appropriation of the products of labour, we would be forced to preserve money, leaving more or less accumulation of wealth according to more or less merit rather than need of individuals.

# If G is any group acting smoothly on the manifold M, then it acts on the vector fields, and the vector space of vector fields fixed by the group is closed under the Lie bracket and therefore also forms a Lie algebra.

If they believe in determinism, they will therefore believe free will to be an illusion, a position known as Hard Determinism.

If, therefore, the legislature pass any laws, inconsistent with the sense the judges put upon the constitution, they will declare it void.

If ( V, ‖·‖) is a normed vector space, the norm ‖·‖ induces a metric ( a notion of distance ) and therefore a topology on V. This metric is defined in the natural way: the distance between two vectors u and v is given by ‖ u − v ‖.

If a deity's power is to be great, it must therefore be over beings that have at least some of their own defenses and agenda.

Psychological egoism has been accused of being circular: " If a person willingly performs an act, that means he derives personal enjoyment from it ; therefore, people only perform acts that give them personal enjoyment.

If we can prevent something bad without sacrificing anything of comparable significance, we ought to do it ; absolute poverty is bad ; there is some poverty we can prevent without sacrificing anything of comparable moral significance ; therefore we ought to prevent some absolute poverty.

If you've got any ideas ''.

If she sensed any unusual preoccupation on the part of her mother, she did not comment upon it.

If any of us miss, they can pick up the pieces.

If the old fool argues about the price, tell him I shall order my husband not to treat him as a patient any longer.

If a work is divided into several large segments, a last-minute drawing of random numbers may determine the order of the segments for any particular performance.

If any had escaped expulsion by hiding, they certainly would not frequent the market-place.

If in any one calculation Ptolemy had had to invoke 83 epicycles all at once, while Copernicus never required more than one third this number, then ( in the sense obvious to Margenau ) Ptolemaic astronomy would be simpler than Copernican.

If `` Jack the Courtier '' is really to be taken as Swift, the following remark is obviously Steele's comment on Swift's change of parties and its effect on their friendship: `` I assure you, dear Jack, when I first found out such an Allay in you, as makes you of so malleable a Constitution, that you may be worked into any Form an Artificer pleases, I foresaw I should not enjoy your Favour much longer ''.

If she wanted to borrow any sum of money in expecting the arrangements of Congress, it would not become a stranger, unknown to her, to offer himself for that purpose.

If any one of them has any power to veto the Secretary General's decisions the nature of the organization will have changed.

Mr. Richard Preston, executive director of the New Hampshire State Planning and Development Commission, in his remarks to the Governors Conference on Industrial Development at Providence on October 8, 1960, warned against the fallacy of attempting to attract industry solely to reduce the tax rate or to underwrite municipal services such as schools when he said: `` If this is the fundamental reason for a community's interest or if this is the basic approach, success if any will be difficult to obtain ''.

If an atom bomb in 1945 could destroy an entire city surely the atomic arsenal we now have is more than adequate to fulfill any military objective required of it.

If any are left, presently, we may expect to see signs specifically prohibiting the feeding of them too.

The Vice President said, `` If you hear of any names that would fix me cheap in return for advertising they decorated the Vice President's home, let me know.

If he had any worries, it was only the small ones, about Mother in New York, and his daughter Edwina and what she might be doing at this hour, with her Aunt Asia, in Philadelphia.

If he bothers you, don't pay him any mind.

If, in the trustee's judgment, `` reasonable market conditions '' did not prevail during any given year, he was to be allowed to petition the court for an extension of time within the ten-year period.

If any official operation abroad begins to go wrong, we shall look to the ambassador to find out why and to get suggestions for remedial action.

If you use a fiscal year, a year ending on the last day of any month other than December, your return is due on or before the 15th day of the 4th month after the close of your tax year.

If the last day ( due date ) for performing any act for tax purposes, such as filing a return or making a tax payment, etc., falls on Saturday, Sunday, or a legal holiday, you may perform that act on the next succeeding day which is not a Saturday, Sunday, or legal holiday.

If you file Form 1040A and the District Director computes your tax, any refund to which you are entitled will be mailed to you.

If you have neither a legal residence nor a principal place of business in any Internal Revenue district, your return should be filed with the District Director of Internal Revenue, Baltimore 2, Md..

If your principal place of abode for the tax year is outside the United States ( including Alaska and Hawaii ), Puerto Rico, or the Virgin Islands and you have no legal residence or principal place of business in any Internal Revenue district in the United States, you should file your return with the Office of International Operations, Internal Revenue Service, Washington 25, D.C..

If your state has no provisions for the numbering of pleasure boats, you must apply for a number from the U.S. Coast Guard for any kind of boat with mechanical propulsion rated at more than 10 horsepower before it can be used on Federal waterways.