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 therefore any prince or other layman shall arrogate to himself the right of disposition, control, or ownership of ecclesiastical goods or properties, let him be judged guilty of sacrilege.

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 a patient could not be cured physically, the Babylonian physicians often relied on exorcism to cleanse the patient from any curses.

If a patient could not be cured physically, the Babylonian physicians often relied on exorcism to cleanse the patient from any curses.

If exorcism failed, some authorities advocated steps such as confinement, beating, and other types of torture to make the body uninhabitable by witches, demons, and spirits.

If the bluff failed and they ran into trouble, Brannon had told the others, they would withdraw -- and he would come after his son another time.

If this capacity had not failed them, they would see that their enemy has made a disastrous miscalculation.

If he failed to reach the riverbank in five minutes, say, then the skiffs would pull away and leave him groping in the mud.

If, however, the figure to be discerned were complicated, composed of several interlocking subfigures, and so on, even the tracing process failed him, and he could not focus even relatively simple shapes among its parts.

If it failed on occasion to elect its candidates for general state offices by majorities, the failure was due to a lingering remnant of the Know-Nothing party, which called itself the American Republican party.

If, however, a layer of paper remains stuck to the surface, the adhesive has not failed.

If a rebellion failed to quickly seize the capital and control of the military for itself, it was normally doomed to a quick destruction.

If a dive is performed which is as submitted but not as ( incorrectly ) announced, it is declared failed and scores zero according to a strict reading of the FINA law.

If Matthew's prime concern was to preserve the Jewish character of the church, he failed: Christianity became a Gentile religion, and Christianity and Judaism came to view each other as opposites.

If the British failed to meet the deadline, the Congress would call upon all Indians to fight for complete independence.

If he claimed the throne, he would throw the country into yet another series of wars, and if he failed, he would be sacrificing everyone and everything he knew.

If all else failed, a besieger could claim the booty of his conquest undamaged, and retain his men and equipment intact, for the price of a well-placed bribe to a disgruntled gate-keeper.

If the number of dies — the integrated circuits that will eventually become chips — etched on a wafer exceeds a failure threshold ( i. e. too many failed dies on one wafer ), the wafer is scrapped rather than investing in further processing.

If the CSM failed, the spacecraft and Saturn IB for the next Skylab mission would have been launched with two astronauts to retrieve the crew ; given Skylab's ample supplies, its residents would have been able to wait up to several weeks for the rescue mission.

If the crew had failed to repair Skylab in time, the plastic insulation inside the station would have melted, releasing poisonous gas and making Skylab completely uninhabitable.

One of Edison's famous quotations about his attempts to make the light globe suggest that perhaps Tesla was right about Edison's methods of working: " If I find 10, 000 ways something won't work, I haven't failed.

If his plans failed, it was almost always out of luck on behalf of the other characters.

Albert Shaw, editor of The Review of Reviews, stated that after Bryan's nomination, many easterners professed not to have heard of him but: " If, indeed, they had not heard of Mr. Bryan before, they had failed to follow closely the course of American politics in the past eight years.

If those boards failed to agree then there was a central board.

In an interview conducted by Nick Spark, George Nichols, another engineer who was present, stated that Murphy blamed the failure on his assistant after the failed test, saying, " If that guy has any way of making a mistake, he will.

If they failed to live up to their obligations, the Admiralty Courts could and did revoke the Letter of Marque, refuse to award prize money, forfeit bonds, even award tort ( personal injury ) damages against the privateer's officers and crew.

If they failed, the provincial government could carry out the necessary measures and charge the costs to the landowner or municipality.

If the champion failed to start the car, it was not eliminated from the round, but was available again if the champion returned the next day.

If the crop failed, payment was deferred, and no interest could be charged for that year.

If the adopted child failed to carry out the filial duty, the contract was annulled in the law courts.