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 ( 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 treaty requires implementing legislation, a state may be in default of its obligations by the failure of its legislature to pass the necessary domestic laws.

If a senator died, resigned, or was expelled, the legislature of the state would appoint a replacement to serve out the remainder of the senator's term.

If the State Legislature was not in session, its Governor could appoint a temporary replacement to serve until the legislature could elect a permanent replacement.

Thomas Jefferson explained in 1785 that there are three main limitations on the power of a court of equity: " If the legislature means to enact an injustice, however palpable, the court of Chancery is not the body with whom a correcting power is lodged.

If a federal or provincial law has been held contrary to the division of power provisions of one of the various constitution acts, the legislature or parliament must either live with the result, amend the law so that it complies, or obtain an amendment to the constitution.

If enough registered voters sign the petition, the Speaker of the legislature announces before the House that the member has been recalled and a by-election follows as soon as possible, giving voters the opportunity to replace the politician in question.

If a legislature lacks a quorum, many jurisdictions allow the members present the power to order a call of the house, which orders the arrest of the members who are not present.

If the legislature seems to have purposefully left out a mental state element ( mens rea ) because they felt mental state need not be proven, it is treated as a strict liability.

If a governor exercises the veto authority after the legislature has adjourned, the veto stands.

If it is deemed necessary, the governor may also convene extraordinary sessions of the state legislature ; and he may convene them anywhere in the state, if Madison, the state capital, is deemed unfit for the purpose due to invasion or contagious disease.

If an elected official commits a crime while in office, the state legislature can hold impeachment trials.

According to Harlan, " If the end which the legislature seeks to accomplish be one to which its power extends, and if the means employed to that end, although not the wisest or best, are yet not plainly and palpably unauthorized by law, then the court cannot interfere.

If the legislature has concluded its yearly session, and the governor does not sign the bill within ten days, it dies.

If the legislature fails to act on the issue within the 90 days, licenses to same-sex couples will resume.

If the legislature is bicameral, and the government is responsible only to the lower house, then " hung parliament " is used only with respect to that chamber.

If the proposition is approved by more than 50 % of all voters who vote, it becomes a part of the state constitution ( if it is a proposed amendment ) or the state's statutes ( if it is a proposed statute ) in the same manner and having the same legal effect as if it had been passed by the state legislature and signed by the governor.

If the legislature had abolished the rule, the Ex Post Facto Clause would not allow that abolishment to apply to Rogers's case.

If there is a dispute in parliament or in a provincial legislature concerning whether or not legislation that has been passed and assented to is constitutional, a third of the members of the body concerned may apply to the Constitutional Court to give a ruling.

If both the Congress and the Maryland state legislature agreed, jurisdiction over the District of Columbia could be returned to Maryland, possibly excluding a small tract of land immediately surrounding the United States Capitol, the White House, and the Supreme Court building.

If both the Congress and the Maryland state legislature agreed, jurisdiction over the District of Columbia could be returned to Maryland, possibly excluding a small tract of land immediately surrounding the United States Capitol, the White House and the Supreme Court building.

If it is thought desirable to make it so, that is a matter for the legislature rather than the courts.

If both the Congress and the Maryland state legislature agreed, jurisdiction over the District of Columbia could be returned to Maryland, excluding a small tract of land immediately surrounding the United States Capitol, the White House and the Supreme Court building which would become known as the " National Capital Service Area ".

If a vacancy occurs while the General Assembly is not in session, the Governor may appoint a successor to serve until thirty days after the commencement of the next session of the legislature.

If the parliamentary group is represented in the legislature, the leader is almost always chosen from among the sitting members ; if the leader does not yet have a seat in the legislature, a sitting member of the group may be expected to resign to make way for him or her.