" If we cannot move on to point B until we have proved point A, and if in order to prove point A we must establish it with absolute certainty, then it looks as though we will have a very hard time proving any point at all.

Many results about plane figures are proved, e. g., If a triangle has two equal angles, then the sides subtended by the angles are equal.

If an isomorphism can be found from a relatively unknown part of mathematics into some well studied division of mathematics, where many theorems are already proved, and many methods are already available to find answers, then the function can be used to map whole problems out of unfamiliar territory over to " solid ground " where the problem is easier to understand and work with.

If it could be proved that it is in either NP-Complete or co-NP-Complete, that would imply NP = co-NP.

If a definite statement is believed plausible by some mathematicians but has been neither proved nor disproved, it is called a conjecture, as opposed to an ultimate goal: a theorem that has been proved.

If the intention behind the executions had been to frighten the anti-unionist party into submission, it proved wholly counterproductive.

If Randall ’ s claim that the host may chase off possible dangers to the cleaner will be proved right, an experiment might be constructed in which reciprocity could be demonstrated.

Catch Me If You Can received favourable reviews and proved to be an international success, becoming Dicaprio's highest-grossing film since Titanic with a total of US $ 351. 1 million worldwide.

If R is a ring, let R denote the ring of polynomials in the indeterminate X over R. Hilbert proved that if R is " not too large ", in the sense that if R is Noetherian, the same must be true for R. Formally,

If she did not prove her case, but was proved to be a bad wife, she was drowned.

If his stay in London was disappointing professionally, however, it proved a fertile period for Semper's theoretical, creative and academic development.

If Smackout proved the Jordan-Quinn union's viability, their next creation proved their most enduring.

If it could be proved that a Scottish parliament were desirable ... Scotsmen should support it.

If indeed that was a part of their thinking it proved to be singularly unsuccessful, as the fighting, drinking, and sinning continued on into the 20th century.

A kind of converse of this theorem was proved by Anne C. Davis: If every order preserving function f: L → L on a lattice L has a fixed point, then L is a complete lattice.

If # 1 Record had a certain pop perfection that brought everything together, Radio City was the sound of everything falling apart, which proved at least as compelling.

If charge is proved Bodhi Pala will have to serve rigorous imprisonment for at least 10 years. Although it was certified by the scientists appointed by the government that live branches of the Holy Bodhi Tree were cut without any approval from concerned authorities during the period when Bodhpala was Monk-in-charge.

If the delator lost his case or refused to carry it through, he was liable to the same penalties as the accused ; he was exposed to the risk of vengeance at the hands of the proscribed in the event of their return, or of their relatives ; while emperors like Tiberius would have no scruples about banishing or putting out of the way those whom he had no further use for and who might have proved dangerous to himself.

If Scottish mine owners had hoped to remove a potential labour agitator from their midst by blacklisting Hardie from work in the mines, their action proved to be a major miscalculation.

If the double-decker train had proved success on these Melbourne suburban train lines, another 19 double decker trains were proposed.

If the axiom of choice ( AC ) is used, it can be further proved that the class of cardinal numbers is totally ordered, and thus is the second-smallest infinite cardinal number.

If these laws or any other work of authority can be proved on any point to be at variance with a single passage of the Veda, their authority is at once overruled.

If such an unlikely transaction were to take place, it would more logically be accomplished by a stock purchase, followed by the prosecution of the claim by the wholly-owned subsidiary, followed by liquidation.

If the franchisor requires purchase from his stores, it may come under anti-trust legislation or equivalent laws of other countries.

If you are unsure if GAP coverage had been purchased, you should check your vehicle lease or purchase documentation.

If a PC company attempted to increase prices above the market level all its customers would abandon the company and purchase at the market price from other companies.

If an item is pawned for a loan, within a certain contractual period of time the pawner may purchase it back for the amount of the loan plus some agreed-upon amount for interest.

If the property is someday sold or subdivided, the proceeds must be distributed equally with no credits given for any excess than any one co-owner may have contributed to purchase the property.

This is because each point on the demand curve is the answer to the question " If this buyer is faced with this potential price, how much of the product will it purchase?

If the scammers believe they are being traced, they discard their mobile phones and purchase new ones.

If the vehicle was purchased to be used for interstate trade, no stamp duty was payable on the purchase price of the vehicle.

If a property purchaser borrows more than 80 % of the property's value, the lender will likely require that the borrower purchase private mortgage insurance to cover the lender's risk.

If the division simply " gave " them away at cost as part of a system purchase, they would never have a real budget of their own.

If the goods were stolen and the rightful owner reclaimed them, he had to prove his purchase by producing the seller and the deed of sale, or witnesses to it ; otherwise, he would be adjudged a thief and die.

If you ask people about their purchasing behavior within the coming six or twelve months, there will be more of those who “ hope to be able to buy ”, than if consumers are asked about what they will purchase in the next three months.

Cognitive dissonance is also useful to explain and manage post-purchase concerns. If a consumer feels that an alternate purchase would have been better, it is likely he will not buy the product again.

If a subsystem is only 50 % reliable, one can concentrate on improving the reliability of that subsystem, design in multiple subsystems in parallel ( 5 in this case would achieve approximately 97 % reliability of that subsystem ), purchase and store spare subsystems for rapid change out, establish repair capability that would get a failed subsystem back in operation in the required amount of time, and / or choose any combination of those approaches to achieve the optimal cost vs. reliability solution.

If the user desired a Beta Hi-Fi model but lacked the funds at the time, he could purchase an " SL-HFRxx " and at a later date purchase the separate Hi-Fi Processor.

If the entertainment value ( or other non-monetary value ) obtained by playing is high enough for a given individual, then the purchase of a lottery ticket could represent a gain in overall utility.

of packaging supplements with a textbook to sell as one unit limits the opportunity students have to purchase less expensive used books .... If publishers continue to increase these investments, particularly in technology, the cost to produce a textbook is likely to continue to increase in the future.

If, for instance, the petty bourgeoisie propose the purchase of the railways and factories, the workers must demand that these railways and factories simply be confiscated by the state without compensation as the property of reactionaries.

If, soon after the " purchase ", that which was expensed holds no value then it is usually identified as an expense.

If it retains value soon and long after the purchase, it will be viewed as capital with life that should be amortized / depreciated and retained on the Balance Sheet.

If a survey respondent were to choose a response of “ definitely ” and an intent probability of. 99 was assigned to that category, then the actual probability of purchase could be read off the vertical axis.

If a survey respondent were to choose a response of “ probably not ” and an intent probability of. 25 was assigned to that category, then the actual probability of purchase could be read off the vertical axis as. 35, indicating the specifiers of the function feel that some people that claim they probably will not purchase will actually purchase.