However, that particular case is a theorem of Zermelo – Fraenkel set theory without the axiom of choice ( ZF ); it is easily proved by mathematical induction.

However, in practice they proved to be too large to manoeuvre well in the close waters of estuaries and rivers, the only places in which a ' naval ' battle could occur.

However, shortly after this positive result, Kurt Gödel published On Formally Undecidable Propositions of Principia Mathematica and Related Systems ( 1931 ), showing that in any sufficiently strong axiomatic system there are true statements which cannot be proved in the system.

However, this design proved to be very difficult to remove from the console and was replaced by a second design.

However, chlorpromazine proved to reduce the effects of psychosis in a more effective and specific manner than lobotomy, even though it was known to be capable of causing severe sedation.

" However, his looks proved to be an asset in his personal and political relationships, as his law partner William Herndon wrote, " He was not a pretty man by any means, nor was he an ugly one ; he was a homely man, careless of his looks, plain-looking and plain-acting.

However, it was ultimately unable to achieve a significant market share and proved commercially unviable for Be Inc.

However, subscription-funded digital terrestrial television ( DTT ) proved less of a competitive threat.

However the rope available at the time proved too susceptible to wear and the system was abandoned in favour of steam locomotives after eight years.

However, the PDP-6 proved to be a " hard sell " with customers, as it offered few advantages over similar machines from the better established vendors like IBM or Honeywell, in spite of its low cost around $ 300, 000.

However, the plantations, both under their previous private ownership and under government administration, proved consistently unprofitable due to the introduction of new oils and lubricants in the international marketplace, and the establishment of vast coconut plantations in the East Indies and the Philippines.

However, Thomas Crow argues that this path " proved to be less a way forward than a cul-de-sac for history painting.

However, when Baden was transferred west to fight the French in 1692 his successors, first Caprara, then from 1696, Frederick Augustus, the Elector of Saxony, proved incapable of delivering the final blow.

However, the facilities proved to be able to barely handle the flood of immigrants that arrived in the years just before World War I.

However, General Niazi proved to be a failure and ineffective ruler.

" However the Times also reflected widespread skepticism as to whether the forward pass could be effectively integrated into the game: " There has been no team that has proved that the forward pass is anything but a doubtful, dangerous play to be used only in the last extremity.

However, according to the late Tomoyuki Tanaka, it proved to be difficult to obtain permission to use King Kong.

However the Eastern Question and the resulting Crimean War proved to be the downfall of his government.

However, the rise of quiz shows proved to be short-lived.

However, a campaign to reconquer Peru and Chile during the Chincha Islands War ( 1864 – 1866 ) proved disastrous and Spain suffered defeat before the determined South American powers.

However, Paul Henreid proved to be the best player.

However, his discoveries were not appreciated by his contemporaries and came into general use only with discoveries of British surgeon Joseph Lister, who in 1865 proved the principles of antisepsis in the treatment of wounds ; However, medical conservatism on new breakthroughs in pre-existing science prevented them from being generally well received during the 19th century.

However, the lamps proved a false dawn because they became unsafe very quickly and provided a weak light.

However, his captaincy proved to be an unhappy one ; he lost form and the team did not do well.

However, relying on funds like this, to support a trade deficit, is unsustainable, and the country may eventually be placed in a position where its currency needs to be devalued.

However, this proved unsustainable and growth scaled back sharply to just 8. 2 % in the Eighties and 8. 9 % in the Nineties.

However, the industry is also known to require generally high working hours of its employees, sometimes at a level seen as unsustainable and destructive.

However, there are different factors that might have caused this declination, including a too-high Total Annual Catch, unsustainable fishing, ecological problems, juvenile catches, or non-registered catches.

However, product heterogeneity, market frictions or high fixed costs ( which make marginal-cost pricing unsustainable in the long run ) can allow for some degree of differential pricing to different consumers, even in fully competitive retail or industrial markets.

However, burning biodiesel and bioethanol ( see biofuel ) when created from virgin material, is increasingly controversial and may or may not be considered sustainable because it inadvertently increases global poverty, the clearing of more land for new agriculture fields ( the source of the biofuel is also the same source of food ), and may use unsustainable growing methods ( such as the use of environmentally harmful pesticides and fertilizers ).

However, one can be said to pursue a strategy of attrition when one makes it the main goal to cause gradual attrition to the opponent eventually amounting to unacceptable or unsustainable levels for the opponent while limiting your own gradual losses to acceptable and sustainable levels.

However, Boris tells Anna that Woburn House is financially unsustainable, as it relies on a finite supply of items taken from Dr. Woburn's collection.

However, Bo's critics called the model of “ red GDP ” – subsidized infrastructure, housing and public works projects – unsustainable and a drain on the city's budget.

However, their lack of consideration and research completed for their influences on the household level can cause the global issue of environmental degradation as farmers may result to unsustainable measures.

However, it has also been criticised as an unsustainable effort, resulting in the growth of capitalistic farming, ignoring institutional reforms and widening income disparities.

However, Yasuni National Park is threatened by oil extraction and the colonization, deforestation, illegal logging, and unsustainable hunting that accompanies oil-access routes.

However, the monetary authorities will continue to be required to pay attention to the consequences of their actions with respect to our international balance of payments position and the outflow of gold, as well as with regard to avoiding the creation of excessive liquidity in the economy, which would delay the effectiveness of monetary policy measures in the next expansion phase of the business cycle.

However, this action did not bolster the economy, and construction of the CPR slowed drastically due to lack of funding.

However, during the 1980s oil glut Alberta's economy suffered.

However, after more than three decades of unbroken growth, the Cypriot economy contracted in 2009.

However, the government's economic policy and democratic security strategy have engendered a growing sense of confidence in the economy, particularly within the business sector.

However the evidence points to a balanced economy for the greater part of Domitian's reign.

However, there was little support toward cleaning up the river, because it would negatively affect Detroit's industrialism and economy.

However, much of the success of Lancaster's " school economy " was that the children were natives of an intensely mercantile culture.

However, they too struggled to keep the economy going, and many merchants were running at a loss.

" However, Whiggery as a political doctrine had little affinity for classical political economy, the tabernacle of the Manchester School and William Gladstone.

However, Erhard suffered a series of decisive defeats in his effort to create a free, competitive economy in 1957 ; he had to compromise on such key issues as the anti-cartel legislation.

However, economic and administrative problems multiplied in Castile, and the weakness of the native economy became evident in the following century.

However, in general the Scottish economy stagnated leading to growing unemployment and political agitation among industrial workers.

However, the devaluation of the Ringgit and the deepening recession caused him to create his own programme, based on protecting Malaysia from foreign investors and reinvigorating the economy through construction projects and the lowering of interest rates.

" However, he also advocated that individuals give to each other their surplus property on the occasion that others have a need for it, without involving trade ( e. g. gift economy ).

However, in the second half of the 1980s, rising stock and real estate prices caused the Japanese economy to overheat in what was later to be known as the Japanese asset price bubble caused by the policy of low interest rate by Bank of Japan.

However since 2000, Kazakhstan's economy grew sharply, aided by increased prices on world markets for Kazakhstan's leading exports — oil, metals and grain.

However, in these countries, collective agreements are not widespread ; only a few businesses and a few sectors of the economy have a strong tradition of finding collective solutions in labour relations.

However, the 1997 Asian Financial Crisis, coupled with the Lao government's own mismanagement of the economy, resulted in spiraling inflation and a steep depreciation of the kip, which lost 87 % of its value from June 1997 to June 1999.

However in recent decades an alternative Marxist-Leninist economy that exists is the Socialist market economy that has been used by the People's Republic of China, Socialist Republic of Vietnam and historically by the People's Republic of Hungary and the Socialist Federal Republic of Yugoslavia.

However by 1924, considerable economic progress had been achieved and by 1926 the economy regained its 1913 production level.

However, Namibia's economy is heavily dependent on the earnings generated from primary commodity exports in a few vital sectors, including minerals, especially diamonds, livestock, and fish.

However, when Puerto Rico changed its mostly agricultural economy to an industrialized one, and the U. S. and Puerto Rican governments started investing heavily in interstate highways and freeways, the railroad business soon collapsed.

However no thought appears to have been given in the recent developments to providing up-market hotel and tourist accommodation which would revitalise the economy of the town.