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, appeals may be costly, and the appellate court must find an error on the part of the court below that justifies upsetting the verdict.

However, it might be produced at significantly lower cost than insulin produced in costly bioreactors.

However, electron microscopes are expensive instruments that are costly to maintain.

However, CCFLs are less energy efficient then LEDs and require a somewhat costly inverter to convert whatever voltage the device uses ( usually 5 or 12v ) to the high voltage needed to light a CCFL.

However, this makes it harder for the community to create mods and it makes the process of porting the game to another platform ( such as GNU / Linux ) more costly.

However, distance and topography make their immediate development costly.

However, Hallet's designs were overly fancy, with too much French influence, and were deemed too costly.

However, the migration path to UMTS is still costly: while much of the core infrastructure is shared with GSM, the cost of obtaining new spectrum licenses and overlaying UMTS at existing towers is high.

However, defects found at this stage are the most costly to fix.

However, Kerckhoffs's principle points out that the things kept secret ought to be those least costly to change if inadvertently disclosed.

However, they tend to be more expensive than conventional heating systems and although more energy efficient, a ground extraction system is more costly.

However, the war was very costly to the Chinese side, as more than just " volunteers " were mobilized, and because of the lack of experience in modern warfare and the lack of modern military technology, China's casualties vastly outnumbered that of the United Nations.

However, when matched against modern weaponry such as automatic firearms, artillery and aircraft, a human wave attack is an extremely dangerous and costly tactic in the face of devastating firepower.

However, the project was abandoned and Periander instead constructed a simpler and less costly overland portage road, named the Diolkos or stone carriageway, along which ships could be towed from one side of the isthmus to the other.

However, AMC faced costly engineering work to bring their Jeeps into compliance with a federal directive for all 4-wheel-drive vehicles to average by 1981.

However, the materials are more costly than types such as the lead acid battery, and the cells have higher self-discharge rates than some other types.

However, fabrication of the OLED substrate is more costly than that of a TFT LCD, until mass production methods lower cost through scalability.

However, they then remember the tale of Sir Swamphole McDuck, who sealed the dungeons ( due to the costly price of running a dungeon ) and find a secret passageway into the dungeons and castle through his fake grave site ( while his skeleton was located inside of his armor ), which had been made in case there was a need of emergency evacuation of the castle.

However, his costly invasions of Burma, Annam, Sakhalin and Champa secured only the vassal status of those countries.

However, colonists began to see indentured servant as too costly, and in 1619, Dutch traders brought the first African slaves to Jamestown.

However, by the end of the 1980s, average disc-pressing prices were over $ 5. 00 per two-sided disc, due to the large amount of plastic material and the costly glass-mastering process needed to make the metal stamper mechanisms.

However, the volumes were slow in appearing and costly.

However, the fuzzy logic paradigm may provide scalability for large control systems where conventional methods become unwieldy or costly to derive.

However, at the end of his higher education he failed to collect his degree certificate, probably because of the expense that this would have entailed ; in Bolivia, obtaining a " certificate for completed studies " was " a very costly and time-consuming affair.