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, according to Josephus, in Antiquities, Book 7, Chapter 1, Joab had forgiven Abner for the death of his brother, Asahel, the reason being that Abner had slain Asahel honorably in combat after he had first warned Asahel and had no other choice but to kill him out of self defense.

However, one can choose the index of the first element by an appropriate choice of the base address B.

However, if none of the larger parties can receive enough votes to form their preferred coalition, a grand coalition might be their only choice for forming a government.

However, the choice of a husband might also provoke political instability or even insurrection.

However the situation is different when the language is of arbitrary large cardinality since then, though the completeness and compactness theorems remain provably equivalent to each other in ZF, they are also provably equivalent to a weak form of the axiom of choice known as the ultrafilter lemma.

However, with the proper choice of h, one can use the technique of rolling hash to compute all those hashes with an effort proportional to k + n.

However, environmental and safety issues exist with the choice of vegans.

However, economic dilution must prevail towards accounting dilution when making the choice.

However, like methadone, buprenorphine produces similar cognitive dehabilitation in multiple areas of mental function in both memory and timed choice task tests, which may persist after cessation of substitution treatment.

However, in certain axiom systems for constructive set theory, the axiom of choice does imply the law of the excluded middle ( in the presence of other axioms ), as shown by the Diaconescu-Goodman-Myhill theorem.

However, some species exhibit sex-role reversed behavior in which it is males that are most selective in mate choice ; the best-known examples of this pattern occur in some fishes of the family Syngnathidae, though likely examples have also been found in amphibian and bird species.

However, combining delegation with first-past-the-post may be a better choice because that way Charlie can give his votes to Fran instead of seeing them handed to Bob following the runoff.

However the Holy Roman Emperor Otto I ( 936 – 973 ) did not approve of the choice and had him deposed after only a month with his acquiescence.

However, Principia Mathematica required, in addition to the basic axioms of type theory, three further axioms that seemed to not be true as mere matters of logic, namely the axiom of infinity, the axiom of choice, and the axiom of reducibility.

However, a 2007 meta-analysis concluded that " pair programming is not uniformly beneficial or effective " because many other factors besides the choice of whether to use pair programming have large effects on the outcome of a programming task.

However, the legislation of Pope Nicholas II ( in the famous decree of 1059 ) pre-empted the choice of the majority of the cardinal priests and cardinal deacons.

However, whereas in Quake, the only option was to change the color of the player's uniform unless third party modifications were used, now the game comes with a selection of three different player models: a male marine, a female marine, and a male cyborg ; choice of player model also affects the speech effects the player's character will make, such as exhaling in effort while jumping or groaning when injured.

However, for the rejected pilot " The Cage " ( 1964 ), Roddenberry went with director Robert Butler's choice of John Hoyt to play Dr. Philip Boyce.

However, the unified nature of spacetime and the freedom of coordinate choice it allows imply that to express the temporal coordinate in one coordinate system requires both temporal and spatial coordinates in another coordinate system.

However, if the relation in question is already well-ordered, one can often use transfinite induction without invoking the axiom of choice.

However, the Court struck down the " choice of expenditure " rule, which required that parties could either make coordinated expenditures for all its candidates, or permit candidates to spend independently, but not both, which they agreed " placed an unconstitutional burden on the parties ' right to make unlimited independent expenditures.

However, these may all be different if the axiom of choice fails.

However, the decomposition of a vector into components is not unique, because it depends on the choice of the axes on which the vector is projected.

However, the choice to become vegetarian for non-medical reasons can sometimes be controversial.

However, in 1816, Thomas lost all of his land in court cases because of faulty property titles.

However, as this was essentially just a 65XE computer with a detachable keyboard, it was able to run most of the home computer titles directly.

However, the Game Gear was backed up by significantly more popular titles and consequently the market became dominated by Nintendo followed by Sega in a distant second and the Lynx in third.

However, the 3DS and its predecessor the DSi lack the Game Boy Advance slot found on the DS and DS Lite, causing certain titles to lose functionality when played on the systems.

However, subsequent models removed this and the " Graphics Synthesizer " GPU, thus removing support for PS2 titles, but still able to play most original PS games.

However, following bad sales of Dark Void and Bionic Commando, the company's management has decided to limit outsourcing to sequels and newer versions of installments in existing franchises, reserving the development of original titles for its in-house teams.

However, custom gives the legislation little strength and all graduates keep their titles, and those with a doctorate are referred as Professor Doutor.

However, after 1919 titles of nobility were banned by law in Austria, and the " von Hayek " family became simply the Hayek family.

However, each of these titles was followed by a first-round playoff elimination, in 1998 by the San Diego Padres and in 1997 and 1999 against the Atlanta Braves.

However, they argue for conditional titles to land, whose private ownership is legitimate only so long as it remains in use or occupation ( which Proudhon called " possession.

However, while names, key events and titles may have been borrowed, Brynley Roberts has argued that " the Arthurian section is Geoffrey's literary creation and it owes nothing to prior narrative.

However, these titles had their own distinct editorial approach, and did not directly imitate Mad.

However, the release of the Wii has been accompanied by a number of even more controversial mature titles, such as Manhunt 2, No More Heroes, The House of the Dead: Overkill and MadWorld, the latter three of which are published exclusively for the console.

However, this small period of independence meant that it was fashioned as its own kingdom and the subsequent kings named their titles as king of Galicia and León, instead of merely king of León, even though Galicia was never to be independent again.

However, with the aid of his crusader knights, Fulk excluded Melisende from granting titles, offering patronage, and of issuing grants, deplomas, and charters.

However, since such surname cannot be passed to a descendant with a rank lower than Son of France, and since Philippe had already given up his French titles, his descendants simply took the name of their royal house as their surname (" de Bourbon ", rendered in Spanish as " de Borbón ").

The Duke of Gloucester, The Duke of Kent, Princess Alexandra, The Hon Lady Ogilvy and Prince Michael of Kent enjoy the titular dignity of Prince or Princess and the style Royal Highness as male-line grandchildren of HM King George V. However, none of their children ( being only great-grandchildren of a monarch ) have royal titles.

However, some princely titles develop in unusual ways, such as adoption of a style for dynasts which is not pegged to the ruler's title, but rather continues an old tradition ( e. g. grand duke in Romanov Russia ), claims dynastic succession to a lost monarchy ( e. g. prince de Tarente for the La Trémoïlle heirs to the Neapolitan throne, or is simply assumed by fiat ( e. g. prince Français by the House of Bonaparte ).

However, the American Bonapartes were not considered part of the dynasty and never used any titles.

However, this hard disk support introduced some incompatibility with existing Amiga software because the memory used for hard disk control prevented some memory intensive titles from launching without adding additional RAM.

However, he then went on a 24-match winning streak, including the Stella Artois Championships, Wimbledon ( equaling Roy Emerson's record of 12 Grand Slam singles titles ), Los Angeles, and Cincinnati.

However, Allen tried several titles over five test screenings, including " Anxiety " and " Alvy and Me ", before settling on " Annie Hall ".

However, when he was ten years old, his father unexpectedly succeeded his grandnephew Charles II of Savoy as duke and head of the Savoy dynasty, which had now also received the titles of the kingdoms of Cyprus, Jerusalem and Armenia.

However, as of 2001, 64 titles in English and 24 in French do not carry the Allianz logo ; Prentice Hall and Macmillan have published the English titles.