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These axiom schemas are chosen to enable one to derive the deduction theorem from them easily.
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These and axiom
These are equivalent in the sense that, in the presence of other basic axioms of set theory, they imply the axiom of choice and are implied by it.
These results might be weaker than, equivalent to, or stronger than the axiom of choice, depending on the strength of the technical foundations.
These axioms are sufficient for many proofs in elementary mathematical analysis, and are consistent with some principles, such as the Lebesgue measurability of all sets of reals, that are disprovable from the full axiom of choice.
These evidently weaker axiom pairs are trivial consequences of A3 and A4.
" These theorems along with their alternative postulates, such as Playfair's axiom, played an important role in the later development of non-Euclidean geometry.
These structures introduced the field of non-Euclidean geometry where Euclid's parallel axiom is denied.
These are essentially the strongest known large cardinal axioms not known to be inconsistent in ZFC ; the axiom for Reinhardt cardinals is stronger, but is not consistent with the axiom of choice.
These conditions are separation axioms that are somewhat stronger than the more familiar Hausdorff axiom T < sub > 2 </ sub >.
These non-polar solvents are themselves lipophilic — the axiom that like dissolves like generally holds true.
These differences could develop because, if one assumes that a topological space satisfies the T < sub > 1 </ sub > axiom, then the various definitions are ( in most cases ) equivalent.
These and schemas
These premises apply to the entire spectrum of guerrilla communication because each tactic intends to disrupt cognitive schemas and thought processing.
These requirements influence the detailed structure of schemas that are produced.
These schemas are a mental structure that represents some aspect of the world.
These objects are modeled using classes defined in database schemas.
These and are
These are just another couple of characters to roll.
These men are not callous.
These things are important to almost all Persians and perhaps most important to the most ordinary.
These are traversed by another line of vaults, and thus rooms, arched on all four sides, are formed.
These songs ( practically all Persian music, for that matter ) are limited to a range of two octaves.
These are not, however, differences in federal structure.
These my grandmother left in their places ( they are still there, more persistent and longer-lived than the generations of man ) and planted others like them, that flourished without careful tending.
These are like the initial ways in which the world forces itself upon the self and thrusts the self into decision and choice.
These are suggestive of Selma Lagerlof.
These assumptions lead to an organization with one man at the top, six directly under him, six under each of these, and so on until there are six levels of personnel.
These ways are absolutely irreconcilable because they offer two different recipes for man's redemption from chaos.
These responses are explicable in terms of characteristics inherent in the crisis.
These discoveries are of present interest chiefly to the scientific community ; ;
These are personality factors ; ;
These are, if the research is done with subtlety and skill, researchable topics, but the research is missing.
These moments are historical events in the lives of individual authors with which the student of comparative literature must be frequently concerned.
These lives are in themselves outside of the moral order and are unburdened with moral responsibility.
These conceptions and the manner in which they were transposed into poetry or engendered by poetic form are intrinsic to western life from the time of Aeschylus to that of Shakespeare.
These biographical analogies are obvious, and far too much time has been spent speculating on their possible implications.
These women whose organization grew out of the old suffrage movement are dedicated to Thomas Jefferson's dictum that one must cherish the people's spirit but `` Keep alive their attention ''.
These are, of course, the same people whose support he has only now rejected to seek the independent vote.
These are New York, Pennsylvania, Michigan, Maryland, Missouri, New Jersey, Illinois and Minnesota.
These men are spenders.
These and chosen
These would be two human beings, whom God had chosen.
These fruit versions sometimes contain colored pearls ( and / or " jelly cubes " as in the related drink taho ), the color chosen to match whatever fruit juice is used.
These numbers also arise in combinatorics, where gives the number of different combinations of b elements that can be chosen from an n-element set.
These are also named by IUPAC, which generally adopts the name chosen by the discoverer.
These differences stem from the threshold chosen for describing an extinction event as " major ", and the data chosen to measure past diversity.
These narrations are randomly chosen and so a different story ( or more details of the same story ) are revealed during multiple playthroughs of the game.
These key stations have been chosen due to high usage, interchange potential, and geographic spread, so that up to 75 % of journeys will be achievable step-free.
These elections are not competitive multiparty elections and most are not multi-candidate elections ; usually a single communist party candidate is chosen to run for office in which voters vote either to accept or reject the candidate.
These can be chosen appropriately in order to obtain a quantitative description of a quantum system.
These words must be factual yet entertaining as the Count states, “ then, it is necessary to arrange what is to be said or written in its logical order, and after that to express it well in words that, if I am not mistaken, should be appropriate, carefully chosen, clear and well formed, but above all that are still in popular use " ( Courtier 77 ).
These team movements are determined by the teams chosen serve receive system, offensive system, coverage system, and defensive system.
( These names were deliberately chosen for contrast, with MACHOs named later than WIMPs.
These types are often chosen as family vessels, especially those in the 26 to 40-feet ( 8 to 12 m ) range.
" These are not necessarily incompatible, however, since while an observer is free to choose the definition of order that they wish to take, once it is chosen that definition applies objectively to any system independently of observation.
These n bits are the remainder of the division step, and will also be the value of the CRC function ( unless the chosen CRC specification calls for some postprocessing ).
These may be chosen to honor a family tradition, to use family heirlooms, to be unique, to be socially responsible ( they are not associated with blood diamonds or the pollution caused by gold mining and cyanide process ), to fit the individual's stylistic preferences, or to manage cost.
These physical objects are often chosen because they are smaller or cheaper than the actual object or system.
These satraps were personally picked by Darius to monitor these provinces, which were divided into sub-provinces with their own governors which were chosen either by the royal court or by the satrap.
These may vary from asphalt mountain passes to rough forest tracks, from ice and snow to desert sand, each chosen to provide an enjoyable challenge for the crew and a test of the car's performance and reliability.
These non-western missionaries often have unparalleled success because they need few western resources and comforts to sustain their livelihood while doing the work they have chosen among a new culture and people.
These colours were chosen for the senior team because they weren't associated with either of the two teams United were merged from.
These films were all chosen primarily due to their use of the 70mm process, which Ebert feels is overlooked.
These responded with the reformation of the Lombard League, which had already defeated his grandfather Frederick Barbarossa in the 12th century, and again Milan was chosen as the league's leader.
These were the staples of the UK take-away business, and indeed of English diets, however, like many national dishes, quality can vary drastically from the commercial or mass produced product to an authentic or homemade variety using more carefully chosen ingredients.
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