If V is a standard model of ZFC and κ is an inaccessible in V, then: V < sub > κ </ sub > is one of the intended models of Zermelo – Fraenkel set theory ; and Def ( V < sub > κ </ sub >) is one of the intended models of Von Neumann – Bernays – Gödel set theory ; and V < sub > κ + 1 </ sub > is one of the intended models of Morse – Kelley set theory.

If H < sub > 1 </ sub > is a separable space ( in particular, if it is a Euclidean space ) the result is true in Zermelo – Fraenkel set theory ; for the fully general case, it appears to need some form of the axiom of choice ; the Boolean prime ideal theorem is known to be sufficient.

Statements such as the Banach – Tarski paradox can be rephrased as conditional statements, for example, " If AC holds, the decomposition in the Banach – Tarski paradox exists.

If amber is heated under the right conditions, oil of amber is produced, and in past times this was combined carefully with nitric acid to create " artificial musk " – a resin with a peculiar musky odor.

Theorem: If K < sub > 1 </ sub > and K < sub > 2 </ sub > are the complexity functions relative to description languages L < sub > 1 </ sub > and L < sub > 2 </ sub >, then there is a constant c – which depends only on the languages L < sub > 1 </ sub > and L < sub > 2 </ sub > chosen – such that

If this etymology is combined with the tradition reported by Geoffrey of Monmouth stating that Ambrosius Aurelianus ordered the building of Stonehenge – which is located within the parish of Amesbury ( and where Ambrosius was supposedly buried ) – and with the presence of an Iron Age hill fort also in that parish, then it may be tempting to connect Ambrosius with Amesbury.

If anything, her poems suggest the opposite – they speak of quietly experienced but intensely felt emotions, hidden from others, without any indication of being requited.

If Acorn had thought to include this small modification in the original Electron design it is likely the machine would have had a much greater impact as it would have nearly doubled the amount of motion possible in games and saved modes 0 – 3 ( including the only 16 colour mode ) from being nearly useless due to contended memory timings.

If during the 14 – 15th centuries armour seldom weighed more than 15 kg, then by the late 16th century it weighed 25 kg.

If this article / noun pair is used as the object of a verb, it ( usually ) changes to the accusative case, which entails an article shift in German – Ich sehe den Wagen.

* If the third baseman fields a ball and throws it to first, it is recorded as a 5 – 3 out.

If the score reaches 20-all, then the game continues until one side gains a two point lead ( such as 24 – 22 ), up to a maximum of 30 points ( 30 – 29 is a winning score ).

But this destruction, coupled with a protracted siege of Rain ( 9 – 16 July ), caused Prince Eugene to lament "... since the Donauwörth action I cannot admire their performances ", and later to conclude " If he has to go home without having achieved his objective, he will certainly be ruined.

* The Lusternik – Schnirelmann theorem: If the sphere S < sup > n </ sup > is covered by n + 1 open sets, then one of these sets contains a pair ( x, − x ) of antipodal points.

If a " clinch " – a defensive move in which a boxer wraps his or her opponents arms and holds on to create a pause – is broken by the referee, each fighter must take a full step back before punching again ( alternatively, the referee may direct the fighters to " punch out " of the clinch ).

If one integrates this picture, which corresponds to applying the fundamental theorem of calculus, one obtains Cavalieri's quadrature formula, the integral – see proof of Cavalieri's quadrature formula for details.

If a cabal can be implicated in a sequence of events, there is always the hope, however tenuous, of being able to break the cabal's power – or joining it and exercising some of that power oneself.

If the corporate President is not the COO ( such as Richard Parsons of Time Warner from 1995 – 2001 ), then many division heads report directly to the CEO themselves, with the President taking on special assignments from the CEO.

If their existence past the K – T boundary can be confirmed, these hadrosaurids would be considered a Dead Clade Walking.

: " If anyone kills person – unless it be ( a punishment ) for murder or for spreading mischief in the land — it would be as if he killed all people.

* If the Venerable was not a martyr – all non-martyrs are " confessors " as they " confessed " or bore witness to their faith by how they lived their lives – it must be proven that a miracle has taken place by his or her intercession: that is, that God has shown a sign that the person is enjoying the Beatific Vision by God performing a miracle in response to the Blessed's prayers.

If Mahzeer was planning to set up the prime minister for Muller he would have to do it in the next few minutes.

If the defendant pleads not guilty, a date is set for a preliminary hearing or a trial.

If they don't then for the algorithm to be effective it must provide a set of rules for extracting a square root.

If we cannot make explicit choices, how do we know that our set exists?

If we try to choose an element from each set, then, because X is infinite, our choice procedure will never come to an end, and consequently, we will never be able to produce a choice function for all of X.

** If S is a set of sentences of first-order logic and B is a consistent subset of S, then B is included in a set that is maximal among consistent subsets of S. The special case where S is the set of all first-order sentences in a given signature is weaker, equivalent to the Boolean prime ideal theorem ; see the section " Weaker forms " below.

** If the set A is infinite, then there exists an injection from the natural numbers N to A ( see Dedekind infinite ).

If the automorphisms of an object X form a set ( instead of a proper class ), then they form a group under composition of morphisms.

* If a set S is in M and S is congruent to T then T is also in M and a ( S )

* Let Q be a set enclosed between two step regions S and T. A step region is formed from a finite union of adjacent rectangles resting on a common base, i. e. S ⊆ Q ⊆ T. If there is a unique number c such that a ( S ) ≤ c ≤ a ( T ) for all such step regions S and T, then a ( Q )

* If M is some set and S denotes the set of all functions from M to M, then the operation of functional composition on S is associative:

* If you want to be happy, set a goal that commands your thoughts, liberates your energy, and inspires your hopes.

If a is algebraic over K, then K, the set of all polynomials in a with coefficients in K, is not only a ring but a field: an algebraic extension of K which has finite degree over K. In the special case where K = Q is the field of rational numbers, Q is an example of an algebraic number field.

If X and Y are Banach spaces over the same ground field K, the set of all continuous K-linear maps T: X → Y is denoted by B ( X, Y ).

If the score is tied the set is halved.

If each player wins a set, or both sets end tied, there is a 3-end tiebreaker to determine a winner.

If a node has fewer than two children, some of the child pointers may be set to a special null value, or to a special sentinel node.

Some authors require in addition that μ ( C ) < ∞ for every compact set C. If a Borel measure μ is both inner regular and outer regular, it is called a regular Borel measure.

In the words of Geoffrey Crowther, then editor of The Economist, " If the economic relationships between nations are not, by one means or another, brought fairly close to balance, then there is no set of financial arrangements that can rescue the world from the impoverishing results of chaos.

If, for instance, an addition operation was requested, the arithmetic logic unit ( ALU ) will be connected to a set of inputs and a set of outputs.

If the addition operation produces a result too large for the CPU to handle, an arithmetic overflow flag in a flags register may also be set.