** 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 S and T are in M then so are S ∪ T and S ∩ T, and also a ( S ∪ T )

* If S and T are in M with S ⊆ T then T − S is in M and a ( T − S ) =

* 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:

* 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.

More formally a k-combination of a set S is a subset of k distinct elements of S. If the set has n elements the number of k-combinations is equal to the binomial coefficient

If f is also surjective and therefore bijective ( since f is already defined to be injective ), then S is called countably infinite.

If the tree contains data, we can simply simultaneously store it in a consecutive array in preorder.

If the cord is twisted up tightly on itself and then the bucket is released, it begins to spin rapidly, not only with respect to the experimenter, but also in relation to the water it contains.

In the words of Henry Chadwick, " If the Consolation contains nothing distinctively Christian, it is also relevant that it contains nothing specifically pagan either ... is a work written by a Platonist who is also a Christian, but is not a Christian work.

If the ideals A and B of R are coprime, then AB = A ∩ B ; furthermore, if C is a third ideal such that A contains BC, then A contains C. The Chinese remainder theorem is an important statement about coprime ideals.

If each pixel on the display actually contains three rectangular subpixels of red, green, and blue, in that fixed order, then things on the screen that are smaller than one full pixel in size can be rendered by lighting only one or two of the subpixels.

If the sheet contains regions where the number of atoms in a ring is different from six, while the total number of atoms remains the same, a topological defect has formed.

If the frame contains areas where nothing has moved, the system simply issues a short command that copies that part of the previous frame, bit-for-bit, into the next one.

If both copies are damaged, the Purkinje layer ( a part of the cerebellum that contains better-connected neurons than any other ) develops abnormally, runting is more common, and pups die within weeks due to inadequate lung development.

If neither A nor B includes the idea of existence, then " some A are B " simply adjoins A to B. Conversely, if A or B do include the idea of existence in the way that " triangle " contains the idea " three angles equal to two right angles ", then " A exists " is automatically true, and we have an ontological proof of A's existence.

If a project plan contains a significant portion of LOE, and the LOE is intermixed with discrete effort, EVM results will be contaminated.

If an explosive molecule contains just enough oxygen to convert all of its carbon to carbon dioxide, all of its hydrogen to water, and all of its metal to metal oxide with no excess, the molecule is said to have a zero oxygen balance.

If a map contains a reducible configuration, then the map can be reduced to a smaller map.

If the packet contains options, this field contains the option type of the next option.

# If V is a set of strings then V * is defined as the smallest superset of V that contains λ ( the empty string ) and is closed under the string concatenation operation.

The inscription on his tomb contains Persian words penned which state: اگرفردوس روی زمین است همین است و همین است و همین است ( If there is a paradise on earth, it is this, it is this, it is this!

If the picture contains an area of the same color, it can be compressed without loss by saying " 200 red dots " instead of " red dot, red dot, ...( 197 more times )..., red dot.

If each group contains less than half the population, then some of the population is exactly equal to the median.

If S contains two elements that are not pairwise orthogonal ( in particular, the set of all quantum states includes such pairs ) then an argument like that given above shows that the answer is no.

If each additional layer contains less oil, the final painting will crack and peel.

# If he contains and sustains all things, would he not be existence itself?

If it is made from durum wheat, the pasta contains protein and gluten.

If nothing else, at least two good songs came out of the project, `` Out Of This World '' and `` June Comes Around Every Year ''.

If it proclaims that the best is yet to be, it always arouses, at least in the young, either a suspicious question or perhaps the exclamation of the Negro youth who saw on a tombstone the inscription, `` I am not dead but sleeping ''.

If Blanche had been honest, she would have yelled, slammed at least a couple of doors, and thrown a few little, valueless things.

If the patient can perceive figure kinesthetically when he cannot perceive it visually, then, it would seem, the sense of touch has immediate contact with the spatial aspects of things in independence of visual representations, at least in regard to two dimensions, and, as we shall see, even this much spatial awareness on the part of unaided touch is denied by the authors.

If, on the other hand, they opted for representation, it had to be representation per se -- representation as image pure and simple, without connotations ( at least, without more than schematic ones ) of the three-dimensional space in which the objects represented originally existed.

If we thus spent our very first day in the midst of a large number of your people honoring a new hero and a great national achievement, our last day, to us at least, was equally impressive and very moving, even though the crowds were absent and there was almost complete silence.

If the master of scops who was most responsible for the poem ever used kennings that were traditional, he was at least partly deprived of free will and not inclined towards shrewd and sophisticated misuse of speech elements.

If we look at the situation this way, we can get an idea of Khrushchev's nightmarish worries -- or, at least, of the worries he might have if his enemies were disposed to exploit their advantage.

If she could not take the children out of this section, at least she could take other children out of their countries and put them on the farms.

If the pencil with the angle u2 is that of the maximum aberration of all the pencils transmitted, then in a plane perpendicular to the axis at O ' 1 there is a circular disk of confusion of radius O ' 1R, and in a parallel plane at O ' 2 another one of radius O ' 2R2 ; between these two is situated the disk of least confusion.

If the Crawford rule is in effect, then another option is the Holland rule, which stipulates that after the Crawford game, a player cannot double until after at least two rolls have been played by each side.

If the cow does react, it would take at least four people to push it over, according to the results.

* If a player's injury causes the play-clock to stop, the player must sit out at least one play ; and

If κ is an infinite cardinal number, then cf ( κ ) is the least cardinal such that there is an unbounded function from it to κ ; and cf ( κ ) = the cardinality of the smallest collection of sets of strictly smaller cardinals such that their sum is κ ; more precisely

If the compromise was a rebuke to the most radical vision of the liberty of the Church, on at least one point its implication was firm and unmistakable: The king, even an emperor, was a layman.

If not " true " crannogs, small occupied islets ( often at least partially artificial in nature ) may be referred to as island duns, although rather confusingly, 22 islet-based sites are classified as ' proper ' crannogs due to the different interpretations of the inspectors or excavators who drew up field reports Canmore search for crannog in the Western Isles Hebridean island dwellings or crannogs were commonly built on both natural and artificial islets, usually reached by means of a stone causeway.

If at least one class of creditors votes against the plan and thus objects, the plan may nonetheless be confirmed if the requirements of cramdown are met.

If we are told that at least one of two statements is true ; and also told that it is not the former that is true ; we can infer that it has to be the latter that is true.

If the test was true, the value one is written to the least significant bit of the destination register to indicate the condition.

If a physical object statement is to be translatable into a sense-data statement, the former must be at least deducible from the latter.

If the four-color conjecture were false, there would be at least one map with the smallest possible number of regions that requires five colors.

If at least one of a set of configurations must occur somewhere in G, that set is called unavoidable.

If the radix point is omitted then it is implicitly assumed to lie at the right ( least significant ) end of the string ( that is, the number is an integer ).

If multiple fermions have the same spatial probability distribution, then at least one property of each fermion, such as its spin, must be different.

If Language A borrowed a word from Language B, or both borrowed the word from a third language or inherited it from a common ancestor, and later the word shifted in meaning or acquired additional meanings in at least one of these languages, a native speaker of one language will face a false friend when learning the other.