** 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 circumstances are faced frankly it is not reasonable to expect this to be true.

If his dancers are sometimes made to look as if they might be creatures from Mars, this is consistent with his intention of placing them in the orbit of another world, a world in which they are freed of their pedestrian identities.

If a work is divided into several large segments, a last-minute drawing of random numbers may determine the order of the segments for any particular performance.

If they avoid the use of the pungent, outlawed four-letter word it is because it is taboo ; ;

If Wilhelm Reich is the Moses who has led them out of the Egypt of sexual slavery, Dylan Thomas is the poet who offers them the Dionysian dialectic of justification for their indulgence in liquor, marijuana, sex, and jazz.

If he is the child of nothingness, if he is the predestined victim of an age of atomic wars, then he will consult only his own organic needs and go beyond good and evil.

If it is an honest feeling, then why should she not yield to it??

If he thus achieves a lyrical, dreamlike, drugged intensity, he pays the price for his indulgence by producing work -- Allen Ginsberg's `` Howl '' is a striking example of this tendency -- that is disoriented, Dionysian but without depth and without Apollonian control.

If love reflects the nature of man, as Ortega Y Gasset believes, if the person in love betrays decisively what he is by his behavior in love, then the writers of the beat generation are creating a new literary genre.

If he is good, he may not be legal ; ;

If the man on the sidewalk is surprised at this question, it has served as an exclamation.

If the existent form is to be retained new factors that reinforce it must be introduced into the situation.

If we remove ourselves for a moment from our time and our infatuation with mental disease, isn't there something absurd about a hero in a novel who is defeated by his infantile neurosis??

If many of the characters in contemporary novels appear to be the bloodless relations of characters in a case history it is because the novelist is often forgetful today that those things that we call character manifest themselves in surface behavior, that the ego is still the executive agency of personality, and that all we know of personality must be discerned through the ego.

If he is a traditionalist, he is an eclectic traditionalist.

If our sincerity is granted, and it is granted, the discrepancy can only be explained by the fact that we have come to believe hearsay and legend about ourselves in preference to an understanding gained by earnest self-examination.

If to be innocent is to be helpless, then I had been -- as are we all -- helpless at the start.

* If the metric space X is compact and an open cover of X is given, then there exists a number such that every subset of X of diameter < δ is contained in some member of the cover.

If A admits a totally ordered cofinal subset, then we can find a subset B which is well-ordered and cofinal in A.

If the XML document depends on parsable external entities ( including the specified external subset, or parsable external entities declared in the internal subset ), it should assert in its XML declaration.

If the XML document type declaration includes any SYSTEM identifier for the external subset, it can't be safely processed as standalone: the URI should be retrieved, otherwise there may be unknown named character entities whose definition may be needed to correctly parse the effective XML syntax in the internal subset or in the document body ( the XML syntax parsing is normally performed after the substitution of all named entities, excluding the five entities that are predefined in XML and that are implicitly substituted after parsing the XML document into lexical tokens ).

If K is a subset of ker ( f ) then there exists a unique homomorphism h: G / K → H such that f = h φ.

If G = GL < sub >*</ sub >( K ), then the set of natural numbers is a proper subset of G < sub > 0 </ sub >, since for each natural number n, there is a corresponding identity matrix of dimension n. G ( m, n ) is empty unless m = n, in which case it is the set of all nxn invertible matrices.

# If A and B are Lebesgue measurable and A is a subset of B, then λ ( A ) ≤ λ ( B ).

# If A is an open or closed subset of R < sup > n </ sup > ( or even Borel set, see metric space ), then A is Lebesgue measurable.

# If A is a Lebesgue measurable set with λ ( A ) = 0 ( a null set ), then every subset of A is also a null set.

A measure μ is continuous from below: If E < sub > 1 </ sub >, E < sub > 2 </ sub >, E < sub > 3 </ sub >, … are measurable sets and E < sub > n </ sub > is a subset of E < sub > n + 1 </ sub > for all n, then the union of the sets E < sub > n </ sub > is measurable, and

A measure μ is continuous from above: If E < sub > 1 </ sub >, E < sub > 2 </ sub >, E < sub > 3 </ sub >, … are measurable sets and E < sub > n + 1 </ sub > is a subset of E < sub > n </ sub > for all n, then the intersection of the sets E < sub > n </ sub > is measurable ; furthermore, if at least one of the E < sub > n </ sub > has finite measure, then

If μ is not a positive measure, then N is μ-null if N is | μ |- null, where | μ | is the total variation of μ ; equivalently, if every measurable subset A of N satisfies μ ( A )

If ( x < sub > α </ sub >) is a net from a directed set A into X, and if Y is a subset of X, then we say that ( x < sub > α </ sub >) is eventually in Y ( or residually in Y ) if there exists an α in A so that for every β in A with β ≥ α, the point x < sub > β </ sub > lies in Y.

If in addition, the subset F is the union of P and − P, we call P a positive cone of F. The nonzero elements of P are called the positive elements of F.

If S and T are subsets of a group G then their product is the subset of G defined by

If G is a finite group and S and T are subgroups of G, then ST is a subset of G of size | ST | given by the product formula:

If A is a subset of B, but A is not equal to B ( i. e. there exists at least one element of B not contained in A ), then

If, we call X self-similar if it is the only non-empty subset of Y such that the equation above holds for.

# If N is a subset of X containing a neighbourhood of x, then N is a neighbourhood of x.

If M is an open subset of R < sup > n </ sup >, then M is a C < sup >∞</ sup > manifold in a natural manner ( take the charts to be the identity maps ), and the tangent spaces are all naturally identified with R < sup > n </ sup >.