** 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 Analytical Engine had been built, it would have been digital, programmable and Turing-complete.

If the author wished, he could apply for a second 14 ‑ year monopoly grant, but after that the work entered the public domain, so it could be used and built upon by others.

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 50 cars had been built, sportscars like the GT40 and the Lola T70 were allowed, with a maximum of 5. 0 L. John Wyer's revised 4. 7 litre ( Bored to 4. 9 litres, and o-rings cut and installed between the deck and head to prevent head gasket failure, a common problem found with the 4. 7 engine.

If they further reflect, they must also recognize that an act of mutual love which impairs the capacity to transmit life which God the Creator, through specific laws, has built into it, frustrates His design which constitutes the norm of marriage, and contradicts the will of the Author of life.

If it is not pleasant it could be relevant to economics because, for example, the house could be built closer to the stream, plumbing could be installed, a person could be employed to fetch water, or investment in a better path to the water might be worth considering.

If he succeeded and built the Holy Temple in its proper place and gathered the dispersed ones of Israel together, this is indeed the anointed one for certain, and he will mend the entire world to worship the Lord together, as it is stated: " For then I shall turn for the nations a clear tongue, so that they will all procalim the Name of the Lord, and to worship Him with a united resolve ( Zephaniah 3: 9 ).

If enough spaceships can be built in time, the human race might be able to make it to Mars and eke out a living there.

If the user opens the protective sliding door that covers the pins, the processor can be exposed to any electric charge the user has built up.

If the user has built up a substantial charge, the discharge can jump around the door's edge or pass through the door itself ( dielectric breakdown ).

If a Nicaraguan canal were built, " it would bring an economic effervescence never seen before in Central America ," Bolaños said.

If the strip was in the same place each year, the soil built up into a ridge, creating the ridge and furrow topography still seen in some ancient fields.

If there is sufficient commercial interest, a hardware implementation of the CPU specification may be built ( e. g., the Pascal MicroEngine or a version of the Java processor ).

If the software is a first release that is later built upon, requirements documentation is very helpful when managing the change of the software and verifying that nothing has been broken in the software when it is modified.

In after-dinner conversation at his home near Versailles, Laboulaye, an ardent supporter of the Union in the American Civil War, is supposed to have said: " If a monument should rise in the United States, as a memorial to their independence, I should think it only natural if it were built by united effort — a common work of both our nations.

If necessary an AIX program may be built directly in PASE using a standard AIX XL C / C ++ compiler, for example if native APIs are to be used.

If further digital communication is desired, it may take place via external data modems or via optional modems built into the ALE terminal.

If a system is to be built from existing system, this approach is more suitable as it starts from some existing modules.

If such a link were built, it would likely be a 17-km-long submerged rail tunnel.

If completely built, it would house up to 240, 000 residents.

If built the development would see replicas of streetscapes and zones replicating locations from the UK, Europe and the USA.

On its mantelpiece is the motto of Thomas Bohier-" S ' il vient à point, me souviendra " ( If the building is finished, it will preserve the memory of the man who built it )-which echoes his coat of arms above the door.

If such an extreme construction would ever be built, it would by far exceed the length of any man-made tunnel in the world today.

If Chanute built his railroad through Fairbury, then Patton would give Chanute half of the town's property.