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** for all f in X ′ there exists x in X with ǁxǁ ≤ 1, so that f ( x ) = ǁfǁ.
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** 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.
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