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We also prove a closely related preparation theorem for f and μ.
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In preparation for this theorem, we introduce the following notation.
As the preparation for proving Theorem 1.2, we first give two auxiliary lemmas.
An Impossibility Theorem for Clustering.
The Spectral Theorem for symmetric real matrices.
A hookup theorem for multilevel security.
The Fundamental Theorem for Line Integrals.
This theorem is called the "prime number theorem for geodesics", because it is exactly analogous to the usual "prime number theorem" for whole numbers, which estimates the number of primes less than a given size.
Yet much of the theory of calculus, including the fundamental theorem of calculus and the mean-value theorem for derivatives, is based on such approximations.
Concerning the undefinability theorem for L ω1,ω1).
Using (2), theorem (4) entails the isomorphism theorem for finitary and finitely algebraizable logics.
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