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Abstract

We present a simple proof of the theorem which says that for a series of extensions of differential fields K ⊂ L ⊂ M, where K ⊂ M is Picard-Vessiot, the extension K ⊂ L is Picard-Vessiot iff the differential Galois group $Gal_{L} M$ is a normal subgroup of $Gal_{K} M$. We also present a proof that the probability function Erf(x) is not an elementary function.