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As a corollary, for a wide class of functions $\\psi$, we show that the operator $-\\psi(A)$ generates a sectorially bounded holomorphic $C_0$-semigroup on a Banach space whenever $-A$ does, and the sectorial angle of $A$ is preserved. When $\\psi$ is a Bernstein function, this was recently proved by Gomilko and Tomilov, but the proof here is more direct. 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