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Equivalently, $f$ is a Fano-Mori contraction associated to an extremal face in $\\overline {NE(X)}_{K_X+\\tau L = 0}$; these maps naturally arise in the context of the minimal model program. We prove that, if $\\tau > (n-3) >0$, the general element $X' \\in |L|$ is a variety with at most terminal singularities. 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