Proof.
Let $p_1, \ldots , p_ n \in X$ be the points where $X \to \mathop{\mathrm{Spec}}(k)$ isn't smooth. Choose an affine open $U_2 \subset X$ containing $p_1, \ldots , p_ n$ and the generic point of every irreducible component of $X$. This is possible by Varieties, Lemma 33.43.3 and Properties, Lemma 28.29.5. Then $X \setminus U_2$ is finite and we can choose an affine open $U_1 \subset X \setminus \{ p_1, \ldots , p_ n\} $ such that $X = U_1 \cup U_2$. Set $U_{12} = U_1 \cap U_2$. Then $U_1$ and $U_{12}$ are smooth affine schemes over $k$. We conclude that
\[ \mathcal{D}\! \mathit{ef}_ X \longrightarrow \mathcal{D}\! \mathit{ef}_{U_2} \]
is smooth by Lemma 93.16.2. Applying Lemmas 93.9.7 and 93.16.1 we win.
$\square$
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