Lemma 10.67.7. Let $A$ be a ring. Let $I$ be an ideal generated by a regular sequence $f_1, \ldots , f_ n$ in $A$. Let $g_1, \ldots , g_ m \in A$ be elements whose images $\overline{g}_1, \ldots , \overline{g}_ m$ form a regular sequence in $A/I$. Then $f_1, \ldots , f_ n, g_1, \ldots , g_ m$ is a regular sequence in $A$.

**Proof.**
This follows immediately from the definitions.
$\square$

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