Lemma 37.10.2. Consider a commutative diagram

of thickenings. Assume

$X'$ is flat over $S'$,

$f$ is flat,

$S \subset S'$ is a finite order thickening, and

$X = S \times _{S'} X'$ and $Y = S \times _{S'} Y'$.

Then $f'$ is flat and $Y'$ is flat over $S'$ at all points in the image of $f'$.

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