Lemma 88.13.6. Let $S$ be a scheme. Let $f : X \to Y$ and $g : Z \to Y$ be morphisms of locally Noetherian formal algebraic spaces over $S$.
If $f$ is flat and $g_{red} : Z_{red} \to Y_{red}$ is locally of finite type, then the base change $X \times _ Y Z \to Z$ is flat.
If $f$ is flat and locally of finite type, then the base change $X \times _ Y Z \to Z$ is flat and locally of finite type.
Proof. Part (1) follows from a combination of Formal Spaces, Remark 87.21.10, Lemma 88.13.2 part (1), Lemma 88.13.5, and Lemma 88.12.7.
Part (2) follows from a combination of Formal Spaces, Remark 87.21.9, Lemma 88.13.2 part (2), Lemma 88.13.5, and Lemma 88.11.5. $\square$
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