In the book Local Rings Nagata states in Theorem 44.1:
If $R$ is a Henselian pseudo-geometric analytically normal ring, then every finite integral extension $R'$ of $R$ is analytically irreducible and is algebraically closed in its completion $\widehat{R'}$ (i.e., every element of $\widehat{R'}$ which is algebraic over $R'$ is already in $R'$).
In the (short) proof he deduces from the fact if some $c$ is integral over $R'$ and $R'[c] \otimes_{R'} \widehat{R'}$ is an integral domain that $c \in R'$.
My question is: How does this follow? I think the Henselian property must be used at this point, because it is not used anywhere else and I believe the statement is wrong without this assumption. But I just don't see how it holds.
Thank you for advice in advance.