Lemma 9.3.716B19C0E622D

Proof. Let $p\in Z$; near points outside $Z$ there is nothing to prove, since $\overline{Z\setminus Y}\subset Z$. Choose representatives $Z_1,\ldots ,Z_ k$ of the branches of $(Z,p)$ on an open neighborhood $U$ of $p$ with $Z\cap U=\bigcup _ iZ_ i$, so small that $Z_ i\subset Y$ whenever $(Z_ i,p)\subset (Y,p)$, and that, for each of the other branches, $Z_ i\setminus Y$ is dense in $Z_ i$; the latter is possible by corollary 9.3.6, applied to $(Z_ i,p)$ and its subgerm $(Z_ i\cap Y,p)$, which differs from $(Z_ i,p)$. Then $(Z\setminus Y)\cap U$ is the union of the sets $Z_ i\setminus Y$ over the branches not contained in $(Y,p)$, and its closure in $U$ is the union of the corresponding sets $Z_ i$, which is analytic in $U$. $\square$