Let $0\in X\subset\mathbb{C}^N$ be a germ of complex space and $0\in Z\subset X$ be a closed analytic subset (globally) defined by holomorphic functions $f_1,\dots,f_r$. Is there a complex analytic stratification (after a possible shrinking of $X$) such that $Z$ is a union of strata and for each stratum $S$ and each point $x\in S$, there is a Lipschitz piecewise smooth homeomorphism $\alpha:U_S\times C_x\simeq U$ such that
(1) $U$ is an open neighborhood of $x$ in $X$. $U_S$ is an open neighborhood of $x$ in $S$. $C_x=X\cap H$ for some linear plane $H$ which intersects transversally with $S$ at $x=\alpha(t_0,0)$.
(2) For every $i=1,\dots,r$ and every $(t,z)\in U_S\times C_x$, $f_i(\alpha(t_0,z))=f_i(\alpha(t,z))$.
