After a linear coordinate change, every nonzero germ is regular in the last variable

Choose a holomorphic representative of $f$ on a small polydisc, and expand it by [L1] as $f(z)={\sum }_{\alpha \in {\mathbb{N}}^m}{c}_{\alpha }{z}^{\alpha }\mathrm{.}$ Because $f$ is nonzero, some coefficient is nonzero. Let $d$ be the smallest total degree $\mathrm{\mid }\alpha \mathrm{\mid }$ for which $c_{\alpha }\mathrm{\ne }0$, and put $P_d(z):={\sum }_{\mathrm{\mid }\alpha \mathrm{\mid }=d}{c}_{\alpha }{z}^{\alpha }\mathrm{.}$ Then $P_d$ is a nonzero homogeneous polynomial of degree $d$.

If $P_d(v)=0$ for every $v\in {\mathbb{C}}^m$, then the polynomial function $P_d$ vanishes identically on all of ${\mathbb{C}}^m$; applying [L2] to that finite power series would force every coefficient $c_{\alpha }$ with $\mathrm{\mid }\alpha \mathrm{\mid }=d$ to be $0$, contradicting step 1.1. Therefore choose $v\in {\mathbb{C}}^m$ with $P_d(v)\mathrm{\ne }0$.

Choose an invertible complex-linear map $T$ sending the last basis vector $e_m$ to $v$. Along the last axis one then has $f\circ T(0,\dots ,0,\zeta )=P_d(v){\zeta }^d+{\sum }_{n>d}{b}_n{\zeta }^n={\zeta }^d(P_d(v)+{\sum }_{n>d}{b}_n{\zeta }^{n-d})\mathrm{.}$ The bracketed factor is holomorphic and nonzero at $0$ because $P_d(v)\mathrm{\ne }0$. Hence [L3] makes $f\circ T$ regular in the last variable of order $d$.