Let f,g:(Rn,0)→(Rm,0)\documentclass[12pt]{minimal}
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\begin{document}$$f,g:({\mathbb {R}}^n,0)\rightarrow ({\mathbb {R}}^m,0)$$\end{document} be Cr+1\documentclass[12pt]{minimal}
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\begin{document}$$C^{r+1}$$\end{document} mappings and let Z={x∈Rn:ν(df(x))=0}\documentclass[12pt]{minimal}
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\begin{document}$$Z=\{x\in \mathbf {\mathbb {R}}^n:\nu (df (x))=0\}$$\end{document}, 0∈Z\documentclass[12pt]{minimal}
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\begin{document}$$0\in Z$$\end{document}, m≤n\documentclass[12pt]{minimal}
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\begin{document}$$m\le n$$\end{document}. We will show that if there exist a neighbourhood U of 0∈Rn\documentclass[12pt]{minimal}
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\begin{document}$$0\in {\mathbb {R}}^n$$\end{document} and constants C,C′>0\documentclass[12pt]{minimal}
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\begin{document}$$C,C'>0$$\end{document} and k>1\documentclass[12pt]{minimal}
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\begin{document}$$k>1$$\end{document} such that for x∈U\documentclass[12pt]{minimal}
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\begin{document}$$x\in U$$\end{document}ν(df(x))≥Cdist(x,Z)k-1,∂s(fi-gi)(x)≤C′ν(df(x))r+k-|s|,\documentclass[12pt]{minimal}
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\begin{document}$$\begin{aligned}&\nu (df(x))\ge C{\text {dist}}(x,Z)^{k-1}, \\&\left| \partial ^{s} (f_i-g_i)(x) \right| \le C'\nu (df(x))^{r+k-|s|}, \end{aligned}$$\end{document}for any i∈{1,⋯,m}\documentclass[12pt]{minimal}
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\begin{document}$$i\in \{1,\dots , m\}$$\end{document} and for any s∈N0n\documentclass[12pt]{minimal}
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\begin{document}$$s \in \mathbf {\mathbb {N}}^n_0$$\end{document} such that |s|≤r\documentclass[12pt]{minimal}
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\begin{document}$$|s|\le r$$\end{document}, then there exists a Cr\documentclass[12pt]{minimal}
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\begin{document}$$C^r$$\end{document} diffeomorphism φ:(Rn,0)→(Rn,0)\documentclass[12pt]{minimal}
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\begin{document}$$\varphi :({\mathbb {R}}^n,0)\rightarrow ({\mathbb {R}}^n,0)$$\end{document} such that f=g∘φ\documentclass[12pt]{minimal}
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\begin{document}$$f=g\circ \varphi $$\end{document} in a neighbourhood of 0∈Rn\documentclass[12pt]{minimal}
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\begin{document}$$0\in {\mathbb {R}}^n$$\end{document}. By ν(df)\documentclass[12pt]{minimal}
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\begin{document}$$\nu (df)$$\end{document} we denote the Rabier function.
