Local $C^r$-right equivalence of $C^{r+1}$ functions

Abstract: Let $$f,g:(\mathbb{R}^{n,0)\rightarrow} (\mathbb{R},0)$$ be $C<sup>{r+1}$ functions, $r\in \mathbb{N}$. We will show that if $\nabla f(0)=0$ and there exist a neigbourhood $U$ of $0\in \mathbb{R}<sup>n$ and a constant $C&gt;0$ such that $$\left|\partial^{m(g-f)(x)\right|\leq} C \left|\nabla f(x)\right|^{r+2-|m|}, \quad x\in U,$$ for any $m\in \mathbb{N}_0<sup>n$ such that $|m|\leq r$, then there exists a $C<sup>r$ diffeomorphism $$\varphi:(\mathbb{R}^{n,0)\rightarrow} (\mathbb{R}<sup>n,0)$$ such that $f=g\circ \varphi$ in a neighbourhood of $0$.