Finitness theorem for Multi-K-Bi-Lipschtiz equivalence of map-germs

Let $P^{k}(n,p)$ be the set of all real polynomial map germs $f = ( f_1 , ..., f_p ) : (\mathbb{R}^{n},0) \rightarrow (\mathbb{R}^{p},0)$ with degree of $f_1 , ...,f_p$ less than or equal to $k \in \mathbb{N}$. The main result of this paper shows that the set of equivalence classes of $P^{k}(n,p)$, with respect to multi-$\mathcal{K}$-bi-Lipschitz equivalence, is finite.