  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.
