On the extension of bi-Lipschitz mappings

Let $X$ be a closed semialgebraic set of dimension $k\mathrm{.}$ If $n\ge 2k+1$, then there is a bi-Lipschitz and semialgebraic embedding of $X$ into ${\mathbb{R}}^n\mathrm{.}$ Moreover, if $n\ge 2k+2$, then this embedding is unique (up to a bi-Lipschitz and semialgebraic homeomorphism of ${\mathbb{R}}^n\mathrm{.}$