A relative Nash theorem

We prove that if $M$ is a closed smooth manifold and ${M_i}$, $i = 1, \ldots ,k$, are transversally intersecting closed smooth submanifolds of $M$, then there exist a nonsingular algebraic set $Z$ and nonsingular algebraic subsets ${Z_i}$, $i = 1, \ldots ,k$, of $Z$ such that $(M;{M_1}, \ldots ,{M_k})$ is diffeomorphic to $(Z;{Z_1}, \ldots ,{Z_k})$. We discuss a generalization and the consequences of this result.