Let $M$ be a smooth manifold and $X\subset M$ is a Whitney object, i.e. a subset with a Whitney stratification $\mathcal{S}$. If $W\subset X$ is a closed subset of $X$ such that for each stratum $S\in\mathcal{S}$ the intersection $W\cap S$ is a smooth submanifold of $M$, then is it true that $W$ is also a Whintey object with the decomposition $\mathcal{S}_{W}=\{S\cap W\mid \forall S\in\mathcal{S}\}$?