Let $M$ be an $n$-dimensional smooth manifold and let $\pi:TM\longrightarrow M$ be the tangent bundle of $M$. Furthermore, let $E$ be a vector subbundle of $TM$ such that $\textrm{dim}(E_p)=k$ for every $p\in M$ where $E_p$ stands for the fiber of $E$ above $p\in M$.
How to show for every $p\in M$ there is a local chart $(U, x_1, \ldots, x_n)$ around $p$ such that $$\displaystyle \left(\frac{\partial}{\partial x_j}\right)_p\in E_p,$$ for every $j=1, \ldots, k$?
I know this has to do with the local version of Frobenius theorem but I don't know how to show this statement.
Thanks.