Let $\pi: E \to M$ be a smooth real vector bundle of rank $k$ over a smooth manifold $M$. For every point $p \in M$, there exists an open neighbourhood $U \subset M$ of $p$ such that the $C^\infty(U)$-module $\Gamma(U, E|_U)$ of smooth sections of the restricted bundle $E|_U \to U$ is free of rank $k$.