Differential subbundles of trivial bundle over a segment.

Question

Let $F \subset (0,1)\times \mathbb R^n$ be a differential vector subbundle of rank $r$ of trivial vector bundle over the segment $(0,1)$. I'd like to prove that, if for every differential section $\sigma$ of $F$, $\frac{d\sigma}{dt}$ is also a section of $F$, then $F$ is a trivial subbundle. I have a hint that I should use constant rank theorem but I can't realize how to use it there.