Let $\pi:E\to M$ be a smooth real vector bundle of finite rank over a paracompact smooth manifold $M$. If $F\subset E$ is a smooth vector subbundle, then there exists a smooth vector subbundle $G\subset E$ such that, for every $p\in M$,
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\begin{align*}
E_p=F_p\oplus G_p.
\end{align*}
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Equivalently, the fibrewise addition map $F\oplus G\to E$ is an isomorphism of smooth vector bundles.