Let $\pi:E\to M$ be a smooth real vector bundle of rank $k$ over a smooth paracompact manifold $M$. Suppose there exists a smooth section $s:M\to E$ such that $s(x)\neq 0$ for every $x\in M$. Then there exists a smooth real vector bundle $F\to M$ of rank $k-1$ and a smooth vector bundle isomorphism over $M$
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\begin{align*}
\underline{\mathbb R}\oplus F \cong E,
\end{align*}
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where $\underline{\mathbb R}=M\times \mathbb R$ is the product real line bundle over $M$.