How to prove "suppose $\pi:E\to M$is a smooth vector bundle of rank k,if$(\sigma_{1},\dots,\sigma _m)$is a linerly independent m-tuple of smooth local sections of $E$ over an open subset $U\subset M$ ,with $1\leq m\leq k$,then for each $p \in U$ there exist smooth sections$\sigma _{m+1},\dots,\sigma _{k}$defined on some neighborhood $V$ of $p$ such that $(\sigma_{1},\dots,\sigma _{k)}$ is a smooth local frame for $E$ over $U\cap V$."?I hope to prove that a maximal linearly independent subset $X$ of a module $\Gamma (U,E)$ over the ring $C^{\infty}(U)$is a basis of $\Gamma (U,E)$,but I encountered difficulties
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