Let $A$ and $B$ be contractible smooth paracompact gluing pieces with a common smooth gluing locus $C$, given by smooth embeddings $i_A:C\to A$ and $i_B:C\to B$, and let $M=A\cup_C B$ be the resulting smooth paracompact manifold with its quotient smooth structure. If two rank-$k$ clutching functions $\gamma_0,\gamma_1:C\to GL(k,\mathbb R)$ are smoothly homotopic, then the corresponding clutched rank-$k$ vector bundles over $M$ are isomorphic. More generally, changing the frames on the two pieces by smooth maps $a:A\to GL(k,\mathbb R)$ and $b:B\to GL(k,\mathbb R)$ replaces $\gamma$ by $a|_C\gamma(b|_C)^{-1}$, and the resulting clutched vector bundle is isomorphic to the original one.