 Stable cancellation for complex vector bundles

Let $E_0,E_1$ be complex vector bundles of rank $d$ over a finite CW complex of dimension at most $2d$. If $E_0\oplus {\mathbb{C}}^r\cong E_1\oplus {\mathbb{C}}^r$ for some $r$, then $E_0\cong E_1$. This follows because the stabilization map $BU(d)\to BU$ is $2d$-connected, so stable equivalence is injective on bundles over such a complex.