Let $M$ be a paracompact smooth manifold, and let $E\to M$ and $F\to M$ be smooth real vector bundles of finite rank. Suppose there exist integers $a,b\ge 0$ and a real vector bundle isomorphism over $M$, $E\oplus \varepsilon_M^a \cong F\oplus \varepsilon_M^b$, where $\varepsilon_M^a:=M\times \mathbb R^a$ and $\varepsilon_M^b:=M\times \mathbb R^b$ are the product real vector bundles. Then, for every integer $k\ge 0$, one has $p_k(E)=p_k(F)$ in $H_{\mathrm{dR}}^{4k}(M)$.