= Local basis criterion for Leray-Hirsch classes
On a trivialized <fiber bundle> over $U$, assume the fiber has finite-dimensional total <cohomology> over a <field>. The <Künneth theorem> identifies the total-space cohomology with a finite free graded module over $H^*(U)$. Classes restricting to a basis on every fiber give an invertible change of generators: its degree-zero coefficient blocks are invertible on each path component, while its positive-base-degree terms strictly decrease fiber degree and are therefore nilpotent. Inverting the diagonal blocks and then using a finite geometric series proves that multiplication by the chosen classes is an <isomorphism> over $U$.
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