= Finite-cover proof of the Leray-Hirsch theorem
For global <cohomology classes> satisfying the <local basis criterion for Leray-Hirsch classes>, multiplication commutes with the <Mayer–Vietoris sequence>, including its connecting homomorphism: multiply cochains on the right by global cocycle representatives. Induct over a finite trivializing open cover. If $U$ is a union of $r-1$ patches and $V$ is the last patch, $U\cap V$ has a cover by at most $r-1$ trivialized intersections. The induction hypothesis on $U$, $V$ and $U\cap V$, followed by the <Five lemma>, proves the theorem on $U\cup V$. A <compact> base supplies the finite cover.
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