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To quote from Differential Forms in Algebraic Topology,

Set $x=c_1(S^*)$. Then $x$ is a cohomology class in $H^2(P(E))$. Since the restriction of the universal subbundle $S$ on $P(E)$ to a fiber $P(E_P)$ is the universal subbundle $\tilde{S}$ of the projective space $P(E_p)$, by the naturality property of the first Chern class (6.39), if follows that $c_1(\tilde{S})$ is the restriction of $-x$ to $P(E_p)$. Hence the cohomology classes $1,x,\ldots,x^{n-1}$ are global classes on $P(E)$ whose restrictions to each fiber $P(E_p)$ freely generate the cohomology on the fiber.

I can't seem to understand the step of going from restricting $-x$ to $P(E_p)$ being freely generated and $1,x,\ldots,x^{n-1}$ being a basis for it.

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The first Chern class of the tautological bundle over $P(E_p)$ is a generator of $H^2(P(E_p))$. Lets call this $a=-i^{*}(x)$ with $i:P(E_p)\rightarrow P(E)$ the inclusion of the fiber. The cohomology ring of $P(E_p)$ is $H^*(P(E_p))=Z[a]/a^{n}$ (note that $P(E_p)\cong \mathbb{C}P^{n-1}$). So an additive basis of the cohomology of the fiber is $1,a,\dots,a^{n-1}$. But of course $1,-a,\ldots, (-a)^{n-1}$ is also a basis.

This now works for every fiber simulatiously. The classes $1,x,\ldots,x^{n-1}$ restrict to a basis of the cohomology of each fiber. One can apply the Leray-Hirsch theorem.

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