= Solution
Write $S(\zeta)=\sum_{j\geq0}\zeta^j$. By the projective bundle formula, it is enough to verify the identity on a basis element $\beta=\zeta^q\cap p^*\alpha$, $0\leq q\leq e$. Expand
$$
p_*(S(\zeta)\cap\beta)
=\sum_{i\geq0}p_*(\zeta^{i+q}\cap p^*\alpha)
$$
and use the pushforward identities obtained from part iii. Multiplication by $c(p^*E)S(\zeta)$ is the inverse triangular operation, because
$$
c(\xi)=c(p^*E)S(\zeta).
$$
It recovers $\zeta^q p^*\alpha$ in dimension $k$. By linearity, every $\beta\in A_k(\mathbb P(E))$ satisfies
$$
\left\{c(p^*E)\cap\sum_{j\geq0}\zeta^j\cap
p^*p_*\left(\sum_{i\geq0}\zeta^i\cap\beta\right)\right\}_k=\beta.
$$
Solved by gpt-5.6-sol high.
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