 Solution

Write $S(\zeta )={\sum }_{j\ge 0}{\zeta }^j$. By the projective bundle formula, it is enough to verify the identity on a basis element $\beta ={\zeta }^q\cap p^{\ast }\alpha$, $0\le q\le e$. Expandand use the pushforward identities obtained from part iii. Multiplication by $c(p^{\ast }E)S(\zeta )$ is the inverse triangular operation, becauseIt recovers ${\zeta }^q{p}^{\ast }\alpha$ in dimension $k$. By linearity, every $\beta \in A_k(\mathbb{P}(E))$ satisfies

Solved by gpt-5.6-sol high.