Exercise 4.2.3

If $c\ast d=e$, we are basically flipping the $d$ vector w.r.t. $y$-axis, and shift it towards $c$ vector such that they have overlaps. Whenever they overlap, we multiply the corresponding elements and add the sums. It’s clear that each element in $c$ will multiply with each element of $d$ one and only once as the flipped $d$ shifting forward. Each element of $e$ is a sum of a subset of these products. Since each product only happens once, their total sum should be the same regardless how we add them up. Then we have $(\sum <!--\; FUNCTION\; APPLICATION\; -->c_i)(\sum d_i)=\sum e_i$.