Consider the expression
$$\partial (f_1\Box f_2)(\bar{x})=\partial f_1(\bar{x_1})\cap\partial f_2(\bar{x_2}).$$
What does the above simplify into when $f_1$ and $f_2$ are indicator functions of convex subsets $C$ and $D$ of $X$, respectively?
Considering that the sub differential of the indicator function is the normal cone I have the following proof. Please confirm for me that I am on the right path with this proof.
proof \begin{align*} \partial(\iota_{C}\Box\iota_{D})(\bar{x})&=\partial\iota_{C}(\bar{x_1})\cap\iota_{D}(\bar{x_2})\\ &=N_{C}(\bar{x_1})\cap N_{D}(\bar{x_1})\\ &=N_{C+D}(\bar{x_1}+\bar{x_2})\\ &=\partial(\iota_{C+D}(\bar{x_1}+\bar{x_2})) \end{align*} Thus, when the functions are the indicator functions we see that the corollary simplifies down to $$\partial(f_1\Box f_2)=\partial(\iota_{C+D}(\bar{x_1}+\bar{x_2}))$$ $\Box$