Using result R4587: and the definition of an indicator function, we have the chain of equivalencies \begin{equation} \begin{split} I_E(a x) = 1 \quad & \iff \quad a x \in E \\ & \iff \quad x \in a^{-1} E \\ & \iff \quad I_{a^{-1} E}(x) = 1 \\ \end{split} \end{equation} whence the first claim follows as a consequence of R2965: Indicator function is uniquely identified by its support. The second claim follows by substituting $a^{-1} \neq 0$ in the place of $a$. $\square$