= Solution
Fix $|t|<t_0$. Since $|ct|<t_0$, $0<M_X(ct)<\infty$. Using <independence of random variables> and the <Tonelli theorem> for the positive exponential integrands gives the extended-real identity
$$
M_X(t)=M_{cX+R}(t)=M_X(ct)M_R(t).
$$
The left side is finite and the first factor is finite and strictly positive. Consequently $M_R(t)$ is finite and
$$
M_R(t)=\frac{M_X(t)}{M_X(ct)}.
$$
The same reasoning applies to $R'$, so their <moment-generating functions> are finite and equal throughout $(-t_0,t_0)$. The <uniqueness theorem for moment-generating functions> yields
$$
\boxed{R\overset d=R'.}
$$
This proves <uniqueness of a self-decomposition remainder> while also checking the required finiteness; no independence between $R$ and $R'$ is needed.
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