Since $E$ is a G-delta set, there are open sets $U_0, U_1, U_2, \ldots \in \mathcal{T}$ such that \begin{equation} E = \bigcap_{n \in \mathbb{N}} U_n \end{equation} By definition, $X \setminus U_n$ is now a closed set in $T$ for each $n \in \mathbb{N}$. Applying result R4168: Difference of set and countable intersection equals union of differences, one has \begin{equation} X \setminus E = X \setminus \bigcap_{n \in \mathbb{N}} U_n = \bigcup_{n \in \mathbb{N}} (X \setminus U_n) \end{equation} We have found an expression for $E$ as a countable union of closed sets, so the proof is complete. $\square$