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. $\square$
