Let $T = (X, \mathcal{T})$ be a D1106: Topological space.
Let $U \subseteq X$ be an D97: Open set in $T$.
Let $F \subseteq X$ be a D98: Closed set in $T$.

Result R50: Set difference equals intersection with complement shows that \begin{equation} F \setminus U = F \cap (X \setminus U) \end{equation} The complement $X \setminus U$ is closed in $T$ since $U$ is open in $T$, whence R75: Intersection of closed sets is closed implies the result. $\square$