I want to prove that $\overline{W \cap S} \supseteq \overline{W} \cap S$ in a frame $L$, where $S$ is a open sublocale of the frame.
My attempt;
Let $y \in \overline{W} \cap S \Longrightarrow y \in {\uparrow} \bigwedge W \cap S \Longrightarrow y \geq \bigwedge W \;\text{and}\; y \in S \Longrightarrow y \geq \bigwedge W \;\text{and}\; y \geq \bigwedge S \;(\text{since for every}\; x \in S, x \geq \bigwedge S)\; \Longrightarrow y \in {\uparrow}(\bigwedge W \cap \bigwedge S) \Longrightarrow y \in {\uparrow}\bigwedge (W \cap S)$. Therefore $y \in \overline{W \cap S}$.
It seems easier than i anticipated and hence im assuming that i am doing something wrong. Also my lattice theory understanding is a bit shakey, please advise.