Gleason cover of a compact regular locale ($\gamma X$)

For a compact regular locale $X$, put $B=\mathcal{O}(X{)}_{\neg \neg }$ and $\mathcal{O}(\gamma X)=\mathrm{Idl}(B)$. This locale is extremally disconnected. Its natural surjection to $X$ has inverse imageThe Joyal extension lemma gives the frame homomorphism; its injectivity follows by recovering $a$ as the join of the regular opens in this ideal.