In Johnstone´s Sketches of an Elephant Volume 2, page 716,

lemma 4.1.8 states that for spatial locales $X$ and $Y$ with $X$ locally compact then the product $X\times Y$ is spatial.

Is this lemma still valid for a family $\{X_{\alpha}\}_{\alpha}$ of locally compact spatial locales ?

Or anyone knows a modification for a countable family of locally compact spatial locales?

Thanks in advanced.