Rectangular neighborhood of a product of compact sets ($K\times L\subseteq U\times V\subseteq W$)

If $K$ and $L$ are compact subsets of two spaces and $W$ is an open neighborhood of their product, there are open sets $U,V$ with $K\times L\subseteq U\times V\subseteq W$. First use compactness of $L$ to produce a rectangle around each slice $\{x\}\times L$, intersecting finitely many first factors and uniting second factors. Then use compactness of $K$, uniting finitely many resulting first factors and intersecting second factors. No Hausdorff assumption is required.