Every paracompact Hausdorff space is normal

For each $x\in E$, regularity supplies an open $U$ containing $x$ with $\overline U\cap F=\varnothing$; therefore the family of all such $U$, together with $X\setminus E$, is an open cover $\mathcal U$ of $X$.

Take a locally finite open cover $\mathcal W$ refining $\mathcal U$, and set $H:=\bigcup\{W\in\mathcal W:W\cap E\ne\varnothing\}$.

The open set $H$ contains $E$, because a member of $\mathcal W$ containing a point of $E$ cannot lie inside $X\setminus E$.

Every member $W$ used in $H$ lies in one of the eligible $U$ and so has $\overline W\cap F=\varnothing$; hence $\overline H=\bigcup\overline W$ is disjoint from $F$ by [L2].

The open sets $H$ and $X\setminus\overline H$ contain $E$ and $F$ respectively and are disjoint, so [F2] proves normality.