Proposition 13

If every point $x$ of a topological space $X$ has a closed neighbourhood which is a regular subspace of $X$, then $X$ is regular.

$X$ is Hausdorff by Proposition 6 of no. 2. Let $x$ be any point of $X$ and let $V$ be a closed regular neighbourhood of $x$. If $U$ is any neighbourhood of $x$ contained in $V$, then $U$ is a neighbourhood of $x$ relative to $V$; hence by hypothesis there is a neighbourhood $W$ of $x$ in $V$ which is closed in $V$ and contained in $U$. But $W$ is a neighbourhood of $x$ in $X$ since $V$ is a neighbourhood of $x$ in $X$, and $W$ is closed in $X$ since $V$ is closed in $X$.