Regular implies semiregular

This article gives the statement and possibly, proof, of an implication relation between two topological space properties. That is, it states that every topological space satisfying the first topological space property (i.e., regular space) must also satisfy the second topological space property (i.e., semiregular space)
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Statement

Any regular space is a semiregular space.

Definitions used

Term	Definition used
regular space	A topological space $X$ is termed regular if the following holds: given any point $x\in X$ and open subset $V\subseteq X$ such that $x\in V$, there exists an open subset $U\subseteq X$ such that $x\in U$ and the closure ${\overline {U}}$ is contained in $V$.
semiregular space	A topological space $X$ is termed regular if the following holds: given any $x\in X$ and any open subset $V\subseteq X$ containing $x$, there exists a regular open subset $U$ of $X$ containing $x$ and contained in $V$. Note that regular open does not mean an open subset that is regular in the subspace topology. Rather, it means a subset that is the interior of its closure.

Proof

Given: A regular space $X$. A point $x\in X$ and any open subset $V\subseteq X$ containing $x$,

To prove: There exists a regular open subset $U$ of $X$ containing $x$ and contained in $V$.

Proof:

Step no.	Assertion/construction	Given data used	Previous steps used	Explanation
1	There exists an open subset $U_{0}$ containing $x$ such that ${\overline {U_{0}}}$ is contained in $V$.	$X$ is regular, $x\in V$, $V$ open in $X$		directly from regularity.
2	Let $U$ be the interior of the closure ${\overline {U_{0}}}$.		Step (1)
3	$U$ contains $U_{0}$ and hence contains $x$.		Steps (1), (2)	$U$ is the largest open subset of ${\overline {U_{0}}}$, hence contains the open subset $U_{0}$.
4	$U$ is a regular open subset.		Step (2)	step-direct
5	$U$ is contained in $V$.		Steps (1), (2)	By Step (2), $U\subseteq {\overline {U_{0}}}$. By Step (1), ${\overline {U_{0}}}\subseteq V$. Combining, $U\subseteq V$.
6	$U$ is the desired open subset.		Steps (3), (4), (5)