Pretty cleanness and filter-regular sequences

Let $K$ be a field and $S=K[x_1,\dots ,x_n]$. Let $I$ be a monomial ideal of $S$ and $u_1,\dots ,u_r$ be monomials in $S$ which form a filter-regular sequence on $S\mathrm{/}I$. We show that $S\mathrm{/}I$ is pretty clean if and only if $S\mathrm{/}(I,u_1,\dots ,u_r)$ is pretty clean.