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