  Let $S={\Bbb K}[x_1,\dots,x_n]$ denote a polynomial ring over a field $\Bbb
K$. Given a monomial ideal $I$ and a finitely generated multigraded $M$ over
$S$, we follow Herzog's method to construct a multigraded free $S$-resolution
of $M/IM$ by using multigraded $S$-free resolutions of $S/I$ and $M$. The
complex constructed in this paper is used to prove the inequality $\Reg(IM)\leq
\Reg(I)+\Reg(M)$ for a large class of ideals and modules. In the case where $M$
is an ideal, under one relative condition on the generators which specially
does not involve the dimensions, the inequality $\Reg(IM)\leq \Reg(I)+\Reg(M)$
is proven.
