  It is known that for any graph $G,$ $\gamma (G\square P_2)\geq \gamma (G)$
where $\gamma$ stands for the domination number, $\square$ for the cartesian
product and $P_2$ is the path graph on two vertices. In an attempt to prove
Vizing's conjecture, Clark and Suen proved in $2000$ that $\gamma (X\square
Y)\geq \frac{1}{2}\gamma (X)\gamma (Y)$ for any pair of graphs $X$ and $Y.$
Combining these two inequalities, we have $\gamma (X\square Y\square P_2)\geq
\frac{1}{2}\gamma (X)\gamma (Y).$ In this paper, we use space projections to
improve this lower bound and show that $\gamma (X\square Y\square P_2)\geq
\frac{2}{3}\gamma (X)\gamma (Y)$ for any pair of graphs $X$ and $Y.$ In
addition, we prove that $\gamma (X\square Y\square P_{n})\geq c_n\gamma
(X)\gamma (Y)\gamma (P_{n}),$ where $c_n$ is almost $\frac{3}{4}$ when $n$ is
big enough.

Combinatorics