  We show that if $(X.A)$ and $(Y,B)$ are two isomorphic Hilbert pro-$C^{\ast}
$-bimodules, then the crossed product $A\times_{X}\mathbb{Z}$ of $A$ by $X$ and
the crossed product $B\times_{Y}\mathbb{Z}$ of $B$ by $Y$ are isomorphic as
pro-$C^{\ast}$-algebras. We also prove a property of "associativity" between "
$\otimes_{\min}$" and "$\times_{X}$" $\ $as well as " $\otimes_{\max}$" and
"$\times_{X}$". As an application of these results we show that the crossed
product of a nuclear pro-$C^{\ast}$ -algebra $A$ by a full Hilbert
pro-$C^{\ast}$-bimodule $X$ is a nuclear pro-$C^{\ast}$-algebra.
