In this paper, we prove the vanishing theorem of Dual Bass numbers (Theorem
5.10). In detail, let $R$ be an U ring, $M$ be an Artinian $R$-module, ${\frak
p}\in\textmd{Cos}_{R}M$, if $\pi_i({\frak p},M)>0$, then
$\textmd{Cograde}_{R_{\frak p}} \textmd{Hom}_R (R_{\frak p},M) \leq i \leq
\textmd{fd}_{R_{\frak p}}\textmd{Hom}_R (R_{\frak p},M)$, where $\pi_i({\frak
p},M) = \textmd{dim}_{k({\frak p})} \textmd{Tor}^{R_{\frak p}}_i (k({\frak
p}),\textmd{Hom}_R (R_{\frak p},M))$ is the $i-\textmd{th}$ Dual Bass number of
$M$ with respect to ${\frak p}$, and $\textmd{fd}_{R_{\frak p}} \textmd{Hom}_R
(R_{\frak p},M)$ might be infinite. Moveover, if $\textmd{Cograde}_{R_{\frak
p}} \textmd{Hom}_{R}(R_{\frak p},M) = s, ~\textmd{fd}_{R_{\frak p}}
\textmd{Hom}_R (R_{\frak p},M) = t < \infty$, then $\pi_s({\frak p},M) > 0$ and
$\pi_t({\frak p},M) > 0$.