  Let $R$ be a commutative Noetherian ring and $\fa$ an ideal of $R$. We intend
to establish the dual of two Faltings' Theorems for local homology modules of
an Artinian module. As a consequence of this, we show that, if $A$ is an
Artinian module over semi-local complete ring $R$ and $j$ is an integer such
that $H_i^{\fa}(A)$ is Artinian for all $i<j$, then the set
$\Coass_R(H_j^{\fa}(A))$ is finite. We also introduce the notion of the $n$th
Artinianness dimension $g_n^\fa(A)=\inf\{g^{\fa R_{\fp}}(^\fp A):
\fp\in\Spec(R) \ \ \text{and} \ \ \dim R/\fp\geq n\}$, for all
$n\in\mathbb{N}_{0}$ and prove that $g_1^\fa(A)=\inf\{i\in\mathbb{N}_0:
H_i^\fa(A) \ \ \text{is not minimax}\}$, whenever $R$ is a semi-local complete
ring. Moreover, in this situation we show that
$\Coass_R(H_{g_1^\fa(A)}^\fa(A))$ is a finite set.
