Let
R\,=\,{{\oplus }_{n\ge 0}}{{R}_{n}}
be a graded Noetherian ring with local base ring
\left( {{R}_{0}},{{\text{m}}_{0}} \right)
and let
{{R}_{+}}\,=\,{{\oplus }_{n>0}}{{R}_{n}}
. Let
M
and
N
be finitely generated graded
R
-modules and let
\mathfrak{a}\,=\,{{\mathfrak{a}}_{0}}\,+\,{{R}_{+}}
an ideal of
R
. We show that
H_{\mathfrak{b}0}^{j}\,\left( H_{\mathfrak{a}}^{i}\left( M,\,N \right) \right)
and
{H_{\mathfrak{a}}^{i}\left( M,\,N \right)}/{{{\mathfrak{b}}_{0}}H_{\mathfrak{a}}^{i}\left( M,\,N \right)}\;
are Artinian for some
i\text{ s}
and
j\,\text{s}
with a specified property, where
{{\mathfrak{b}}_{o}}
is an ideal of
{{R}_{0}}
such that
{{\mathfrak{a}}_{0}}\,+\,{{\mathfrak{b}}_{0}}
is an
{{\mathfrak{m}}_{0}}
-primary ideal.