  In this paper, we prove some well-known results on local cohomology with
respect to a pair of ideals in graded version, such as, Independence Theorem,
Lichtenbaum-Harshorne Vanishing Theorem, Basic Finiteness and Vanishing
Theorem, among others. Besides, we present a generalized version of Melkersson
Theorem about Artinianess of modules and a result concerning Artinianess of
local cohomology modules.
