  In this paper, we study finite projective differential modules on $p$-adic
polyannuli satisfying the Robba condition. Christol and Mebkhout proved the
decomposition theorem (the $p$-adic Fuchs theorem) of such differential modules
on one dimensional $p$-adic annuli under certain non-Liouvilleness assumption
and Gachet generalized it to higher dimensional cases. On the other hand,
Kedlaya proved a generalization of the $p$-adic Fuchs theorem in one
dimensional case. We prove Kedlaya's generalized version of $p$-adic Fuchs
theorem in higher dimensional cases.
