Sufficient conditions for separation of analytic singularities in $C^n$ and a basis for a space of holomorphic functions

It is proved that every holomorphic function of $n$ variables which has singularities on analytic surfaces, whose equations are linearly dependent, can be represented as the sum of functions, each of which has less than one singular surface. This fact is used to construct a basis for the space of functions which are holomorphic in the domain
$$ C^n\setminus\bigcup_{j=1}^N\left\{z:\sum_{\nu=1}^n c_{j\nu}z_\nu+c_{j0}=0\right\}. $$