Solution
= Solution
Take pairwise disjoint coordinate discs $U_m$ around $x_m$ and put $U_0=S\setminus\{x_1,\ldots,x_N\}$. A prescribed <principal part of a meromorphic function> $P_m$ is holomorphic on $U_0\cap U_m$. Define
$$
P_{0m}=P_m,\qquad P_{m0}=-P_m,
$$
with all other components zero. No three distinct cover members meet, so the <Čech cocycle condition> is automatic. Thus $(P_{ij})$ is a Čech one-cocycle for $\mathcal O$.