Lemma 9.4.226BA2ABFD727

Proof. By corollary 9.3.13(1), $X_{\mathrm{reg}}\cap U$ is the disjoint union of the sets $X_ i\cap X_{\mathrm{reg}}=(X_ i)_{\mathrm{reg}}\setminus \bigcup _{j\neq i}X_ j$, each of which is open in $X_{\mathrm{reg}}$. The analytic subset $(X_ i)_{\mathrm{reg}}\cap \bigcup _{j\neq i}X_ j$ of the manifold $(X_ i)_{\mathrm{reg}}$ is nowhere dense in it, by corollary 9.3.6, applied to $(X_ i,x)$ and its subgerm $(X_ i\cap \bigcup _{j\neq i}X_ j,x)$, which differs from $(X_ i,x)$ because the branches are irredundant. Hence a bounded holomorphic function on $X_ i\cap X_{\mathrm{reg}}\cap V$ extends uniquely across this analytic subset to a holomorphic function on $(X_ i)_{\mathrm{reg}}\cap V$, by lemma 9.2.9 applied on connected coordinate balls of $(X_ i)_{\mathrm{reg}}$, and the extension is bounded by the same bound. Applied near each point of $X_ i\cap V$, this shows that the restriction of $f$ has a unique extension $f_ i$, which is weakly holomorphic on $X_ i\cap V$. Conversely, if $f_ i$ is weakly holomorphic on $X_ i\cap V$ for each $i$, the function equal to $f_ i$ on $X_ i\cap X_{\mathrm{reg}}\cap V$ is holomorphic and locally bounded on $X_{\mathrm{reg}}\cap V$, hence weakly holomorphic on $X\cap V$; by the uniqueness of the extensions, the two constructions are inverse to each other. Passing to germs at $x$ gives the last assertion. $\square$