I am going through the proof of Riemann's removable singularity theorem from the link https://en.wikipedia.org/wiki/Removable_singularity. Here I don't understand a fact that how can the power series $g(z)=c_2+c_3(z-a)+\cdots$ becomes holomorphic on $D$. Since $h$ is holomorphic at $a$ so it has a local power series expansion around $a$. That's fine. But then $\exists$ a $r>0$ such that $\forall z \in B(a;r)$, $h(z)= \sum_{n=0}^{\infty} c_n(z-a)^n$. But since $h(a)=h'(a)=0$ so $h(z) = (z-a)^2 \sum_{n=2}^{\infty} c_n (z-a)^{n-2}$, $\forall z \in B(a;r)$ i.e. $f(z)= \sum_{n=2}^{\infty} c_n (z-a)^{n-2}$ , $\forall z \in B'(a;r)$. But from here how can one say the holomorphy about $g$ on $D$. It is clear that $g$ is holomorphic on $B(a;r)$. But how can we assume more than that?
Please help me in understanding this.
Thank you in advance.