complex analysis - Theorems on the Cauchy integral operator

While reading on Cauchy's integral formula, I found the following two theorems on the Cauchy integral operator:

The first theorem states: Let $f$ be a complex function that is holomorphic in a region $U$. Let $\Delta_r(c)$ be a disk, which along with its boundary $\delta \Delta_r(c)$ is contained in $U$. For all $z \in \Delta_r(c)$ the following is true:
\begin{equation*}
f(z) = \frac{1}{2 \pi i} \int_{\delta \Delta_r(c)} \frac{f(\zeta)}{\zeta - z} d\zeta
\end{equation*}

The second theorem states: Let $f: \overline{\Delta_1(0)} \rightarrow \mathbb{C}$ be a continous function which is also holomorphic in $\Delta_1(0)$. It holds for every $z \in \Delta_1(0)$ that
\begin{equation*}
f(z) = \frac{1}{2 \pi i} \int_{\delta \Delta_1(0)} \frac{f(\zeta)}{\zeta - z} d\zeta
\end{equation*}

The second theorem is given without proof. My question is whether the second statement holds for general disks and whether it can be proven using the first one?