Let $f:C_0\to \mathbb{C}$ be holomorphic, with $c_0\in C_0$ the center of the disk. Let ${\gamma }_1,{\gamma }_2:[t_0,t_1]\to U$ be a continuous path which starts at $c_0$ and end at the same point $z\in U$. Suppose $f$ can be analytically continued along both ${\gamma }_1$ and ${\gamma }_2$. Then the results coincide if 1;; ${\gamma }_1\simeq {\gamma }_2$(they are homotopic)

PROOF:

1. Fix a homotopy $H:{\gamma }_1\simeq {\gamma }_2$ ($H:[t_0,t_1]\times [0,1]\to U$).
2. Show that the curves $H(t,s)$ produce the same analytic continuation for a small variation in $s$.
3. Then use compactness of $[0,1]$.
4. I think here we are also using some uniform continuity property or something.
5. Choose disks of the new chain to be contained in the disks of the original chain

STEP 1: Cover path with epsilon neighborhoods, and respective disks containing this neighborhood.

STEP 2: Choose a $H(t,s)$ such that it is contained in the epsilon neighborhood of $H(t,s_0)$. (Here we choose a delta)

STEP 3: Somehow use compactness of sets of the form $\{s_0\}\times [t_0,t_1]$.

STEP 4: Use containment of the disk chains to finish the result