Clarifications on a proof on analytic continuations

Question

In the book Function of one complex variable of John B. Conway, the proof of this lemma on analytic continuations :

Lemma on Analytic Continuations

rest of the proof :

Proof of the preceding lemma

states that if ($f_t, D_t$) is an analytic continuation there exist an $\epsilon$ such that for every path $\epsilon$ near of the first and analytic continuation ($g_t, B_t$) along this path starting at $a$ and such that $[f_0]_a = [g_0]_a$ then under some constraints,if $G$, the intersection of $B_t, B_s, D_s, D_t$ is not empty, knowing that the center of $B_s$ is in $G$ and that G admits a limit point in $B_s \cap D_s$ is enough to affirm that on $B_s \cap D_s$, $f_s$ and $g_s$ agree (by definition of $T$). I don't understand how to deduce this fact, any help or hints would be very appreciated.

Answer 1

I found the answer : Two analytic function are equal on a region of $\mathbb{C}$ iff the set of points on which they are equal admits a limit point in this region.

This fact is a corollary of the fact that an analytic function is null on a region if its zero set admits a limit point in this region.