# Clarifications on a proof on analytic continuations

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.

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.