I'm confused about a Lemma in Ian Stewart & David Tall's Complex Analysis:
Lemma 14.1: Let $f , g$ be analytic on a domain $D$. Suppose that $P$ and $Q$ are open sets and $∅\ne P ∩ Q ⊆ D$. Suppose that $p(z) = q(z)$ for all $z ∈ P ∩ Q$, and $f, g$ are analytic functions defined on $D$ such that $f(z) = p(z)$ for $z ∈ P$ and $g(z) = q(z)$ for $z ∈ Q$. Then $f (z) = g(z)$ for all $z ∈ D$.
First question: is there an error in the first sentence? Should it not be "let $p,q$ be analytic on a domain $D$"?
Second question: what is the point of introducing both pairs $p,q$ and $f,g$ of functions? It seems to me the result can be restated as follows:
Lemma 14.1: Let $p,q$ be analytic on a domain $D$. Suppose that $P$ and $Q$ are open sets and $∅\ne P ∩ Q ⊆ D$. Suppose that $p(z) = q(z)$ for all $z ∈ P ∩ Q$, then $p=z$ on $D$.