Kitaptaki ifadeler aşağıdaki gibidir:
Theorem 1.1 Let $U$ be a connected open set, and let $f$ be a holomorphic function on $U$. If $f'=0$ then $f$ is constant.
Proof. Let $\alpha, \beta$ be two points in $U$, and suppose first that $\gamma$ is a curve joining $\alpha$ to $\beta$, so that $$\gamma(a)=\alpha \hspace{20px} \mbox{and} \hspace{20px} \gamma(b)=\beta.$$
The function $$t\mapsto f(\gamma(t))$$ is differentiable, and by the chain rule, its derivative is $$f'(\gamma(t))\gamma'(t)=0.$$ Hence this function is constant, and therefore $$f(\alpha)=f(\gamma(a))=f(\gamma(b))=f(\beta).$$Next, suppose that $\gamma=\{\gamma_1, \dots, \gamma_n\}$ is a path joining $\alpha$ to $\beta$, and let $z_j$ be the end point of $\gamma_j$, putting $$z_0=\alpha \hspace{20px} \mbox{and} \hspace{20px} z_n=\beta.$$ By what we have just proved, we have $$f(\alpha)=f(z_0)=f(z_1)=f(z_2)=\dots =f(z_n)=f(\beta),$$ thereby proving the theorem.