diff --git a/src/op/banach/fc.tex b/src/op/banach/fc.tex
index 5bb2472..fd61bbb 100644
--- a/src/op/banach/fc.tex
+++ b/src/op/banach/fc.tex
@@ -28,7 +28,7 @@
     The mapping $f \mapsto f(x)$ is the \textbf{holomorphic functional calculus} of $x$. 
 \end{definition}
 \begin{proof}[Proof, {{\cite[Proposition I.2.7]{Takesaki1}}}. ]
-    (Definition): Let $U, V \in \cn_\complex(\sigma_A(x))$ such that $\ol V \subset U$, then by \autoref{proposition:existence-curves}, there exists closed rectifiable curves $\seqf{\gamma_j}$ on $V \setminus \sigma_A(x)$ such that 
+    (Definition): Let $U, V \in \cn_\complex(\sigma_A(x))$ such that $\ol V$ is a compact subset of $U$, then by \autoref{proposition:existence-curves}, there exists closed rectifiable curves $\seqf{\gamma_j}$ on $V \setminus \sigma_A(x)$ such that 
     \begin{enumerate}[label=(\alph*)]
         \item For all $f \in H(V; \complex)$ and $z_0 \in \sigma_A(x)$, 
         \[