Let \(\Omega\subseteq\CC\) be a connected open set and let \(S\subseteq\Omega\) have a limit point in \(\Omega\).
If \(f\) and \(g\) are analytic on \(\Omega\) and \(f(z)=g(z)\) for all \(z\in S\), then \(f(z)=g(z)\) for all \(z\in\Omega\).
If \(F\colon\Omega\times\Omega\to\CC\) is holomorphic in each variable separately and \(F(z,w)=0\) for all \(z,w\in S\), then \(F(z,w)=0\) for all \(z,w\in\Omega\).
For (i), the zeros of the analytic function \(f-g\) have a limit point in the connected open set \(\Omega\), so \(f-g\equiv0\). For (ii), fix \(w\in S\); the function \(z\mapsto F(z,w)\) is holomorphic on \(\Omega\) and vanishes on \(S\), so by (i) it vanishes on \(\Omega\). Now fix \(z\in\Omega\); the function \(w\mapsto F(z,w)\) is holomorphic on \(\Omega\) and vanishes on \(S\), so by (i) it vanishes on \(\Omega\).