  We claim that valuations \(w\) and \(w'\) are the same, which means that \eqref{eq9} and \eqref{eq11} are contradictory. Since both are \(x\)-variants of \(v\), it is enough to show that \(w(x) = w'(x)\), that is, \(v(f, \Gamma ) = v(f, \Delta )\), which is intuitively what local rigidity says. Proceeding formally, \(v(f, \Gamma ) = w(x) = w''(x)\) since \(w''\) is a \(y\)-variant of \(w\) and so they agree on \(x\). We also have \(v(f, \Delta ) = w''(y)\). And finally, \(w''(x) = w''(y)\) by \eqref{eq14}.