
\begin{proof}~
\begin{align*}
  \operatorname{Vol}(\OO^{\times}, dx^{\times}) &= \frac{1}{1 - q^{-1}} \int_{\OO^{\times}} \frac{dx}{|x|} \\
  &= \frac{1}{1 - q^{-1}} \int_{\OO^{\times}} dx \\
  &= \frac{1}{1 - q^{-1}} \operatorname{Vol}(\OO^{\times}, dx) \\
  &= \frac{1}{1 - q^{-1}} \big( \operatorname{Vol}(\OO, dx) -  \operatorname{Vol}(\varpi \OO, dx) \big)\\
  &= \frac{1}{1 - q^{-1}} \big( \operatorname{Vol}(\OO, dx) -  |\varpi|\operatorname{Vol}(\OO, dx) \big)\\
  &= \operatorname{Vol}(\OO, dx)
\end{align*}
\end{proof}


\begin{proof}~
\begin{align*}
  \operatorname{Vol}(\OO^{\times}, dx^{\times}) &= \frac{1}{1 - q^{-1}} \int_{\OO^{\times}} \frac{dx}{|x|} \\
  &= \frac{1}{1 - q^{-1}} \int_{\OO^{\times}} dx \\
  &= \frac{1}{1 - q^{-1}} \operatorname{Vol}(\OO^{\times}, dx) \\
  &= \frac{1}{1 - q^{-1}} \big( \operatorname{Vol}(\OO, dx) -  \operatorname{Vol}(\varpi \OO, dx) \big)\\
  &= \frac{1}{1 - q^{-1}} \big( \operatorname{Vol}(\OO, dx) -  |\varpi|\operatorname{Vol}(\OO, dx) \big)\\
  &= \operatorname{Vol}(\OO, dx)
\end{align*}
\end{proof}
