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\newcommand{\inner}[2]{\langle #1, #2 \rangle}
\DeclareMathOperator{\Div}{Div}

\begin{lemma}[cf.\ Lemma D.4.2]\label{lem:height}
Let $K$ be a number field and $P \in \mathbb{P}^n(\overline{K})$. Then
\begin{equation}\label{eq:bound}
  h(P) \le \frac{1}{d}\, h(\phi(P)) + C_\phi .
\end{equation}
\end{lemma}

\begin{proof}
I think the argument needs $H(P)$ to be \emph{integral}, which is not assumed:
\begin{align*}
  \inner{x}{y} &= \sum_{v \in M_K} n_v \log\max_i |x_i|_v \\
               &\ne \Div(f) \qedhere
\end{align*}
\end{proof}

By Lemma~\ref{lem:height} and \eqref{eq:bound}, the claim follows.
Physics also works: $\dv{f}{x}$, $\ket{\psi}$, and $\ce{H2O}$.

\begin{itemize}
  \item A typo: $\mathscr{O}_X$ should be $\mathcal{O}_X$.
  \item An unknown command: \foo.
\end{itemize}