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\begin{proposition}\label{trivial}
$x = x$.
\end{proposition}
\begin{proposition}\label{irrelevant}
$z = z$.
\end{proposition}
\begin{proposition}\label{alsotrivial}
$y = y$.
\end{proposition}
\begin{proof}
\begin{align*}
y
&= y
\\
&= y
\explanation{by \cref{trivial}}
\end{align*}
\end{proof}
\begin{proposition}\label{trivial_biconditionals}
$y = y$.
\end{proposition}
\begin{proof}
\begin{align*}
y = y
&\iff \top
\\
&\iff y = y
\explanation{by \cref{trivial}}
\end{align*}
\end{proof}
\begin{proposition}\label{bounded_calc}
For all $x \in A$ we have $x = x$.
\end{proposition}
\begin{proof}
For all $x \in A$ we have
\begin{align*}
x
&= x
\end{align*}
\end{proof}
\begin{proposition}\label{bounded_calc_such_that}
For all $x \in A$ such that $x = x$ we have $x = x$.
\end{proposition}
\begin{proof}
For all $x \in A$ such that $x = x$, we have
\begin{align*}
x
&= x
\end{align*}
\end{proof}
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