\subsection{Formal proofs}
Given a language like $\cL = \{ \oplus, \otimes, +, \times \}$, and derivation rules
\begin{multicols}{2}
    \begin{itemize}
        \item $\alpha$: If $+$, then $\otimes$
        \item $\beta$: If $+$, then $\times$
        \item $\gamma$: If $\otimes$ and $\times$, then $\oplus$
        \item $\delta$: $+$ holds
    \end{itemize}
    or displayed using graphical notation:
    \begin{align*}
        \frac{+}{\otimes} \; \alpha
        \qquad \frac{+}{\times} \; \beta \\
        \frac{\otimes \quad \times}{\oplus} \; \gamma
        \qquad \frac{}{+} \; \delta
    \end{align*}
\end{multicols}

Rules like $\delta$ above are also commonly referred to as an \textit{axiom}.

To prove $\oplus$ in this language, we can either write the following or draw a derivation tree:
\begin{multicols}{2}
    \begin{itemize}
        \item $+$ holds by $\delta$
        \item $\otimes$ holds by $\alpha$ with 1.
        \item $\times$ holds by $\beta$ with 1.
        \item $\oplus$ holds by $\gamma$ with 2 and 3
    \end{itemize}
    Or as derivation tree
    \[
        \begin{prooftree}
            % Left branch
            \infer0[$\delta$]{+}
            \infer1[$\alpha$]{\otimes}
            % Right branch
            \infer0[$\delta$]{+}
            \infer1[$\beta$]{\times}
            \infer2[$\gamma$]{ \oplus }
        \end{prooftree}
    \]
\end{multicols}