<html><head><meta name="color-scheme" content="light dark"></head><body><pre style="word-wrap: break-word; white-space: pre-wrap;">\documentclass{article}

\usepackage{proof}
\usepackage{utf8}

\begin{document}

Proof of $âŠ¢ ((P âˆ§ Q) â†’ R) â†’ (P â†’ (Q â†’ R))$:
\begin{displaymath}
\infer[â†’ intro]{âŠ¢ ((P âˆ§ Q) â†’ R) â†’ (P â†’ (Q â†’ R))}{
  \infer[â†’ intro]{(P âˆ§ Q) â†’ R âŠ¢ P â†’ (Q â†’ R)}{
    \infer[â†’ intro]{(P âˆ§ Q) â†’ R, P âŠ¢ Q â†’ R}{
      \infer[â†’ elim]{(P âˆ§ Q) â†’ R, P, Q âŠ¢ R}{
        \infer[assum]{\dots, (Pâˆ§Q)â†’R âŠ¢ (Pâˆ§Q) â†’ R}{}
        &amp;
        \infer[âˆ§ intro]{\dots âŠ¢ Pâˆ§Q}{
          \infer[assum]{\dots, P âŠ¢ P}{}
          &amp;
          \infer[assum]{\dots, Q âŠ¢ Q}{}
        }
      }
    }
  }
}
\end{displaymath}

\newpage{}

Proof of $âŠ¢ Â¬(Pâˆ§Q) â†’ (Â¬P âˆ¨ Â¬Q)$:
\begin{displaymath}
\infer[â†’ intro]{âŠ¢ Â¬(Pâˆ§Q) â†’ (Â¬P âˆ¨ Â¬Q)}{
  \infer[âˆ¨ elim]{Â¬(Pâˆ§Q) âŠ¢ Â¬P âˆ¨ Â¬Q}{
    \infer[excl~mid]{\cdots âŠ¢ Pâˆ¨Â¬P}{}
    &amp;
    \infer[âˆ¨ elim]{\cdots, P âŠ¢ Â¬P âˆ¨ Â¬Q}{(see~below)}
    &amp;
    \infer[âˆ¨ intro]{\cdots, Â¬P âŠ¢ Â¬P âˆ¨ Â¬Q}{
      \infer[assum]{\cdots, Â¬P âŠ¢ Â¬P}{}
    }
  }
}
\end{displaymath}

\begin{displaymath}
\infer[âˆ¨ elim]{\cdots, P âŠ¢ Â¬P âˆ¨ Â¬Q}{
  \infer[excl~mid]{\cdots âŠ¢ Qâˆ¨Â¬Q}{}
  &amp;
  \infer[absurd]{\cdots, Q âŠ¢ Â¬P âˆ¨ Â¬Q}{(see~below)}
  &amp;
  \infer[âˆ¨ intro]{\cdots, Â¬Q âŠ¢ Â¬P âˆ¨ Â¬Q}{
    \infer[assum]{\cdots, Â¬Q âŠ¢ Â¬Q}{}
  }
}
\end{displaymath}

\begin{displaymath}
\infer[absurd]{\cdots, Q âŠ¢ Â¬P âˆ¨ Â¬Q}{
  \infer[âˆ§ intro]{\cdots âŠ¢ Pâˆ§Q}{
    \infer[assum]{\dots,P âŠ¢ P}{}
    &amp;
    \infer[assum]{\dots,Q âŠ¢ Q}{}
  }
  &amp;
  \infer[assum]{\dots, Â¬(Pâˆ§Q) âŠ¢ Â¬(Pâˆ§Q)}{}
}
\end{displaymath}
\end{document}

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