1. Agatha hates Charles
\\[0ex]2. Agatha hates Agatha
\\[0ex]3. $\forall$$p$:Person. $\neg$$p$ is richer than Agatha $\Rightarrow$ The Butler hates $p$
\\[0ex]4. $\forall$$p$:Person. Agatha hates $p$ $\Rightarrow$ $\neg$Charles hates $p$
\\[0ex]5. $\forall$$p$:Person. Agatha hates $p$ $\Rightarrow$ The Butler hates $p$
\\[0ex]6. $\forall$$p$:Person. $\neg$$p$ hates Agatha $\vee$ $\neg$$p$ hates The Butler $\vee$ $\neg$$p$ hates Charles
\\[0ex]7. $\forall$$p$, $q$:Person. $p$ kills $q$ $\Rightarrow$ $\neg$$p$ is richer than $q$
\\[0ex]8. $\forall$$p$, $q$:Person. $p$ kills $q$ $\Rightarrow$ $p$ hates $q$
\\[0ex]$\vdash$  $\neg$The Butler kills Agatha \& $\neg$Charles kills Agatha