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\begin{document}
Idiotites Analogiwn
\begin{eqnarray}
\frac{a}{b} &=& \frac{y}{z} \iff \frac{a}{z} &=& \frac{b}{y}
\end{eqnarray}
Dunameis
\begin{eqnarray}
a^0 &=& 1 \\
a^1 &=& a \\
a^ya^z &=& a^{y+z} \\
\frac{a^y}{a^z} &=& a^{y-z} \\
(a^y)^z &=& a^{yz} \\
a^{-y} &=& \frac{1}{a^y} \\
(\frac{a}{b})^{-y} &=& (\frac{b}{a})^y \\
\\
a^yb^y &=& (ab)^y \\
\frac{a^y}{b^y} &=& (\frac{a}{b})^y
\end{eqnarray}
\end{document}