
\documentstyle{csart}

\begin{document}

Let
\begin{eqnarray*}
P_1 (x) & = &  \exists y \cdot (x_1 x_2 = y_1 y_2) \land x \neq y \\
P_2 (x) & = & (\exists y \cdot (x_1 + x_2 = y_1 + y_2) \land x \neq y) \land
              (\forall y \cdot (x_1 + x_2 = y_1 + y_2) \Rightarrow P_1 (y)) \\
P_3 (x) & = & \forall y \cdot ((x_1 x_2 = y_1 y_2) \land P_2 (y))
                               \Rightarrow x = y \\
P_4 (x) & = & \forall y \cdot ((x_1 + x_2 = y_1 + y_2) \land P_3 (y))
                               \Rightarrow x = y
\end{eqnarray*}
Find (the unique) $x$ such that
\begin{displaymath}
   P_1 (x) \land P_2 (x) \land P_3 (x) \land P_4 (x)
\end{displaymath}
Here all variables are quantified over
{\em unordered} pairs of integers in the range [2, 99] (take $x_1 \leq x_2$
say).


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