vNM Expected Utility Theorem Proof
Let's write
\begin{align}
  x&amp;=ap+(1-a)r\\
  y&amp;=aq+(1-a)r
\end{align}
You want to show that $x\sim y$. But
\begin{equation}
x\sim y\quad\Leftrightarrow\quad x\not\succ y\;\text{ and }\;y\not\succ x.
\end{equation}
So the given proof tries to establish that $x\not\succ y$ (and in a similar way, $y\not\succ x$) is true. 

To show this, use proof by contradiction. Suppose the contrary is true: $x\succ y$, and try to derive a contradiction. This is what the given proof is doing.