vNM Expected Utility Theorem Proof

Question

I have been asked by my professor for Decisions Theory to refer to David M. Kreps' Notes on Theory of Choice for the proof of the vNM theorem of expected utility. Before the proof of the theorem actully starts, there are two Lemmas stated which are used in the proof of the theorem. I am unable to understand the proof of one part of one of the Lemmas. Specifically, the proof for part (c) of the below Lemma. I understand the intuition behind it. Its basically the indpendence axiom but with equivalence relation, however I do not get the proof at all. Any help is much apprciated.

And this is the proof for part (c). When the author talks about applying (5.2), he is referring to the independence axiom and when he talks about applying (5.3) he is referring to the Archemedean or the continuity axiom.

Answer 1

Let's write \begin{align} x&=ap+(1-a)r\\ y&=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.