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I'm studying a proof on this book http://www.math.ucsd.edu/~atparris/papers/book.pdf I can't copy here the proof I'm studying because it is a little bit long. I only need help on a little statement. I don't understand in the page 180 of the book why we can choose $\alpha,\alpha^\prime\in\{\alpha_1,\alpha_2,\alpha_3\}$ distinct such that $x_\alpha-x_{\alpha^\prime}\not\in\{x_\beta-x_{\beta^\prime},x_{\beta^\prime}-x_\beta\}$.

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    @Jacob: If you click on the "edited xxx ago", you can see the changes made in each edit. For instance, if you click on [the corresponding link](http://math.stackexchange.com/posts/45345/revisions) for the other question, you will see the only change I made was to the title.2011-06-27

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If you accept that there are at least two positive (distinct) values for x_\alpha - x_{\alpha'} then the claim follows since \{x_\beta - x_{\beta'}, x_{\beta'} - x_\beta\} contains at most one positive value. Indeed, take one of the positive values for x_\alpha - x_{\alpha'}. If it is not in \{x_\beta - x_{\beta'}, x_{\beta'} - x_\beta\} then chose it, otherwise chose the other one.

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    wlog assume that x_{\alpha_1} < x_{\alpha_2} < x_{\alpha_3}. Then $x_{\alpha_3} - x_{\alpha_1}$ and $x_{\alpha_3} - x_{\alpha_2}$ are two distinct values.2011-06-08