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The proof is fine. I just gave an overview of it to give you an idea of how it worked. Typing out the whole proof with all it's rigour would have been annoying. Besides, I figured you could figure it out. I guess I'm not used to such an adversarial attitude...
For instance, it's not that you can only have one or two "partitions," it's that you are forced to have, at least, one or two partitions. The addition of extra partitions doesn't remove the properties of the first few partitions that the proof was relying on.
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