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the Pythagorean property are retained, and noting that for any triple larger than exactly one of the inverse transition matrices yields a new triple with all positive entries (and a smaller hypotenuse). By induction, this new valid triple itself leads to exactly one smaller valid triple, and so forth. By the finiteness of the number of smaller and smaller potential hypotenuses, eventually is reached. This proves that does in fact occur in the tree, since it can be reached from by reversing the steps; and it occurs uniquely because there was

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