Suppose that all of
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,
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, and
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are distinct.
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,
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, and
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. Multiplying these all together yields
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.
Rearrange to get
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(possible since
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. Since
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is a polynomial with integer coefficients,
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. Since their product is one,
If one of
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,
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, and
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is -1 (without loss of generality, let us suppose that
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then
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. But
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and
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, so
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, yielding
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, contradicting our assumption that all the variables were distinct.
It follows that each of
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,
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, and
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is equal to 1, so we have
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,
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, and
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. Substituting,
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,
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, and
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. Rearrange to get
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,
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, and
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. These equalities quickly yield
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, which again contradicts our assumption that
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are distinct.