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This should follow from the Salamander lemma, which, as you found out in your previous MO question,previous MO question, can be proved without the use of elements. It is a nice exercise, so I won't spell it out for you.

This should follow from the Salamander lemma, which, as you found out in your previous MO question, can be proved without the use of elements. It is a nice exercise, so I won't spell it out for you.

This should follow from the Salamander lemma, which, as you found out in your previous MO question, can be proved without the use of elements. It is a nice exercise, so I won't spell it out for you.

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Chris Schommer-Pries
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This should follow from the Salamander lemma, which, as you found out in your previous MO question, can be proved without the use of elements. It is a nice exercise, so I won't spell it out for you.