Consider the following statement:

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> If $f\_1(x\_1^\*) = f\_n(x\_n^\*)$ for all $n \geq 1$, then $x\_1^\* = x\_n^\* = 1$ for all $n \geq 1$.
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> 
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I would like to write down the contrapositive statement. Here is my thinking:

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> Suppose that there exists an $n \geq 1$ such that $x\_n^\* \neq 1$, then $f\_1(x\_1^\*) \neq f\_n(x\_n^\*)$.
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> 
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My question is the following: Do I really need to include the quantifier for the statement after "then"? ;i.e., do I have to say
$f\_1(x\_1^\*) \neq f\_n(x\_n^\*)$ for **some** $n$ ? If so, then I think I have "two $n$" flows around which confuses me a lot. Or can I just recycle the $n$ used in the sentence of "suppose..." Is this the correct way to think of?

Any suggestion is appreciated.