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I need a proof of this well known fact:

If a group $G$ is given as a quotient of a free group $F/R$ then its pro-finite completion is given by the quotient $\hat{F}/ \bar{R}$ where $\bar{R}$ is the closure of $R$ in $\hat{F}$. (since $F$ embeds into $\hat{F}$ densely we can think of $R$ inside $\hat{F}$)

Thanks in advance.

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    The reason for which you can think of $R$ inside of $\hat F$ is not that $F$ embeds in $\hat F$ densely but simply that it injects there.2010-10-31

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