= Free conjugate bases in a rank-two free group
{title2=$b^{-i}ab^i$}
In $F(a,b)$, any set of distinct conjugates $b^{-i}ab^i$ freely generates its subgroup. Combine adjacent powers of the same conjugate in a reduced word. After expansion, distinct successive indices leave nonzero intervening powers of $b$, and the intervening nonzero powers of $a$ cannot cancel in the <free product> $\langle a\rangle*\langle b\rangle$. The same argument works with $a,b$ interchanged and for finite or countably infinite sets of indices.
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