I am looking for a reference for the following

**Fact 1:** if $A$ and $B$ are finitely generated subgroups of infinite index in a finitely generated free group $F$ then there exists $f \in F$ such that $fAf^{-1} \cap B=\{1\}$.

I know several proofs for this, but surely this should be classical, so a reference would be desirable. Modulo a known result of W. Neumann (Neumann, Walter D. `On intersections of finitely generated subgroups of free groups.' Groups—Canberra 1989, 161–170, Lecture Notes in Math., 1456, Springer, Berlin, 1990), this fact would immediately follow from the following

**Fact 2:** with the assumptions of Fact 1, $F$ cannot be covered by finitely many double cosets of the form AgB, $g \in F$.

I proved a generalization of Fact 2 for hyperbolic groups some time ago, but I do not know of an explicit (classical) reference for it.

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