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In particular, denoting by $L$ and $\\widetilde L$ the bi-Lipschitz constants of $u$ and $v$, with our construction one has $\\widetilde L \\leq C L^4$ (being $C$ an explicit geometrical constant). The same result was proved in 1980 by Tukia (see \\cite{Tukia}), using a completely different argument, but without any estimate on the constant $\\widetilde L$. 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