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We prove that the rescaled function $\\4{u}(x,s)=(T-t)^{-N/(N-2)}u(x/(T-t)^{-1/(N-2)},t)$, $s=-\\log (T-t)$, converges uniformly on $\\R^N$ to the rescaled Barenblatt solution $\\4{B}_{k_0}(x)=2(N-2)/(k_0+|x|^2)$ for some $k_0>0$ as $s\\to\\infty$. 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