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Averaging linear functional on the space continuous functions of the group ofdiffeomorphisms of interval is found. Amenability of several discrete subgroupsof the group of diffeomorphisms $\Diff^3([0,1])$ of interval is prove. Inparticular, a solution of the problem of amenability of the Thompson's group$F$ is given.

About amenability of subgroups of the group of diffeomorphisms of the interval [0,1]