  We prove the Eigenstate Thermalisation Hypothesis (ETH) for local observables
in a typical translation invariant system of quantum spins with mean field
interaction. This mathematically verifies the observation made in [L.Santos and
M.Rigol, Phys.Rev.E 82, 031130 (2010,
https://journals.aps.org/pre/abstract/10.1103/PhysRevE.82.031130)] that ETH may
hold for systems with additional translation symmetries for a naturally
restricted class of observables. We also present numerical support for the same
phenomenon for Hamiltonians with local interactions.
