Translation invariant Gibbs states for the Ising model

We prove that all the translation invariant Gibbs states of the Ising model are a linear combination of the pure phases $\mu^+_\gb,\mu^-_\gb$ for any $\gb \not = \gb_c$. This implies that the average magnetization is continuous for $\gb >\gb_c$. Furthermore, combined with previous results on the slab percolation threshold this shows the validity of Pisztora's coarse graining up to the critical temperature