# $AdS 5 imes S^5$ mirror TBA equations from Y-system and discontinuity relations

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Using the recently proposed set of discontinuity relations we translate the AdS-CFT Y-system to TBA integral equations and quantization conditions for a large subset of excited states from the sl(2) sector of the $AdS 5 \times S^5$ string sigma-model. Our derivation provides an analytic proof of the fact that the exact Bethe equations reduce to the