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For a NIP theory T, a sufficiently saturated model 
${\mathfrak C}$ of T, and an invariant (over some small subset of 
${\mathfrak C}$) global type p, we prove that there exists a finest relatively type-definable over a small set of parameters from 
${\mathfrak C}$ equivalence relation on the set of realizations of p which has stable quotient. This is a counterpart for equivalence relations of the main result of [2] on the existence of maximal stable quotients of type-definable groups in NIP theories. Our proof adapts the ideas of the proof of this result, working with relatively type-definable subsets of the group of automorphisms of the monster model as defined in [3].
