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We prove that for every equivalence relation on a barrier on the space$\langle FIN^{[\infty]}_k,\leq,r\rangle$, for any $k$, there exists $Y\inFIN_k^{[\infty]}$ so that the restriction of the coloring on $\langle Y\rangle$is canonical.

Canonical equivalence relations on fronts on $FIN_k$