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Lemma 28 ([Two-size determinant]). Let ν'>0 be the exponent in Corollary 25 ([Contraction of antisymmetric products]). There are constants C,c>0 such that, for sufficiently large m , 100m≤R≤R'≤2R , and almost every 0<|y|≤m , |K_R(y)f(R')-K_ R' (y)f(R)| ≤C(m/R)^ 2λ+ν' +Ce^ -cm^2 . The constants are independent of the distance from y to the origin.