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We prove the Trung's conjecture about Segre's upper bound for s equimultiplefat points not on a linear (r-1)-space, s\le r+3, by algebraic method used in[3]. This method also may used to research other cases of fat points.

On the regularity index of $s$ equimultiple fat points not on a linear $(r-1)$-space, $s \le r+3$