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We present a new form and a short full proof of explicit two-sided estimatesfor the distribution function F_{n,p}(x) of the binomial law from the paperpublished by D.Alfers and H.Dinges in 1984. These inequalities are universal(valid for all binomial distribution and all values of argument) and exact(namely, the upper bound for F_{n,p}(k) is the lower bound for F_{n,p}(k+1)).By means of such estimates it is possible to bound any quantile of the binomiallaw by 2 subsequent integers.

A full proof of universal inequalities for the distribution function of the binomial law