ω1...,ω"m". Because of this high order vanishing it can be shown that a high-order derivative of "F" takes a value of small size one of the ω"i"s, "size" here referring to an algebraic property of a number. Using the maximum modulus principle Lang also found a separate way to estimate the absolute values of derivatives of "F", and using standard results comparing the size of a number and its absolute value he showed that these estimates were contradicted unless the claimed bound on "m" holds.