From the differentiability of "f" and "g" on formula_67, Cauchy's mean value theorem ensures that for any two distinct points "x" and "y" in formula_67 there exists a formula_73 between "x" and "y" such that formula_77. Consequently, formula_78 for all choices of distinct "x" and "y" in the interval. The value "g"("x")-"g"("y") is always nonzero for distinct "x" and "y" in the interval, for if it was not, the mean value theorem would imply the existence of a "p" between "x" and "y" such that "g' "("p")=0.