The fundamental theorem of calculus for gauge integrals

A basic theorem of calculus for gauge integrals was proven by describing gauge integrals and incorporating the concept of the parametric derivative. One of the conditions for which F can be considered as a parametric derivative of f on {a,b} is that a differentiable function phi should exist and map an arbitrary interval {alpha, beta} onto {a,b}. The theorem stipulates that a function f(x) is gauge integrable over an interval {a,b} and that the integral from a to b of f(x)dx is equal to F(b) F(a), provided that f(x) is a parametric derivative of F(x) on the same interval.