Gaufis integral theorem in two dimensions

To prove Theorem B.5 in the subsequent Appendix B.2.4, we first need to prove another theorem from Gaufi. Suppose we are given two fimctions / (x, y) and g (x, y). We assume both / (x, y) and g (x, y) as well as their first derivatives to be continuous in a simply connected domain D where D is bounded by a piece-wise continuous curve C. Specifically we assume D to bo roprosonted by the set 

Noticing that the function g x, y) is the antiderivative of the integrand on the right side of the previous expression, rue have 

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