  The result is established for a Jordan measurable region with rectifiable
boundary. The integrand F for the new plane integral to be used is a function
of axis-parallel rectangles, finitely additive on non-overlapping ones, hence
unambiguously defined and additive on "figures" (i.e. finite unions of
axis-parallel rectangles). Define its integral over Jordan measurable S as the
limit of its value on the figures, which contain a subfigure of S and are
contained in a figure containing S, as the former/complements of the latter
expand directedly to fill out S/the complement of S. The integral over every
Jordan measurable region exists when additive F is "absolutely continuous" in
the sense of converging to zero as the area enclosed by its argument does, or
with F the circumferential line integral $\oint P dx + Q dy$ for $P$, $Q$
continuous at the rectifiable boundary of S and integrable along axis -
parallel line segments. Thus the equality of this area integral with the line
integral around the boundary, to be proved, follows for the various integrals
of divergence presented in: Pfeffer, W.F. The Riemann Approach to Integration,
Cambridge Univ. Press, New York, 1993.
