Q: Extending Green's theorem from very special regions to more general regions Green's theorem
Let $C$ be a positively oriented and consists of a finite union of disjoint,piecewise smooth simple closed curve in a plane, and let $D$ be the region bounded by $C$. If $P$ and $Q$ are functions of $(x, y)$ defined on an open region containing $D$ and have continuous partial derivatives there, then
$$\oint_C (Pdx+Qdy)=\iint_D (\frac{\partial Q }{\partial x}-\frac{\partial P}{\partial y})dxdy$$where the path of integration along $C$ is anticlockwise.
In almost all textbooks, authors only proved it when $D$ is a simple region or the region $D$ can be cut up into finitely many simple regions without common interiors.My questions is how to
 generalize those special regions to more general regions as Green's theorem gives? I need a rigorous complete proof.Likewise Green's theorem,the divergence theorem and Stokes's theorem have the same situation requiring some rigorous proofs from very special regions to more general regions.
