Green's theorem says the following:
Let $R$ be a region in $\mathbb{R}^2$ whose boundary is a simple closed curve $C$ which is piecewise smooth. Let $\mathbf{f}(x,y)=P(x,y)\mathbf{i}+Q(x,y)\mathbf{j}$ be a smooth vector field defined on both $R$ and $C$. Then
$$ \oint_C P\text{d}x + Q\text{d}y={\int\int}_R\left(\dfrac{\partial Q}{\partial x}-\dfrac{\partial P}{\partial y}\right)\text{d}x\text{d}y.\tag{*} $$
However, the proof that I have seen in many books only shows that this is true for a simple region $R$. That is, regions $R$ such that any line parallel to the $x$-axis or $y$-axis only intersects $R$ in a single line segment. However, Green's theorem works for all regions $R$ satisfying its hypotheses - not just simple ones.
Green's theorem for union of simple regions
The image above shows a more general region which is the union of simple regions $D_1$, $D_2$, and $D_3$. I understand how Green's theorem can be shown to work for this specific case. We can apply (*) to each of the simple regions $D_i$. Summing the right hand sides up will give us the required double integral over the whole region $D=D_1+D_2+D_3$. Summing the left hand sides up will also give us the required line integral over $C=C_1+C_2+C_3$ because the line integral over $D_1$ has a $-E_1-E_2$ component, while the line integrals over $D_2$ and $D_3$, respectively, have an $E_1$ and an $E_2$ component - so everything cancels nicely.
However, I do not see how to generalize this approach to all regions of this kind. I would like a rigorous proof that Green's theorem works for regions that are the union of simple regions.