= Solution
Write $f=u+iv$ and $z=x+iy$. For a positively oriented, piecewise smooth simple closed <contour> bounding a region where $u,v$ have continuous first <partial derivatives>, <Green's theorem> gives
$$
\begin{aligned}
\oint_C f(z)\,dz
&=\oint_C(u\,dx-v\,dy)+i\oint_C(v\,dx+u\,dy)\\
&=\iint_S(-v_x-u_y)\,dx\,dy+i\iint_S(u_x-v_y)\,dx\,dy.
\end{aligned}
$$
The <Cauchy-Riemann equations> are $u_x=v_y$ and $u_y=-v_x$, so both integrands vanish:
$$
\boxed{\oint_C f(z)\,dz=0.}
$$
This proves <Cauchy's integral theorem> under the standard regularity assumptions for <Green's theorem>. More general admissible <contours> follow by subdivision; every enclosed region must be free of singularities.
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