= Solution
The allowed theorem makes $f'$ <holomorphic> and hence <continuous>; applying it repeatedly gives <continuous> <derivatives> of every order. Since $u_x=\Re f'$, $v_x=\Im f'$, $u_y=-\Im f'$ and $v_y=\Re f'$, the real and imaginary parts have <continuous> second <partial derivatives>. Differentiate the <Cauchy-Riemann equations>[Cauchy–Riemann equations] and commute mixed <derivatives>:
$$
u_{xx}+u_{yy}=(v_y)_x-(v_x)_y=0,
\qquad
v_{xx}+v_{yy}=-(u_y)_x+(u_x)_y=0.
$$
Thus $\boxed{\Delta u=\Delta v=0}$, with the required $C^2$ regularity, proving harmonicity.
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