= Solution
At $z=x+iy$, compute the <complex derivative> first along real increments and then along imaginary increments. <Holomorphy> gives
$$
f'(z)=u_x+iv_x=\frac{u_y+iv_y}{i}=v_y-iu_y.
$$
Equating real and imaginary parts proves the <Cauchy-Riemann equations>:
$$
\boxed{u_x=v_y,\qquad u_y=-v_x.}
$$
The partial derivatives exist because these directional difference quotients are limits of the <holomorphic> <derivative>. This argument proves the necessary equations; it does not claim that their pointwise existence alone is sufficient for <holomorphy>.
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