= Solution
A real <harmonic function> on a planar open set is a twice continuously differentiable function satisfying $f_{xx}+f_{yy}=0$. The ordered pair $(f,g)$ consists of <harmonic conjugates> when $f+ig$ is holomorphic; the corresponding <Cauchy-Riemann equations> are $f_x=g_y$ and $f_y=-g_x$.
To prove <composition of harmonic conjugate pairs>, denote $P=p(u,v)$ and $Q=q(u,v)$, and assume the inner pair's image lies in the outer pair's domain. Their equations are $u_x=v_y$, $u_y=-v_x$, $p_u=q_v$ and $p_v=-q_u$. The <chain rule> gives
$$
P_x=p_uu_x+p_vv_x=p_uu_x-q_uv_x=Q_y,
$$
$$
P_y=p_uu_y+p_vv_y=-p_uv_x-q_uu_x=-Q_x.
$$
Thus the composite satisfies the Cauchy-Riemann equations. Its components are $C^2$, and differentiating gives $P_{xx}+P_{yy}=Q_{yx}-Q_{xy}=0$ and similarly $Q_{xx}+Q_{yy}=0$. \b[The composite pair is harmonic conjugate], including when the inner map has a critical point.
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