Composition of harmonic conjugate pairs

If $u+iv$ and $p+iq$ are holomorphic functions, with the image of the first inside the domain of the second, then $p(u,v)+iq(u,v)$ is holomorphic by the chain rule. Directly, their Cauchy-Riemann equations imply $[p(u,v){]}_x=[q(u,v){]}_y$ and $[p(u,v){]}_y=-[q(u,v){]}_x$. Differentiating these relations proves harmonicity of both parts. No nonvanishing derivative hypothesis is required.