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\end{align*}  But since $u$ and $v$ are harmonic, $u_{xx}+u_{yy}=0$ and $v_{xx}+v_{yy}=0$. Using this fact and combining like terms gives us,  \begin{align*}  -u^{2}u^{2}_x+u^{2}v^{2}_x+v^{2}u^{2}_x-v^{2}_v^{2}_x-4uvu_xv_x-u^{2}u^{2}_y+u^{2}v^{2}_y+v^{2}u^{2}_y-v^{2}v^{2}_y-4uvu_yv_y $$-u^{2}u^{2}_x+u^{2}v^{2}_x+v^{2}u^{2}_x-v^{2}_v^{2}_x-4uvu_xv_x-u^{2}u^{2}_y+u^{2}v^{2}_y+v^{2}u^{2}_y-v^{2}v^{2}_y-4uvu_yv_y  \end{align*}  \end{proof}