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&=\frac{1}{(u^{2}+v^{2})^{2}}[(uu_x + uu_{xx} +v_xv_x + vv_{xx})(u^{2}+v^{2})-(uu_x + vv_x)(2uu_x+2vv_x)\\   &+ (uu_y + uu_{yy} +v_yv_y + vv_{yy})(u^{2}+v^{2})-(uu_y + vv_y)(2uu_y+2vv_y)]  \end{align*}  We will show that the numerator expression will be equal to zero and thus the Laplace equation will equal zero. zero.\  \begin{align*}  \end{align*}  \end{proof}