$ A=(ax_{1}+bx_{2}+cx_{3}+dx_{4})^{2}\leq(ab+cd)(\frac{(ax_{1}+bx_{2})^{2}}{ab}+\frac{(cx_{3}+dx_{4})^{2}}{cd})$ $B=(ay_{4}+by_{3}+cy_{2}+dy_{1})^{2}\leq(ab+cd)(\frac{(ay_{4}+by_{3})^{2}}{ab}+\frac{(cy_{2}+dy_{1})^{2}}{cd})$ $\frac{(ax_{1}+bx_{2})^{2}}{ab}\leq \frac{(ax_{1}+bx_{2})^{2}+(ay_{1}-by_{2})^{2}}{ab}=\frac{a^{2}+b^{2}+2ab(x_{1}x_{2}-y_{1}y_{2})}{ab}$ similarly $\frac{(cy_{2}+dy_{1})^{2}}{cd}\leq\frac{c^{2}+d^{2}+2cd(y_{1}y_{2}-x_{1}x_{2})}{cd}$ do it for two other terms and use $ab+cd=1$ then $A+B\leq2(\frac{a^{2}+b^{2}}{ab}+\frac{c^{2}+d^{2}}{cd})$

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