Let $n\ge2$. For any two $n$-vectors $\vec x=(x_1,\ldots,x_n)$ and $\vec y=(y_1,\ldots,y_n)$, we define $$f\left(\vec x,\vec y\right)=x_1\overline{y_1}-\sum_{i=2}^nx_i\overline{y_i}.$$Prove that if $f\left(\vec x,\vec x\right)\ge0$, and $f\left(\vec y,\vec y\right)\ge0$, then $\left|f\left(\vec x,\vec y\right)\right|^2\ge f\left(\vec x,\vec x\right)f\left(\vec y,\vec y\right)$.