For any two complex numbers \(z_1\) and \(z_2\), prove that \[ \text{Re} \left(\overline{z_1} z_2\right) = \text{Re} \left(z_1\right) \text{Re} \left(z_2\right) - \text{Im} \left(z_1\right) \text{Im} \left(z_2\right) \]