CVP5

Show that $\overline{z_1z_2}=\overline{z_1}\,\overline{z_2}\,$
Let $z_1 = x_1 + iy_1\,$ and $z_2 = x_2 + iy_2\,$. Then
 $\overline{z_1z_2}$ $=\overline{(x_1 + iy_1)(x_2 + iy_2)}$ $= \overline{x_1x_2 + x_1y_2i + x_2y_1i - y_1y_2}$ $=\overline{(x_1x_2 - y_1y_2) + i(x_1y_2 + x_2y_1)}$ $=(x_1x_2 - y_1y_2) - i(x_1y_2 + x_2y_1)\,$ $=x_1x_2 - x_1y_2i - x_2y_1i - y_1y_2\,$ $=(x_1 - y_1i)(x_2 - y_2i)\,$ $=\overline{(x_1 + y_1i)}\,\overline{(x_2 + y_2i)}$ $=\overline{z_1}\,\overline{z_2}$