\overline{a + b} &=& \rm{Re}(a) + \rm{Re}(b) - (\rm{Im}(a) + \rm{Im}(b)) \; i \\
&=& \rm{Re}(a) -  \rm{Im}(a) \; i + \rm{Re}(b) -  \rm{Im}(b) \; i \\
&=& \overline{a} + \overline{b}
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