Sum of Complex Number with Conjugate

Theorem
Let $$z \in \mathbb{C}$$ be a complex number.

Let $$\overline {z}$$ be the complex conjugate of $$z$$.

Let $$\Re \left({z}\right)$$ be the real part of $$z$$.

Then $$z + \overline z = 2 \Re \left({z}\right)$$.

Proof
Let $$z = x + \imath y$$.

Then:

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