Complex Conjugation/Rules/Fact

For the complex conjugation the following rules hold (for arbitrary ${}z,w\in \mathbb {C}$).

1. ${}{\overline {z+w}}={\overline {z}}+{\overline {w}}$.
2. ${}{\overline {-z}}=-{\overline {z}}$.
3. ${}{\overline {z\cdot w}}={\overline {z}}\cdot {\overline {w}}$.
4. For ${}z\neq 0$ we have ${}{\overline {1/z}}=1/{\overline {z}}$.
5. ${}{\overline {\overline {z}}}=z$.
6. We have ${}{\overline {z}}=z$ if and only if ${}z\in \mathbb {R}$ holds.

Write a proof