Complex numbers/Conjugation/Real part imaginary part/Fact

For a complex number ${}z$ the following relations hold.

1. ${}{\overline {z}}=\operatorname {Re} \,{\left(z\right)}-{\mathrm {i} }\operatorname {Im} \,{\left(z\right)}$.
2. ${}\operatorname {Re} \,{\left(z\right)}={\frac {z+{\overline {z}}}{2}}$.
3. ${}\operatorname {Im} \,{\left(z\right)}={\frac {z-{\overline {z}}}{2{\mathrm {i} }}}$.

Write a proof