How to prove that $(z^{-1})^{-1} = z$ and $(zw)^{-1} = z^{-1}w^{-1}$? I need to prove that $(z^{-1})^{-1} = z$ and $(zw)^{-1} = z^{-1}w^{-1}$ but the only thing I can think about is to consider
$$z = a+bi, w = c+di$$
and then prove it algebraically using laws of multiplication for complex numbers. Any ideas to prove it structuraly?

Generally, $(z^a)^b=z^{ab}$, so $(z^{-1})^{-1}=z$.
We have $(zw)(zw)^{-1}=1.$ Pre-multiply by $z^{-1}$ and then $w^{-1}$ to get the result.
