How can I show $-(-C) = C$ and $(C^{-1})^{-1}=C$? If C is a Dedekind cut.
I showed it using the distributive property and the fact $$C-C=C(0)\Rightarrow -(C-C)=C(0) \Rightarrow -C+(-(-C))=C(0)$$ $∴ -(-C) = C$ (I did a similar process for the other) but I would like to know if there is another alternative-proof, e.g. showing that $-(-C)\subset C$ and $C\subset -(-C)$ or $(C^{-1})^{-1} \subset C$ and $C \subset (C^{-1})^{-1}$. I tried to do it, but now, I'm stagnant.