For every number $ a \ne 0 $, there is a number $ a ^{-1} $ such that $$ a . a^{-1} = a^{-1} . a = 1. $$
This line is from Calculus by Michael Spivak.
I wanted to prove $ (ab)^{-1} = a^{-1}b^{-1} $ using the basic properties of numbers and i did this same solution as xbh(1st ans) in this link.
This solution uses the idea that each inverse is unique, hence proving both term is equal.
Since I am just relying on the basic properties and definitions, how do I know that the inverse is unique. Are the words that I quoted from Spivak enough to say every number has a unique inverse or there is something I missed?
