Proof that $ax =1$ with no division and subtraction

Question

I need to proof that $ax = 1$ has one solution with no division and subtraction. It means that I can use only commutative,associative and distributive laws of multiplication and addition. Any suggestions on how to start?

Answer 1

This is basically asking about unicity of (right) multiplicative inverse.

Assume $a$ has two inverses $b$ and $c$, i.e., $ab=ac=1$. Then, using associativity and commutativity of the product

$b=1\cdot b=(c\cdot a)\cdot b=(a\cdot c)\cdot b=c\cdot(a\cdot b)=c\cdot 1=c$.