Proof of the cancellation laws

Proof

1. Let $a$, $b$ and $c$ be elements of a group $G$ with identity $e$, and suppose that $ab = ac$.
2. Multiply both sides on the left by the inverse $a^{-1}$. This gives $a^{-1}(ab) = a^{-1}(ac)$.
3. By associativity, $(a^{-1}a)b = (a^{-1}a)c$, so $eb = ec$ and $b = c$.
4. Now suppose that $ba = ca$. Multiply both sides on the right by $a^{-1}$.
5. By associativity, $b(aa^{-1}) = c(aa^{-1})$, so $be = ce$ and $b = c$.