Let m, n e N. If n >= m, then there is a unique b e N such that n = m + b.
How do I prove that b is unique.
$\displaystyle m,n\in\mathbb{N},n\geqslant m\implies\exists!\,b\in\mathbb{N}:n=m+b$
Suppose not. Let $\displaystyle m,n\in\mathbb{N}$ and let $\displaystyle b,c\in\mathbb{N}$ such that $\displaystyle n=m+b$ and $\displaystyle n=m+c$. By transitivity, we see that $\displaystyle m+b=m+c$. Thus, by cancellation, we have $\displaystyle b=c$. This shows that $\displaystyle b$ is unique.
Does this make sense?