Ordinal division algorithm ($\alpha =\delta q+r,r<\delta$)

For ordinals $\alpha$ and $\delta >0$, there are unique ordinals $q,r$ with the displayed relation. The function $q\mapsto \delta q$ is strictly increasing, continuous at limit ordinals and unbounded. Its first value above $\alpha$ occurs at a successor, giving the largest $q$ with $\delta q\le \alpha$. The ordered tail of $\alpha$ after that initial segment gives $r$, and maximality gives $r<\delta$.