 Division by an additively indecomposable ordinal

For every ordinal $\gamma$ and $\lambda ={\omega }^{\alpha }$, there are unique ordinals $\beta ,\delta$ such thatChoose the greatest initial multiple $\lambda \beta \le \gamma$; if its remainder were at least $\lambda$, the next multiple would still fit. Uniqueness follows because every remainder below $\lambda$ lies before $\lambda (\beta +1)$.