We have an assumption $9|10^j-1$ and $j\in\mathbb{N}$ (inductive assumption)
At this point the transformation occurs:
$10^{j+1}-1=10^j\cdot10-1=10^{j}-1+9\cdot10^j$ from where we know: $9|10^{j+1}-1$
I would like to ask you to explain how we pass from one form to another and how we get $9|10^{j+1}-1$ from the final transformation.I understand the transition from the first form to the second because $10^{j+1}=10^j\cdot10$, which is obvious thanks to the rules of operations on powers, but I cannot understand the next two steps despite sitting on it for a long time.