Proof for a^n - 1 = 1.

Question

I'm reading this book:

https://doc.lagout.org/science/0_Computer%20Science/2_Algorithms/The%20Art%20of%20Computer%20Programming%20(vol.%201_%20Fundamental%20Algorithms)%20(3rd%20ed.)%20%5BKnuth%201997-07-17%5D.pdf

And the first exercise is the following:

Let a be any positive number. For all positive integers n, we have a^n - 1 = 1.

Proof.

Base case: n = 1 a^1 - 1 = a^0 = 1.

This is true.

Now, when he tries to prove that a^(n + 1) - 1 = 1, I get confused in the third equality of the formula:

a^(n + 1) - 1 = a^n = ((a^n - 1) * ((a^n - 1)) / ((a^n - 1) - 1) = 1

The idea of the exercise is to prove that the proof is wrong. Now, where did he get the terms ((a^n - 1) * ((a^n - 1)) and ((a^n - 1) - 1) from? Why is he dividing one by the other?

Thank you beforehand for any help.