This is like the third time the author has brought up something without explanation in this section...
Anyway, in one example right after showing how every permutation can be written as the product of transpositions, the author writes this to demonstrate that the factorizations are not unique: $$ (2 \space 3)(1 \space 2)(2 \space 5)(1 \space 3)(2 \space 4) = (1 \space 2 \space 4 \space 5) = (1 \space 5)(1 \space 4)(1 \space 2) $$
Right now, I'm only familiar with factorizations like below, where it is clear where the factors came from. $$ (1 \space 2 \space 3 \space 4 \space 5) = (1 \space 2)(2 \space 3)(3 \space 4)(4 \space 5) $$
So what's up with the factors like $(1 \space 4)$ and $(1 \space 5)$ (in the first equation)? I know that you can multiply permutations so that they behave like a composition. So is this the same thing that's happening here? I'm confused about how the products on the sides are equal to the permutation in the middle, so I would appreciate it if someone would show me step-by-step on how to deduce that the products are indeed equal.