Unit Vector proof. Odd notation

Question

I am reading this proof for the unit vector length and this is odd to me:

How did the book go from the double usage of the length symbol to the broken out usage of two single length symbols? How did they go from the second step (highlighted in yellow) to the third step?

Answer 1

We have

$$\left\|\frac{v}{a} \right\|= \frac1{|a|}\left\| v\right\|$$

here $a=\left\|v\right\|$ is a scalar and we don't have to write the absolute value is $a$ is positive.