Click the image for my response.Note that it says "the previous" and not "a previous".
They've omitted the step of generalizing eqn 8 to an expression in terms of T(n-k). Look at eqns 6, 7, 8 and see if you can see the pattern and write it as
T(n) = T(n-k) + ....
Then pick the value of k needed to reduce it to the for of the bottom equation.