First, here is the proof out of the book. I will explain what is causing confusion below the image but I will point out that my question is in regards to the highlighted statement.
Now, the reason I am stuck has to do with the projection $\mathcal P_\mathcal L$. Oksendal does not seem to explicitly state the projection is orthogonal. If it were, I would have no problem with the highlighted statement.
My question is how to justify the highlighted statement. Does the projection $\mathcal P_\mathcal L$ have to be orthogonal for this to hold?