I am trying to understand the proof for the generalised case of the Inclusion-Exclusion Principle as set out in 9.2.1 and 9.2.2 (Theorem 9.2) of the resource I have linked below.
http://www.maths.manchester.ac.uk/~mrm/Teaching/DiscreteMaths/LectureNotes/InclusionExclusion.pdf
On page 5, where it supposes that $x$ is an element contained in exactly $L$ subsets of the union of all $X$ sets: I don't understand why we need to this. From that point onward I am quite lost and don't understand what is happening.
I am trying to understand what is going on in the proof, however, I can't seem to get my head around what it is trying to say. I have tried using other resources which have either taken an identical or similar approach yet I don't really seem to be making much progress.
Thanks in advance.