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Set Theory - Partitions and Equivalence Relations

I'm having trouble understanding something that was in lectures on set theory. Unfortunately I am not experienced enough in latex to be able to write it out, so have attached a pdf.

(page 10) I am struggling with the theorem:

Theorem 3.1.8. Let A be any set. Then to each equivalence relation R on A there

corresponds a partition PR, and to each partition P of A there corresponds an equivalence relation RP.

If anyone could help me understand the proof I would be most grateful. I get to about line 5 before I just loose track of whats going on completely.