Reading Hacking's intro to probability & inductive logic

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Hi all. I'm a philosophy student and trying to learn some logic of induction on my own. I have only some rudimentary math and logic skills. I've been reading Hacking's Introduction to Probability and Inductive Logic, and having some difficulties on the 6th chapter , "The Basic Rules of Probablity".

To begin with, can you explain to me in simple terms how Hacking draws these identities (screenshot). From "Since Pr(A) =..." on... What do I need to know to perform these? Thanks.
Hacking__logical consequence.jpg
 

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