Degenerate Perturbation Theory

xdrgnh
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http://farside.ph.utexas.edu/teaching/qm/lectures/node53.html

So I was reading this and I don't understand how he goes from 658 to 661 using the completeness relation. In 661 if you use the completeness relaton can you get rid of the I n,l''>s by doing the outer product and ignoring the inner product first? I mean if you get rid of I n,l''>s you get <n,l'I H_1 In,l1>which equals lamba etc.
 
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Too hard to read the dense notation, sorry.
 

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