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In this simple linear algebra problem, take left side of equation

|a+b|^2 + |a-b|^2

|a+b|^2 = |a|^2 + |b|^2 + 2|a||b| either a>=b or a<b

|a-b|^2 = |a|^2 + |b|^2 - 2|a||b| either a>=b or a<b

Add these two and result is:

2(|a|^2+|b|^2) = Right hand side

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