An explanation of how $(m+n)^2 -(m'+n')^2 = n'-n$ entails $|m+n+m'+n'|\cdot|m+n-m'-n'|=|n'-n|$

Question

I'm self-studying the Royden-Fitzpatrick book on Real Analysis, and just have a question about what I think should be an elementary step in the proof of Corollary 4 in chapter 1, which I don't seem to be able to fully work out on my own. It says that from $(m+n)^2 -(m'+n')^2 = n'-n$ we can infer $|m+n+m'+n'|\cdot|m+n-m'-n'|=|n'-n|$. I can sort of see how this follows but I must have missed the technique you would use to show this. I was wondering if anyone could break down the inference in detail? Thanks!

Answer 1

Using $a^2-b^2=(a+b)(a-b)$ we get $$(m+n)^2 -(m'+n')^2=(m+n+m'+n')(m+n-m'-n')$$ and so $$|(m+n)^2 -(m'+n')^2|=|m+n+m'+n'||m+n-m'-n'|.$$ Then $$(m+n)^2 -(m'+n')^2 = n'-n$$ implies $$|(m+n)^2 -(m'+n')^2| = |n'-n|$$ which is equivalent to $$|m+n+m'+n'||m+n-m'-n'|=|n'-n|.$$