How did Spivak get this equality? I realise my last question omitted alot of info, so here is the context! ![enter image description here]( I am wondering how Spivak got the following equality: ![enter image description here]( I don't see how the two are equal; wouldn't $$Mi' <= Mi$$ if $$-mi <= Mi$$ ?

Nevermind, I get it now! If when absoluted -mi is still less than Mi on the interval, then Mi' must be equal to Mi as Mi is still the sup value on the interval. If $$-mi > Mi$$ then Mi' = -mi and you refer to the previous equality.