This is all seems fairly obvious to me and proving it is a bit awkward. In the previous exercise we proved that if x<y and n is odd, then x^n<y^n. Spivak argues that, from the previous exercise, x<y would imply that x^n<y^n and y<x would imply that y^n<x^n. That's his whole proof. Is he simply saying that, by analogy, this relationship must hold? I mean qualitatively it makes sense, because anything raised to an odd exponent retains it's original sign. Thus for x^n=y^n to hold, either x=y or -x=-y (which is obviously the same as x=y).