Hmm, someone got something backwards somewhere! We want to make the inequalities point the right way: we want

\[ \mathcal{X}(x,x) \le 0 \]

so that \\( \mathcal{X}(x,x) = 0\\) and

\[ \mathcal{X}(x, z) \le \mathcal{X}(x, y) + \mathcal{X}(y, z) \]

Normally it's me who gets things backward, but I thought I just copied the definitions from Fong and Spivak. I'll straighten it out somehow...

\[ \mathcal{X}(x,x) \le 0 \]

so that \\( \mathcal{X}(x,x) = 0\\) and

\[ \mathcal{X}(x, z) \le \mathcal{X}(x, y) + \mathcal{X}(y, z) \]

Normally it's me who gets things backward, but I thought I just copied the definitions from Fong and Spivak. I'll straighten it out somehow...