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Proof of lemma 2.4

Hi, I was doing some self reading on Gromov geometry and I have difficulty accepting the proof given above for lemma 2.4. While I can understand that $(x|z)\ge \max\{(x|w),(y|z)\}$, there is no assumption imposed whatsoever on $(y|w)$. The last line of the proof seem to suggest that $(x|z)+(y|w)\ge (y|z)+(x|w)$ 1which may not be true for the reason stated above.

Is there something wrong with the proof above or am I missing something here?

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