# Proof of a Lemma regarding absolute values

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Hi all,

There's this proof that I've been trying to wrap my head around but it just doesn't seem to sink in. I've attached a screenshot below. Many thanks in advance!

Consider Case 1. There is a step that goes
$$\text{Then} \ |r| = r$$
$$Then -|r| \leq |r| \ \text{and} \ r \leq |r|$$
Why is this the case? This seems to imply that because ##|r|=r##, then ##r \leq |r|##. Is this because of the "generalisation" rule of inference that goes
$$p$$
$$\text{Therefore} \ p \vee q$$
Where ##p = |r| = r## and ##q = |r| > r##? If so, why not write ##q = |r| < r## and get a completely different result altogether?

#### Attachments

• Screen Shot 2018-10-03 at 6.55.13 PM.png
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