Can anyone please help me with this example from Spivak? I am math autodidact.

I have to express this without absolute value: $$a-|a-|a||$$ The answer is $$\begin{cases}a&a\ge0\\ 3a&a<0\end{cases}$$ All examples went OK, but I dont know this one. I might understand answer by intuition but I can't do it rigorously. Thanks.

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    $\begingroup$ If $a>0$ then $|a|=a$ so your expression is $a-|a-a|=a$. The case $a<0$ is similar. $\endgroup$ – lulu Oct 25 '16 at 11:15
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    $\begingroup$ $$a\gt 0\Rightarrow |a|=a\Rightarrow a-|a-a|=a-0=a$$ $$a\lt 0\Rightarrow |a|=-a\Rightarrow a-|a-(-a)|=a-|2a|=a-(-2a)=3a$$ $\endgroup$ – Piquito Oct 25 '16 at 11:29

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