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I have to prove some monotonic laws for ordinals. It's quite comfortable for me to show monotonic laws of ordinal addition (e.g. $\beta\leq\gamma\Rightarrow\alpha+\beta\leq\alpha+\gamma$). But when it comes to laws with subtraction, then I'm not sure where to start.

Maybe it's because of definition of subtraction for ordinals $\alpha-\beta=\gamma$, if $\alpha=\beta+\gamma$, which is not constructive.

So, maybe someone can give me a hint on how to prove those:

$\alpha,\beta,\gamma$ - ordinals.

  1. $\alpha>\beta\Rightarrow \gamma(\alpha-\beta)=\gamma\alpha-\gamma\beta$

  2. $\alpha>\beta>\gamma \Rightarrow \alpha-\gamma>\beta-\gamma$

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  • $\begingroup$ You can define $\alpha-\beta$ as the unique ordinal isomorphic to the set $\alpha\setminus\beta$. Is this method "more" constructive? $\endgroup$ – Asaf Karagila May 2 '15 at 9:17
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HINT: If you know the laws are true for addition, this is not a difficult proof. For example, $\gamma\beta+\gamma\eta=\gamma(\beta+\eta)$.

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