My Math Forum monotonic laws for ordinal subtraction

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 May 2nd, 2015, 02:14 AM #1 Member   Joined: Jan 2010 Posts: 44 Thanks: 0 monotonic laws for ordinal subtraction 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. (i) $\alpha>\beta\Rightarrow \gamma(\alpha-\beta)=\gamma\alpha-\gamma\beta$ (ii)$\alpha>\beta>\gamma \Rightarrow \alpha-\gamma>\beta-\gamma$
 May 2nd, 2015, 05:40 AM #2 Math Team   Joined: Dec 2013 From: Colombia Posts: 7,136 Thanks: 2380 Math Focus: Mainly analysis and algebra Well you can always use $p \gt q \implies p \lt (-q)$ and $p - q = p + (-q)$.
 May 4th, 2015, 10:17 AM #3 Member   Joined: Jan 2010 Posts: 44 Thanks: 0 I haven't seen $\displaystyle -q$ defined for an ordinal $\displaystyle q$...

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