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May 2nd, 2015, 01:14 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, 04:40 AM  #2 
Math Team Joined: Dec 2013 From: Colombia Posts: 7,313 Thanks: 2447 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, 09:17 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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