The solution can't be correct without the transitivity:
If AB > BA and BC > CB, then we have AC > CA.
But how to prove it?
What I have is:
1) AB > BA
2) BC > CB
from 2) we have
3) BAC > CAB
from 1) we have
4) BAC > CAB > CBA
from 1) we also have
5) ABC > BAC > CAB > CBA
from 5 and "take out" the B we have
6) AC > CA
Glad you asked this, even after showing this property, I'm still not sure why this "sorting" solution works.
EDIT: Just realized I didn't prove that AB > BA => ACB > BCA. I believe that inequality is true. Don't want to prove it at the moment.
EDIT2 : Realized what I said above is wrong too. My intuition is really not helping me for this problem.