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## Ordering

The total ordering defined in can be extended into a total ordering in .

Definition 6.2.13   Let and be two integers. Then we denote if, and only if, there exists a non negative integer such that .

The proof that is an ordering is similar to the proof of theorem 1.28; therefore wo do not develop it here. On , this ordering coincides with the ordering studied above, in subsection 1.3. Moreover we have:

Denote and for some natural numbers and ; then the following hold:

As , we are done. The ordering in is compatible with the addition and with the multiplication. The proofs are similar to the corresponding proofs in (v.s. Propositions 1.29 and 1.30).

root 2002-06-10