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The total ordering defined in
can be extended into a total ordering in
The proof that is an ordering is similar to the proof of theorem 1.28; therefore wo do not develop it here.
, this ordering coincides with the ordering studied above, in subsection 1.3. Moreover we have:
be two integers. Then we denote
if, and only if, there exists a non negative integer
for some natural numbers and ; then the following hold:
, 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).