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  \T Rational Numbers t\
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\b       Rational Numbers \


\b Ordered Field b\

\bNotation.b\
Set "satisfying to" above axioms will be called "Ordered Field".

|D will denote any Ordered field which includes natural numbers.
Such field necessarily includes integer, and rational numbers.
This can be observed if starting from natural numbers, repeat building
integer and rational numbers as it has been done in sections |Z and |Q.

Before study the question, can |D contain "more" numbers, 
consider some properties of |D.

\bProperties of |D b\

\b'Inter' relation:b\
\bDefinition. Number b lies between a and c, b^ac, \ iff
             (a &lt; b and b &lt; c) or
             (c &lt; b and b &lt; a).
This definition is equivalent to 
\bDefinition'b\ b^ac : (b-a)(c-b)>0
which is symmetric by a -&gt; b 'interchange' and 

First, this relation is not empty: 2^1,3.
Second, this relation is symmetric for a-b interchange, so this is relation
between b and unordered pair {a,c}.
Third, for any of three 'non-equal' numbers, one and only one of them is between others:
  a. it is exclusive, if it is true for b^ac, then no a^bc and no c^ba.
  b. for any of three 'non-equal' numbers a,b,c one is always between others.







Properies of the "norm."

\bTriangle inequaility.b\ |a| + |b| &gt;= |a + b|, and "=" iff ab&gt;=0

\bProof.b\
|ab| >= ab, and "=" iff ab >=0
This "inequality" is equivalent to
a\]2 \ + 2|ab| + b\]2 \ &gt;= a\]2 \ + 2ab + b\]2 
wich is equivalent to the property.\bHb\

Metric, |a-b|, posesses similar property:
\bTriangle inequility.b\ |b-a| + |c-b| &gt;= |c-a|, 
and "=" iff (b-a)(c-b)&gt;=0  (*)
which is direct consequence of |.| triangle property.

Condition (*) has geometrical interpretation. Indeed

This implies that all three numbers are different and
is equivalent to
   (c-b)(b-a)&gt;0 (&gt;)
For given three not equal numbers, 
 


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