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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" and denoted as |F.

Any |F includes natural, zero, and rational numbers.
This can be observed if starting from 1 repeat building
this numbers as it has been done in sections |N, |Z, and |Q.

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

\bProperties of |F b\

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
\bDefinition. Number b lies between a and c b\ iff
   (a &lt; b and b &lt; c) or
   (c &lt; b and b &lt; a).

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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