Sorted nature of the real number

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Sorted nature of the real number
a linear sequence of real number system through the creation of a collection compiled in R, called the assemblage positive real number.
Definition.
a nonempty subset P of R called positive real number :
1. for each a,b  element P apply a+b element P
2. for each a,b  element P apply a.b element P
3. for each a element R,  exactly one statement met a element P or -a element P or a= O.
    (trichotomy)
furthermore if a element P said a positive number and writed a>0. If a element P Union {0} said  a non negative number and writed a>= 0. If -a element P said a negative number and writed a<0 , if -a element p union {0} said a non negative number and writed a<= 0.

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