[1] **viXra:1412.0001 [pdf]**
*replaced on 2015-12-28 21:16:51*

**Authors:** Pith Xie

**Comments:** 28 Pages.

The reference [2] contructs the Operator axioms to deduce number systems. In this paper, we slightly improve on the syntax of the Operator axioms and construct a semantics of the Operator axioms. Then on the basis of the improved Operator axioms, we define two fundamental operator functions to study the analytic properties of the Operator axioms. Finally, we prove two theorems about the fundamental operator functions and pose some conjectures. Real operators can give new equations and inequalities so as to precisely describe the relation of mathematical objects or scientific objects.

**Category:** Functions and Analysis