If   ; , the relative velocity  no change ，so 

                                                              (3)

   contrast     with     

                                        (4)              

                                                             （5）

                                               （6）

3.  The definition of space-time observation 

       For the sake of clean show the relationship that the different observational results and the lengths and the times in the different coordinates

 mutual contrast, add three sign on characters. For example: 

              Definition: Left sign means the coordinate that the target are observed in there, middle sign means the coordinate that standard length unit or time 

     unit are selected in there by observer , right sign means the coordinate that observer is observing in there. So:

        and  is proper time,   and  is proper length.  

　 4.  The process of reasoning  

             There are    

                                (6.1)

           First, inspect Euclidean geometry. The aspect of the observer, in the coordinate of  Euclidean geometry’s two- dimensional space, the  

    standard unit of length  are observed and the standard unit of  length  are observed in the coordinate of Euclidean geometry’s two-dimensional 

    space. The

    observer is unable to distinguish   and   about length. Then  

                                      （7）

                            This is Euclidean geometry’s observer hypothesis, and conform experience.

                          （8）

                      The general relativity invariance principle determine that there is (8) formula in any coordinate of four-dimensional space-time.

     one  kind of coordinate of four-dimensional space-time: In the time when observer is observing,  space-time metric is time translation invariability and 

     space  translation invariability, let

                      （9）

   It is always this kind of coordinate if space-time value is enough small, it means . The kind of coordinate  are general. In the four-dimensional

 space-time,at random get two coordinate of this kind and  , observer obser- ves the units of space-time that is  in the   coordinate and

 in the   coordinate, he is unable to distinguish   that he is observing and   that he is observing about space-time if observer doesn’t

 contrast other coordinates. ( observer should not contrast in the  coordinate when he is observing   in the   coordinate. He only separately

 observes  and  , then contrasts   and   )  This is four-dimensional curve space- time’s observer hypothesis. It means that observers are

 unable to distinguish proper time and proper length in difference coordinate of this kind. 

        So, observer gets                                                      （10）

    Let  ，  ，so that :

                                （11）

                        （11.1）

   Because (9) formula definition, observer thinks that the coordinate that he is there, it always satisfy:

                                                   （12）

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