Can we prove the difference of phase between the voltage of three alternating current is 120 degree with "math formula"?
In the three-phase electric power, we will assume the difference of phase between the voltage of three alternating current is 120 degree.


No, the phase relationship is not assumed, it is designed.



Figure 1. Source: TES. (Original author unknown.)

As shown in Figure 1, the poles are mechanically laid out in a 120&deg; spacing so the voltage outputs are offset by 120&deg;.


  I want to ask why do we assume the 120 degree is the best option for the difference of three voltage phase? Can it be proved with math formula ?


The maths should be fairly straight forward. The three phases balance out when set at 120&deg;. If you can prove that \$ sin(a) + sin(a - \frac {2}{3}\pi) + sin(a - \frac {4}{3}\pi) = 0 \$ you will have shown that the phases are perfectly balanced for all values of a. Thus the current in on any one phase is perfectly balanced by the current out on the other two. (Hence we don't require a neutral connection for a balanced load.)