Abstract: This paper proves Wigner Rotation Theorem with a rigorous yet accessible calculation.It derives the properties of velocity composition in special relativity and discusses the monotonicity of physical quantities and several special cases for intuitive understanding.Based on this theorem, it demonstrates the internal consistency of special relativity and derives the conclusion on Thomas precession in atomic physics and quantum mechanics.Given the limited coverage of Wigner Rotation Theorem in textbooks, this paper supplements undergraduate learning in special relativity and quantum mechanics.

Key words:  special relativity, Wigner rotation, Lorentz transformation, rotation matrix, Thomas precession