In other words, if particles and fields are moved by the amount x 2 and then by the amount x 1 , overall they have been moved by the amount x 1 + x 2 . For a mathematical proof, one can look at what these operators do to a particle in a position eigenstate: T ^ ^ ( x 1 ) T ^ ^ ( x 2 ) | r ⟩ ⟩ = T ^ ^ ( x 1 ) | x 2 + r ⟩ ⟩ = | x 1 + x 2 + r ⟩ ⟩ = T ^ ^ ( x 1 + x 2 ) | r ⟩ ⟩ Since the operators T ^ ^ ( x 1 ) T ^ ^ ( x 2 ) and T ^ ^ ( x 1 + x 2 ) have the same effect on every state in an eigenbasis, it follows that the operators are equal.