A characterization of plane Lorentz transformations

Denote (x_i,y_i=ct_i), i=1,2, by X_i and (x₂–x₁)²–(y₂–y₁)² by F(X₁,X₂). Then our result is the following: Given a fixed real number 0 and given a bijection of M=IR² such that F(X₁,X₂) = iff F(X _in1 ^su , _in2 ^su ) =p for all X₁, X₂ M. Then must be a Lorentz transformation (time reversal and inhomogeneity included).