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An Algebraic Theorem Related to the Theory of Relativity

In a common development of the special theory of relativity one considers two events P and Q, P being the emission of a flash of light and Q the reception of the light. We suppose Q to be infinitesimally close to P and, by using the principle of the invariance of c, the velocity of light, argue that the infinitesimal coordinate differences between P and Q must satisfy I and II where dx; etc. are the coordinate differences in one inertial frame whilst dx' etc. are those in a second inertial frame (e.g. [1], p. 16).