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One way is to consider the invariance of light velocity. If we postulate ##k^\mu## to be a four-vector then the scalar product of ##k^\mu\,x_\mu=\vec{k}\cdot\vec{r}-\frac{\omega}{c}\,t## is invariant, if it constant in one frame it will also be constant in any other inertial frame.

However this is only a sufficient contition. In general it does not have to be the same constant value to be consistent with the same velocity of light.