Laws of physics in four-vector form. A question.

[lslusa]

Please explain me in mathematical way, why the laws of physics have to be written in four-vector form to remain unchanged in every inertial frame.

[pervect]

The Lorentz interval is invariant. Using the 4-vector form makes the invariance of the Lorentz interval part of the formalism of the description of the laws of physics. It's possible to use other formalisms that don't have Lorentz invariance "built in", so in that sense you don't "have to" write the laws of physics in 4-vector form. It's just very convenient. An alternate description that got the same result would just "hide" the Lorentz invariance, rather than making it manifest.

[Janitor]

Please explain me in mathematical way, why the laws of physics have to be written in four-vector form to remain unchanged in every inertial frame.

As written, your question is a little too restrictive, I think. There are laws pertaining to electromagnetism, for instance, that can be written in terms of a certain antisymmetric tensor, which is not the same as a four-vector. (It so happens that the tensor in question can be written in terms of derivatives of a four-vector potential, but that's an issue for another day.) Maybe your question should be re-phrased as "Why do the laws have to be written in covariant tensor form?" But then, as pervect has pointed out, they don't absolutely have to be written that way. In fact, for decades electromagnetism used the fields E, D, B, and H, which in fact are three-vectors and three-pseudovectors, and thus are not covariant.