Nature of Electric and Magnetic Fields; How the Fields Transform

In this paper the proofs are given that the electric and magnetic fields are properly defined vectors on the four-dimensional (4D) spacetime (the 4-vectors in the usual notation) and not the usual 3D fields. Furthermore, the proofs are presented that under the mathematically correct Lorentz transformations (LT), e.g., the electric field vector transforms as any other vector transforms, i.e., again to the electric field vector; there is no mixing with the magnetic field vector B, as in the usual transformations (UT) of the 3D fields. The derivations of the UT from some well-known textbooks are discussed and objected.