Check Eq. 12.29, using Eq. 12.27. [This only proves the invariance of the scalar product for transformations along the x direction. But the scalar product is also invariant under rotations, since the first term is not affected at all, and the last three constitute the three-dimensional dot product a.b. By a suitable rotation, the x direction can be aimed any way you please, so the four-dimensional scalar product is actually invariant under arbitra.rv Lorentz transformations.]