The quartet spaces of G. 't Hooft

In 1964, G. 't Hooft postulated three axioms, and proved that every nonempty finite model of them has 4n elements. This note confirms this by showing that every nonempty model can be made into a vector space over the field with four elements. For every pair of different elements x and y, the quartet of x and y is the affine line through x and y in this vector space.