Quasi finitely axiomatizable totally categorical theories

As was shown in [2], totally categorical structures (i.e. which are categorical in all powers) are not finitely axiomatizable. On the other hand, the most simple totally categorical structures: infinite sets, infinite projective or affine geometries over a finite field, are quasi finitely axiomatizable (i.e. axiomatized by a finite number of axioms and the schema of infinity, we will use the abbreviation ‘qfa’. Since all totally categorical structures are ‘built up’ from these simple structures, it was conjectured in [2] that all totally categorical structures are quasi finitely axiomatizable (which from now on means: being interdefinable with a qfa structure). We prove in this paper