The proof of the generalized fundamental theorem of moment invariants (GFTMI) is presented for n-dimensional pattern recognition. On the basis of GFTMI, the moment invariants of affine transformation and subgroups of affine transformation are constructed. Using these invariants, the conceptual mathematical theory of recognition of geometric figures, solids, and their n-dimensional generalizations is worked out. By means of this theory, it is possible for the first time to analyze scenes consisting not only of polygons and polyhedra, but also scenes consisting of geometric figures and solids with curved contours and surfaces, respectively. In general, it is the author's opinion that this theory is a useful step toward the essential development of robot vision and toward creating machine intelligence-to make machines able to think by means of geometric concepts of different generalities and dimensions, and by associations of these concepts.