reserve U for Universe;
reserve x for Element of U;
reserve U1,U2 for Universe;

theorem
  for o being Element of FinSETS
  for n being non zero Nat holds
  the carrier of nMatrixFieldCat(F_Complex,o,n) is trivial &
  the carrier of nMatrixFieldCat(F_Complex,o,n) is FinSETS-set &
  not nMatrixFieldCat(F_Complex,o,n) is FinSETS-small Category &
  not nMatrixFieldCat(F_Complex,o,n) is FinSETS-locally_small Category &
  nMatrixFieldCat(F_Complex,o,n) is SETS-small Category &
  nMatrixFieldCat(F_Complex,o,n) is SETS-locally_small Category
  proof
    let o be Element of FinSETS;
    let n be non zero Nat;
    thus the carrier of nMatrixFieldCat(F_Complex,o,n) is trivial;
    thus the carrier of nMatrixFieldCat(F_Complex,o,n) is FinSETS-set by Th18;
    now
      assume nMatrixFieldCat(F_Complex,o,n) is FinSETS-small Category;
      then
A1:   nMatrixFieldCat(F_Complex,o,n) is FinSETS-element;
      the carrier of (n-G_Matrix_over F_Complex)
        = n-Matrices_over F_Complex by MATRIX_1:def 7
       .= n -tuples_on (n -tuples_on the carrier of F_Complex)
         by MATRIX_1:def 1;
      then n -tuples_on the carrier of F_Complex is finite by A1,Th6;
      hence contradiction by Th6;
    end;
    hence not nMatrixFieldCat(F_Complex,o,n) is FinSETS-small Category;
    reconsider x = o as Object of nMatrixFieldCat(F_Complex,o,n)
      by TARSKI:def 1;
A2: now
      thus Hom(x,x) c= the carrier' of nMatrixFieldCat(F_Complex,o,n);
      now
        let oo be object;
        assume oo in the carrier' of nMatrixFieldCat(F_Complex,o,n);
        then reconsider o9 = oo as Morphism of nMatrixFieldCat(F_Complex,o,n);
        dom o9 = o & cod o9 = o;
        hence oo in Hom(x,x) by CAT_1:1;
      end;
      hence the carrier' of nMatrixFieldCat(F_Complex,o,n) c= Hom(x,x);
    end;
    not Hom(x,x) is FinSETS-set
    proof
      assume
A3:   Hom(x,x) is FinSETS-set;
      the carrier of (n-G_Matrix_over F_Complex)
        = n-Matrices_over F_Complex by MATRIX_1:def 7
       .= n -tuples_on (n -tuples_on the carrier of F_Complex)
         by MATRIX_1:def 1;
      then n -tuples_on the carrier of F_Complex is finite
        by A3,A2,Th6;
      hence contradiction by Th6;
    end;
    hence not nMatrixFieldCat(F_Complex,o,n) is FinSETS-locally_small Category
      by Def36;
A4: o in FinSETS & FinSETS c= SETS by CLASSES2:69;
    COMPLEX is Element of SETS & the carrier of F_Complex = COMPLEX
      by CLASSES4:21,55,COMPLFLD:def 1;
    hence nMatrixFieldCat(F_Complex,o,n) is SETS-small Category &
    nMatrixFieldCat(F_Complex,o,n) is SETS-locally_small Category by A4,Th111;
  end;
