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

theorem Th111:
  for K being Field for o being Element of U
  for n being non zero Nat st the carrier of K is Element of U holds
  the carrier of nMatrixFieldCat(K,o,n) is trivial &
  nMatrixFieldCat(K,o,n) is U-small Category &
  nMatrixFieldCat(K,o,n) is U-locally_small Category
  proof
    let K be Field;
    let o be Element of U;
    let n be non zero Nat;
    assume
A1: the carrier of K is Element of U;
    thus the carrier of nMatrixFieldCat(K,o,n) is trivial;
    now
A2:   n -tuples_on the carrier of K in U by A1,CLASSES4:58;
      thus the carrier of nMatrixFieldCat(K,o,n) in U by Th18;
      the carrier of (n-G_Matrix_over K)
      = n-Matrices_over K by MATRIX_1:def 7
      .= n -tuples_on (n -tuples_on the carrier of K) by MATRIX_1:def 1;
      hence the carrier' of nMatrixFieldCat(K,o,n) in U by A2,CLASSES4:58;
    end;
    then nMatrixFieldCat(K,o,n) is U-element;
    hence nMatrixFieldCat(K,o,n) is U-small Category &
      nMatrixFieldCat(K,o,n) is U-locally_small Category by Th96;
  end;
