reserve i, j, k, c, m, n for Nat,
  a, x, y, z, X, Y for set,
  D, E for non empty set,
  R for Relation,
  f, g for Function,
  p, q for FinSequence;

theorem Th21:
  for f being homogeneous PartFunc of X*, X holds
  f is quasi_total non empty iff dom f = (arity f)-tuples_on X
proof
  let f be homogeneous PartFunc of X*, X;
  per cases;
  suppose
    X is empty;
    then f = {};
    hence thesis by Th5,Th17;
  end;
  suppose
    X is non empty;
    hence thesis by Lm1;
  end;
end;
