reserve A,X for non empty set;
reserve f for PartFunc of [:X,X:],REAL;
reserve a for Real;

theorem
  for X being finite non empty Subset of REAL, f being Function of [:X,X
  :],REAL st (SmallestPartition X) in fam_class(f) & f is symmetric nonnegative
  holds fam_class(f) is Strong_Classification of X
proof
  let X be finite non empty Subset of REAL, f be Function of [:X,X:],REAL such
  that
A1: (SmallestPartition X) in fam_class(f) and
A2: f is symmetric nonnegative;
A3: fam_class(f) is Classification of X by Th31;
  {X} in fam_class(f) by A2,Th30;
  hence thesis by A1,A3,Def2;
end;
