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

theorem Th14:
  for S being Subset of PARTITIONS(A) st S = {{A},SmallestPartition A} holds
  S is Classification of A
proof
  let S be Subset of PARTITIONS(A);
  assume
A1: S = {{A},SmallestPartition A};
  let X,Y be a_partition of A such that
A2: X in S and
A3: Y in S;
  per cases by A1,A2,TARSKI:def 2;
  suppose
A4: X = {A};
    per cases by A1,A3,TARSKI:def 2;
    suppose
      Y = {A};
      hence thesis by A4;
    end;
    suppose
      Y = SmallestPartition A;
      hence thesis by A4,Th11;
    end;
  end;
  suppose
A5: X = SmallestPartition A;
    per cases by A1,A3,TARSKI:def 2;
    suppose
      Y = SmallestPartition A;
      hence thesis by A5;
    end;
    suppose
      Y = {A};
      hence thesis by A5,Th11;
    end;
  end;
end;
