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

theorem
  {{A}} is Classification of A
proof
  {A} is a_partition of A by EQREL_1:39;
  then {A} in PARTITIONS(A) by PARTIT1:def 3;
  then reconsider S = {{A}} as Subset of PARTITIONS(A) by ZFMISC_1:31;
  S is Classification of A
  proof
    let X be a_partition of A;
    thus thesis by Lm1;
  end;
  hence thesis;
end;
