theorem Th12:
  On Rank A = A
proof
  thus On Rank A c= A
  proof
    let x be object;
    assume
A1: x in On Rank A;
    then reconsider B = x as Ordinal by ORDINAL1:def 9;
    x in Rank A by A1,ORDINAL1:def 9;
    then the_rank_of B in A by CLASSES1:66;
    hence thesis by CLASSES1:73;
  end;
  On A c= On Rank A by CLASSES1:38,ORDINAL2:9;
  hence thesis by ORDINAL2:8;
end;
