reserve W,X,Y,Z for set,
  f,g for Function,
  a,x,y,z for set;
reserve u,v for Element of Tarski-Class(X),
  A,B,C for Ordinal,
  L for Sequence;

theorem Th38:
  A c= Rank A
proof
  defpred P[Ordinal] means $1 c= Rank $1;
A1: P[0] by XBOOLE_1:2;
A2: P[B] implies P[succ B]
  proof
    assume B c= Rank B;
then  B in Rank succ B by Th32;
    then
 B c= Rank succ B & {B} c= Rank succ B by ORDINAL1:def 2,ZFMISC_1:31;
    hence thesis by XBOOLE_1:8;
  end;
A3: for A st A <> 0 & A is limit_ordinal & for B st B in A holds P[B]
  holds P[A]
  proof
    let A such that
    A <> 0 and
A4: A is limit_ordinal and
A5: for B st B in A holds B c= Rank B;
    let x be object;
    assume
A6: x in A;
    then reconsider B = x as Ordinal;
A7: B c= Rank B by A5,A6;
A8: succ B c= A by A4,A6,ORDINAL1:28,def 2;
A9: B in Rank succ B by A7,Th32;
 Rank succ B c= Rank A by A8,Th37;
    hence thesis by A9;
  end;
 for B holds P[B] from ORDINAL2:sch 1(A1,A2,A3);
  hence thesis;
end;
