theorem Th69:
  A is limit_ordinal iff for B,n st B in A holds B+^ n in A
proof
  thus A is limit_ordinal implies for B,n st B in A holds B+^ n in A
  proof
    assume
A1: A is limit_ordinal;
    let B,n;
    defpred P[Nat] means B+^ $1 in A;
    assume B in A;
    then
A2: P[0] by ORDINAL2:27;
A3: P[k] implies P[k+1]
    proof Segm(k+1) = succ Segm k by NAT_1:38;
      then B+^(k+1) = succ (B+^ k) by ORDINAL2:28;
      hence thesis by A1,ORDINAL1:28;
    end;
    P[k] from NAT_1:sch 2(A2,A3);
    hence thesis;
  end;
  assume
A4: for B,n st B in A holds B+^ n in A;
  now
    let B;
    assume B in A;
    then B+^ 1 in A by A4;
    hence succ B in A by ORDINAL2:31;
  end;
  hence thesis by ORDINAL1:28;
end;
