theorem Th72:
  A is limit_ordinal implies A *^ succ 1 = A
proof consider n such that
A1: A*^2 = A+^ n by Th71;
  assume A is limit_ordinal;
  then
A2: A+^ n is limit_ordinal by A1,ORDINAL3:40;
  now
    assume n <> 0;
    then consider k being Nat such that
A3: n = k+1 by NAT_1:6;
    reconsider k as Element of NAT by ORDINAL1:def 12;
    Segm n = succ Segm k by A3,NAT_1:38;
    then A+^ n = succ(A+^ k) by ORDINAL2:28;
    hence contradiction by A2,ORDINAL1:29;
  end;
  hence thesis by A1,ORDINAL2:27;
end;
