theorem Th4:
  gr {a} is cyclic Group
proof
  ex a1 being Element of gr {a} st gr {a}=gr {a1}
  proof
    a in gr {a} by Th2;
    then reconsider a1=a as Element of gr {a} by STRUCT_0:def 5;
    take a1;
    thus thesis by Th3;
  end;
  hence thesis by GR_CY_1:def 7;
end;
