theorem Th9:
  s divides k implies a|^k in gr {a|^s}
proof
  assume s divides k;
  then consider p be Nat such that
A1: k=s*p by NAT_D:def 3;
  ex i st a|^k=a|^s|^i
  proof
    reconsider p9=p as Integer;
    take p9;
    thus thesis by A1,GROUP_1:35;
  end;
  hence thesis by GR_CY_1:5;
end;
