reserve x,y for object, X for set;

theorem Th30:
  for G be Group, z be Element of G, d,l be Element of NAT st G is
  finite & ord z=d*l holds ord (z|^d)=l
proof
  let G be Group, z be Element of G, d,l be Element of NAT;
  assume that
A1: G is finite and
A2: ord z=d*l;
  set m = d*l;
  reconsider H=gr {z} as strict Subgroup of G;
  reconsider H as finite strict Subgroup of G by A1;
  z in gr{z} by GR_CY_2:2;
  then reconsider z1=z as Element of H;
A3: gr{z} =gr{z1} by GR_CY_2:3;
  card H = m by A1,A2,GR_CY_1:7;
  then ord (z1|^d) = l by A3,GR_CY_2:8;
  hence thesis by A1,GROUP_8:3,5;
end;
