
theorem GRCY26: ::: improve GR_CY_2:6
  for G being finite Group, a, b be Element of G
  holds b in gr {a} iff ex p be Element of NAT st b = a |^p
  proof
    let G be finite Group, a, b be Element of G;
    reconsider a0 = a as Element of gr{a} by GR_CY_2:2, STRUCT_0:def 5;
    X1: gr{a0} = gr{a} by GR_CY_2:3;
    hereby
      assume b in gr{a};
      then reconsider b0 = b as Element of gr{a};
      consider p be Element of NAT such that
      A1: b0 = a0|^p by X1, GR_CY_2:6;
      b= a |^p by GROUP_4:2, A1;
      hence ex p be Element of NAT st b = a|^p;
    end;
    given p be Element of NAT such that
    A1: b = a|^p;
    b = a0|^p by GROUP_4:2, A1;
    hence thesis;
  end;
