
theorem GRCY112:
  for p being Nat, G being finite Group, x, d be Element of G
  st ord d = p & p is prime & x in gr{d}
  holds x = 1_G or gr{x} = gr{d}
  proof
    let p be Nat, G be finite Group, x, d be Element of G;
    assume
    A1: ord d = p & p is prime;
    assume x in gr{d};
    then
    X1: gr{x} is strict Subgroup of gr{d} by GRCY212;
    X2: card (gr{d}) = p by A1, GR_CY_1:7;
    gr{x} = (1).(gr{d}) implies x = 1_G
    proof
      assume
      X3: gr{x} = (1).(gr{d});
      x in the carrier of gr{x} by GR_CY_2:2, STRUCT_0:def 5;
      then x in {1_(gr{d})} by X3, GROUP_2:def 7;
      then x = 1_(gr{d}) by TARSKI:def 1;
      hence x = 1_G by GROUP_2:44;
    end;
    hence thesis by GR_CY_1:12, A1, X1, X2;
  end;
