
theorem Th13:
  for G being finite commutative Group,
  m be Nat,
  A be Subset of G
  st A ={x where x is Element of G: x|^m = 1_G}
  holds
  A <> {}
  &
  (for g1,g2 be Element of G
  st g1 in A & g2 in A holds g1 * g2 in A) &
  for g be Element of G st g in A holds g" in A
  proof
    let G be finite commutative Group,
    m be Nat,
    A be Subset of G;
    assume A1: A ={x where x is Element of G: x|^m = 1_G};
    (1_G) |^m = 1_G by GROUP_1:31;
    then
    A2:1_G in A by A1;
    A3: for g1,g2 be Element of G
    st g1 in A & g2 in A holds g1 * g2 in A
    proof
      let g1,g2 be Element of G;
      assume A4: g1 in A & g2 in A;
      then
      A5: ex x1 be Element of G st g1=x1 & x1|^m = 1_G by A1;
      A6: ex x2 be Element of G st g2=x2 & x2|^m = 1_G by A1,A4;
      (g1 * g2) |^m = g1 |^m * g2 |^m by GROUP_1:38
      .= 1_G by GROUP_1:def 4,A5,A6;
      hence g1 * g2 in A by A1;
    end;
    for g be Element of G st g in A holds g" in A
    proof
      let g be Element of G;
      assume g in A;
      then
      A7: ex x be Element of G st g=x & x|^m = 1_G by A1;
      g" |^ m = (g |^ m)" by GROUP_1:37
      .= 1_G by GROUP_1:8,A7;
      hence g" in A by A1;
    end;
    hence thesis by A2,A3;
  end;
