 reserve G, A for Group;
 reserve phi for Homomorphism of A,AutGroup(G);
 reserve G, A for Group;
 reserve phi for Homomorphism of A,AutGroup(G);
reserve G1,G2 for Group;

theorem Th117:
  for n being odd non zero Nat
  for g1 being Element of INT.Group n st g1 = 1
  for x being Element of Dihedral_group n st x = <*g1,1_(INT.Group 2)*>
  for i being Nat st i < n
  holds (i = 0 or x |^ i <> x |^ (n - i))
proof
  let n be odd non zero Nat;
  let g1 be Element of INT.Group n;
  assume A1: g1 = 1;
  let x be Element of Dihedral_group n;
  assume A2: x = <*g1,1_(INT.Group 2)*>;
  let i be Nat;
  assume A3: i < n;
  assume A4: i <> 0;
  A5: for j being Nat holds g1 |^ j = j mod n by A1,LmINTGroupOrd3;

  g1 |^ i <> g1 |^ (n - i)
  proof
    assume B1: g1 |^ i = g1 |^ (n - i);
    B2: g1 |^ i = i mod n by A5
               .= i by A3, NAT_D:24;
    B3: 0 < n - i & n - i < n
    proof
      i - i < n - i by A3, XREAL_1:9;
      hence 0 < n - i;
      i > 0 by A4;
      hence n - i < n by XREAL_1:44;
    end;
    then n - i in NAT by INT_1:3;
    then g1 |^ (n - i) = (n - i) mod n by A5
                      .= n - i by B3, NAT_D:24;
    then i = n - i by B1,B2;
    then 2*i = n + 0;
    then n is even;
    hence contradiction;
  end;
  then g1 |^ i <> g1 |^ (n - i);
  then <* (g1 |^ i), 1_(INT.Group 2) *> <>
    <* (g1 |^ (n - i)), 1_(INT.Group 2) *> by FINSEQ_1:77;
  then x |^ i <> <* (g1 |^ (n - i)), 1_(INT.Group 2) *> by A2, Th25;
  hence x |^ i <> x |^ (n - i) by A2,Th25;
end;
