 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 :: TH119
  for n being even non zero Nat
  for k being Nat st n = 2*k
  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) *>
  holds (x |^ k) |^ 2 = 1_(Dihedral_group n)
proof
  let n be even non zero Nat;
  let k be Nat;
  assume A1: n = 2*k;
  let g1 be Element of INT.Group n;
  assume A2: g1 = 1;
  let x be Element of Dihedral_group n;
  assume A3: x = <* g1, 1_(INT.Group 2) *>;
  A4: (g1 |^ n) = n mod n by A2, LmINTGroupOrd3
               .= 0 by NAT_D:25
               .= 1_(INT.Group n) by GR_CY_1:14;
  A5: x |^ n = <* (g1 |^ n), 1_(INT.Group 2) *> by A3, Th25
            .= <* 1_(INT.Group n), 1_(INT.Group 2) *> by A4
            .= 1_(Dihedral_group n) by Th17;
  thus (x |^ k) |^ 2 = x |^ (k * 2) by GROUP_1:35
                    .= x |^ n by A1
                    .= 1_(Dihedral_group n) by A5;
end;
