 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 Th103:
  for n being non zero Nat
  for g1 being Element of INT.Group n
  for x being Element of Dihedral_group n
  st x = <*g1,1_(INT.Group 2)*>
  holds x" = x |^ (n - 1)
proof
  let n be non zero Nat;
  let g1 be Element of INT.Group n;
  let x be Element of Dihedral_group n;
  assume A1: x = <*g1,1_(INT.Group 2)*>;
  A2: x * (x |^ (n - 1)) = x |^ (1 + (n - 1)) by GROUP_1:34
                        .= 1_(Dihedral_group n) by A1,Th101;
  (x |^ (n - 1)) * x = x |^ ((n - 1) + 1) by GROUP_1:34
                    .= 1_(Dihedral_group n) by A1,Th101;
  hence x" = x |^ (n - 1) by A2, GROUP_1:5;
end;
