 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 :: TH104
  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)*>
  for j being Nat
  holds (x |^ j)" = x |^ (n - j)
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)*>;
  let j be Nat;
  A2: (g1 |^ j) " = g1 |^ (n - j)
  proof
    (g1 |^ (n - j)) * (g1 |^ j) = g1 |^ ((n - j) + j) by GROUP_1:33
                               .= g1 |^ (card (INT.Group n))
                               .= 1_(INT.Group n) by GR_CY_1:9;
    hence g1 |^ (n - j) = (g1 |^ j) " by GROUP_1:5;
  end;
  (x |^ j) = <* (g1 |^ j), 1_(INT.Group 2) *> by A1,Th25;
  hence (x |^ j) " = <* g1 |^ (n - j), 1_(INT.Group 2) *> by A2, Th24
                  .= (x |^ (n - j)) by A1, Th25;
end;
