 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 :: TH110
  for n being non zero Nat
  for a2 being Element of INT.Group 2
  for y being Element of Dihedral_group n
  st y = <*(1_(INT.Group n)),a2*>
  holds y*y = 1_(Dihedral_group n)
proof
  let n be non zero Nat;
  let a2 be Element of INT.Group 2;
  let y be Element of Dihedral_group n;
  assume A1: y = <*(1_(INT.Group n)),a2*>;
  A2: a2 |^ 2 = a2 |^ (card (INT.Group 2))
             .= 1_(INT.Group 2) by GR_CY_1:9;
  thus y*y = y |^ 2 by GROUP_1:27
          .= <* 1_(INT.Group n), (a2 |^ 2) *> by A1,Th28
          .= <* 1_(INT.Group n), 1_(INT.Group 2) *> by A2
          .= 1_(Dihedral_group n) by Th17;
end;
