 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 Th101:
  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 |^ n = 1_(Dihedral_group n)
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: g1 |^ n = g1 |^ (card (INT.Group n))
             .= 1_(INT.Group n) by GR_CY_1:9;
  thus x |^ n = <*(g1|^n),1_(INT.Group 2)*> by A1, Th25
             .= <* 1_(INT.Group n), 1_(INT.Group 2) *> by A2
             .= 1_(Dihedral_group n) by Th17;
end;
