 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 Th120:
  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 in center 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) *>;
  1 in INT.Group 2 by EltsOfINTGroup2;
  then reconsider a2=1 as Element of INT.Group 2;
  reconsider y = <* 1_(INT.Group n), a2 *>
    as Element of Dihedral_group n by Th9;
  set z = x |^ k;
  A4: y * z = (x |^ (n - k))*y by A3,Th106
           .= (x |^ ((2*k) - k)) * y by A1
           .= z * y;
  for i being Nat holds (x |^ i)*z = z*(x |^ i)
  proof
    let i be Nat;
    thus (x |^ i)*z = x |^ (i + k) by GROUP_1:33
                   .= (x |^ k)*(x |^ i) by GROUP_1:33
                   .= z*(x |^ i);
  end;
  hence z in center Dihedral_group n by A2,A3,A4,Th114;
end;
