 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 Th114:
  for n being non zero Nat
  for g1 being Element of INT.Group n st g1 = 1
  for a2 being Element of INT.Group 2 st a2 = 1
  for x,y,z being Element of Dihedral_group n
  st x = <*g1,1_(INT.Group 2)*> & y = <*(1_(INT.Group n)),a2*>
  holds z in center Dihedral_group n
  iff (y*z = z*y & for i being Nat holds (x |^ i)*z = z*(x |^ i))
proof
  let n be non zero Nat;
  let g1 be Element of INT.Group n;
  assume A1: g1 = 1;
  let a2 be Element of INT.Group 2;
  assume A2: a2 = 1;
  let x,y,z be Element of Dihedral_group n;
  assume A3: x = <*g1,1_(INT.Group 2)*>;
  assume A4: y = <*(1_(INT.Group n)),a2*>;
  thus z in center Dihedral_group n implies
    (y * z = z * y & for i being Nat holds (x |^ i)*z = z*(x |^ i))
      by GROUP_5:77;
  assume A6: y*z = z*y;
  assume A7: for i being Nat holds (x |^ i)*z = z*(x |^ i);
  for g being Element of Dihedral_group n
  holds z * g = g * z
  proof
    let g be Element of Dihedral_group n;
    consider k being Nat such that
    B1: g = (x |^ k)*y or g = x |^ k by A1,A2,A3,A4,Th107;
    per cases by B1;
    suppose B2: g = (x |^ k)*y;
      hence z * g = (z * (x |^ k)) * y by GROUP_1:def 3
                .= ((x |^ k) * z) * y by A7
                .= (x |^ k) * (z * y) by GROUP_1:def 3
                .= (x |^ k) * (y * z) by A6
                .= ((x |^ k) * y) * z by GROUP_1:def 3
                .= g * z by B2;
    end;
    suppose B3: g = x |^ k;
      hence z * g = (x |^ k) * z by A7
                .= g * z by B3;
    end;
  end;
  hence z in center Dihedral_group n by GROUP_5:77;
end;
