 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 Th109:
  for n being non zero Nat
  for g1 being Element of INT.Group n
  for a2 being Element of INT.Group 2 st a2 = 1
  for x,y being Element of Dihedral_group n
  st x = <*g1,1_(INT.Group 2)*> & y = <*(1_(INT.Group n)),a2*>
  for i,j being Nat
  holds ((x |^ i)*y) * (x |^ j) = (x |^ (n + i - j))*y
proof
  let n be non zero Nat;
  let g1 be Element of INT.Group n;
  let a2 be Element of INT.Group 2;
  assume A1: a2 = 1;
  let x,y be Element of Dihedral_group n;
  assume A2: x = <*g1,1_(INT.Group 2)*>;
  assume A3: y = <*(1_(INT.Group n)),a2*>;
  let i,j be Nat;
  thus ((x |^ i)*y) * (x |^ j) = (x |^ i) * (y * (x |^ j)) by GROUP_1:def 3
                              .= (x |^ i)* ((x |^ (n - j))*y) by A1,A2,A3,Th106
                              .= ((x |^ i) * (x |^ (n - j)))*y by GROUP_1:def 3
                              .= (x |^ (i + (n - j)))*y by GROUP_1:33
                              .= (x |^ (n + i - j))*y;
end;
