 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 Th100:
  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*>
  holds y*x = (x |^ (n - 1))*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*>;
  A5: x " = <* g1", 1_(INT.Group 2)*> by A2, Th24
         .= <* g1 |^ (n - 1), 1_(INT.Group 2)*> by Th86
         .= x |^ (n - 1) by A2, Th25;
  thus y*x = (x ")*y by A1,A2,A3,Th99
          .= (x |^ (n - 1))*y by A5;
end;
