 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 Th99:
  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 ")*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;
  set a1 = 1_(INT.Group 2);
  set g2 = 1_(INT.Group n);
  assume A2: x = <*g1,a1*>;
  assume A3: y = <*g2,a2*>;
  reconsider phi1 = ((inversions INT.Group n) . a1),
    phi2 = ((inversions INT.Group n) . a2)
  as Homomorphism of (INT.Group n), (INT.Group n) by AUTGROUP:def 1;
  A4: phi2 = inverse_op (INT.Group n) by A1, DefInversions;
  x" = <* g1 ", a1 *> by A2,Th24;
  then A5: (x") * y = <* (g1 ") * (phi1 . g2), a1 * a2 *> by A3,Th14
                   .= <* (g1 ") * g2, a1 * a2 *> by GROUP_6:31
                   .= <* (g1 "), a1 * a2 *> by GROUP_1:def 4
                   .= <* (g1 "), a2 *> by GROUP_1:def 4;
  thus y * x = <* g2 * (phi2 . g1), a2 * a1 *> by A2,A3,Th14
            .= <* phi2.g1, a2 * a1 *> by GROUP_1:def 4
            .= <* phi2.g1, a2 *> by GROUP_1:def 4
            .= <* g1 ", a2 *> by A4, GROUP_1:def 6
            .= (x ")*y by A5;
end;
