 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);

theorem Th28:
  for i being Integer
  for x being Element of semidirect_product (G, A, phi)
  for a being Element of A
  st x = <* 1_G, a*>
  holds x |^ i = <* 1_G, (a |^ i) *>
proof
  let i be Integer;
  let x be Element of semidirect_product (G, A, phi);
  let a be Element of A;
  assume A1: x = <* 1_G, a*>;
  defpred P[Integer] means x |^ $1 = <* 1_G, (a |^ $1) *>;
  A2: P[0]
  proof
    thus x |^ 0 = 1_(semidirect_product (G, A, phi)) by GROUP_1:25
               .= <* 1_G, 1_A *> by Th17
               .= <* 1_G, (a |^ 0) *> by GROUP_1:25;
  end;
  A3: for i being Integer st P[i] holds P[i - 1] & P[i + 1]
  proof
    let i be Integer;
    assume B1: P[i];
    B2: x " = <* 1_G, a " *> by A1, Th27;
B3: <* 1_G, (a |^ i)*(a ") *> is
      Element of semidirect_product (G, A, phi) by Th9;
    x |^ (i - 1) = (x |^ (i + (- 1)))
                .= (x |^ i)*(x |^ (- 1)) by GROUP_1:33
                .= (x |^ i)*(x ") by GROUP_1:32
                .= <* 1_G, ((a |^ i)*(a ")) *> by B1, B2, B3, Th26
                .= <* 1_G, ((a |^ i)*(a |^ (- 1))) *> by GROUP_1:32
                .= <* 1_G, (a |^ (i + (- 1))) *> by GROUP_1:33
                .= <* 1_G, (a |^ (i - 1)) *>;
    hence P[i - 1];
B4: <* 1_G, (a |^ i)*a *> is Element of semidirect_product (G, A, phi) by Th9;
    x |^ (i + 1) = (x |^ i)*x by GROUP_1:34
                .= <* 1_G, ((a |^ i)*a) *> by A1, B1,B4,Th26
                .= <* 1_G, (a |^ (i + 1)) *> by GROUP_1:34;
    hence P[i + 1];
  end;

  for i being Integer holds P[i] from INT_1:sch 4(A2,A3);
  hence x |^ i = <* 1_G, (a |^ i) *>;
end;
