 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 Th25:
  for x being Element of semidirect_product (G, A, phi)
  for g being Element of G
  st x = <*g,1_A*>
  for i being Integer
  holds x |^ i = <* (g |^ i), 1_A *>
proof
  let x be Element of semidirect_product (G, A, phi);
  let g be Element of G;
  assume A1: x = <*g,1_A*>;
  defpred P[Integer] means x |^ $1 = <* (g |^ $1), 1_A *>;
  A2: P[0]
  proof
    thus x |^ 0 = 1_(semidirect_product (G, A, phi)) by GROUP_1:25
               .= <* 1_G, 1_A *> by Th17
               .= <* (g |^ 0), 1_A *> 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 P[i];
    then B1: x |^ i = <* (g |^ i), 1_A *>;
    B2: x " = <* g ", 1_A *> by A1,Th24;
    B3: <* (g |^ i) * (g "), 1_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
                .= <* ((g |^ i)*(g ")), 1_A *> by B1, B2, B3, Th23
                .= <* ((g |^ i)*(g |^ (- 1))), 1_A *> by GROUP_1:32
                .= <* (g |^ (i + (- 1))), 1_A *> by GROUP_1:33
                .= <* (g |^ (i - 1)), 1_A *>;
    hence P[i - 1];
    B4: <* (g |^ i) * g, 1_A *> is Element of semidirect_product (G, A, phi)
      by Th9;
    x |^ (i + 1) = (x |^ i)*x by GROUP_1:34
                .= <* ((g |^ i)*g), 1_A *> by A1, B1,B4,Th23
                .= <* (g |^ (i + 1)), 1_A *> by GROUP_1:34;
    hence P[i + 1];
  end;
  for i being Integer holds P[i] from INT_1:sch 4(A2,A3);
  hence for i being Integer holds x |^ i = <* (g |^ i), 1_A *>;
end;
