reserve G for Group,
  a,b for Element of G,
  m, n for Nat,
  p for Prime;

theorem Th8:
  for G being Group, N being normal Subgroup of G,
      a being Element of G, S being Element of G./.N holds
      S = a * N implies
  for n holds S |^ n = (a |^ n) * N
proof
  let G be Group;
  let N be normal Subgroup of G;
  let a be Element of G;
  let S be Element of G./.N;
  assume
A1: S = a * N;
  defpred P[Nat] means for n holds S |^ $1 = (a |^ $1) * N;
A2: S |^ 0 = 1_ (G./.N) by GROUP_1:25;
  (a |^ 0) * N = 1_G * N by GROUP_1:25
              .= carr N by GROUP_2:109;
  then
A3: P[0] by A2,GROUP_6:24;
A4: now
    let n;
    assume
A5: P[n];
 S |^ (n + 1) = S |^ n * S by GROUP_1:34
                .= @ (S |^ n) * @ S by GROUP_6:def 3
                .= ((a |^ n) * N) * (a * N) by A1,A5
                .= ((a |^ n) * a) * N  by GROUP_11:1
                .= (a |^ (n + 1)) * N by GROUP_1:34;
    hence P[n+1];
    end;
  for n holds P[n] from NAT_1:sch 2(A3,A4);
  hence thesis;
end;
