reserve n for Element of NAT;
reserve i for Integer;
reserve G,H,I for Group;
reserve A,B for Subgroup of G;
reserve N for normal Subgroup of G;
reserve a,a1,a2,a3,b,b1 for Element of G;
reserve c,d for Element of H;
reserve f for Function of the carrier of G, the carrier of H;
reserve x,y,y1,y2,z for set;
reserve A1,A2 for Subset of G;
reserve N for normal Subgroup of G;
reserve S,T1,T2 for Element of G./.N;
reserve g,h for Homomorphism of G,H;
reserve h1 for Homomorphism of H,I;

theorem Th36:
  g.(a |^ n) = (g.a) |^ n
proof
  defpred Q[Nat] means g.(a |^ $1) = (g.a) |^ $1;
A1: for n being Nat st Q[n] holds Q[n + 1]
  proof
    let n be Nat;
    assume
A2: Q[n];
    thus g.(a |^ (n + 1)) = g.(a |^ n * a) by GROUP_1:34
      .= (g.a) |^ n * g.a by A2,Def6
      .= (g.a) |^ (n + 1) by GROUP_1:34;
  end;
  g.(a |^ 0) = g.(1_G) by GROUP_1:25
    .= 1_H by Th31
    .= (g.a) |^ 0 by GROUP_1:25;
  then
A3: Q[0];
  for n being Nat holds Q[n] from NAT_1:sch 2(A3,A1);
  hence thesis;
end;
