
theorem
  for R being unital associative non empty multMagma, a being Element
  of R, n,m being Nat holds (a|^n)|^m = a|^(n * m)
proof
  let R be unital associative non empty multMagma, a be Element of R, n,m be
  Nat;
  defpred P[Nat] means power(R).(a|^n,$1) = power(R).(a,n * $1);
A1: now
    let m be Nat;
    assume P[m];
    then power(R).(a|^n,m+1) = a|^(n * m) * (a|^n) by GROUP_1:def 7
      .= a|^(n * m + n) by Th10
      .= power(R).(a,n * (m + 1));
    hence P[m+1];
  end;
  power(R).(a|^n,0) = 1_R by GROUP_1:def 7
    .= power(R).(a,n * 0) by GROUP_1:def 7;
  then
A2: P[0];
  for k being Nat holds P[k] from NAT_1:sch 2(A2,A1);
  hence thesis;
end;
