
theorem Th12:
  for L being unital associative non empty multMagma, x being
  Element of L, n,m being Nat holds (power L).(x,n*m) = (power L).((
  power L).(x,n),m)
proof
  let L be unital associative non empty multMagma, x be Element of L,
      n be Nat;
  defpred P[Nat] means (power L).(x,n*$1) = (power L).((power L).(x,n),$1);
  set pL = power L;
  reconsider nn= n as Element of NAT by ORDINAL1:def 12;
A1: for m being Nat st P[m] holds P[m+1]
  proof
    let m be Nat such that
A2: P[m];
     reconsider nm= n*m, mm = m as Element of NAT by ORDINAL1:def 12;
     pL.(x,n*(m+1)) = pL.(x,n*m+n*1) .= pL.(x,nm)*pL.(x,nn) by POLYNOM2:1
      .= pL.(pL.(x,nn),mm+1) by A2,GROUP_1:def 7;
   hence thesis;
  end;
  pL.(x,n*(0 qua Nat)) = 1_L by GROUP_1:def 7
    .= pL.(pL.(x,nn),0) by GROUP_1:def 7;
  then
A3: P[0];
  thus for m being Nat holds P[m] from NAT_1:sch 2(A3,A1);
end;
