
theorem Th3:
  for L being associative well-unital non empty multLoopStr for x
being Element of L for k1,k2 being Element of NAT holds power(L).(x,k1) * power
  (L).(x,k2) = power(L).(x,k1+k2)
proof
  let L be associative well-unital non empty multLoopStr, x be Element of L,
  k1,k2 be Element of NAT;
  defpred P[Nat] means ex j being Element of NAT st j = $1 & power(L).(x,k1) *
  power(L).(x,j) = power(L).(x,k1+j);
A1: now
    let j be Nat;
     reconsider jj=j as Element of NAT by ORDINAL1:def 12;
    assume
A2: P[j];
    power(L).(x,k1) * power(L).(x,j+1) = power(L).(x,k1) * (power(L).(x,jj)
    * x) by GROUP_1:def 7
      .= (power(L).(x,k1) * power(L).(x,jj)) * x by GROUP_1:def 3
      .= power(L).(x,(k1+j)+1) by A2,GROUP_1:def 7
      .= power(L).(x,k1+(j+1));
    hence P[j+1];
  end;
  1_L = 1.L;
  then power(L).(x,k1) * power(L).(x,0) = power(L).(x,k1) * 1.L by
GROUP_1:def 7
    .= power(L).(x,k1+0);
  then
A3: P[0];
  for k be Nat holds P[k] from NAT_1:sch 2(A3,A1);
  then
  ex j be Element of NAT st j = k2 & power(L).(x,k1) * power(L).(x,j) =
  power(L).(x,k1+j);
  hence thesis;
end;
