
theorem Th27:
  for L being associative commutative well-unital distributive
  almost_left_invertible non empty doubleLoopStr, x being Element of L, i,k
  being Element of NAT st 1 <= k holds pow(x, i*(k-1)) = pow(x|^i, k-1)
proof
  let L be associative commutative well-unital distributive
  almost_left_invertible non empty doubleLoopStr;
  let x be Element of L;
  let i,k be Element of NAT;
  assume 1 <= k;
  then 0 < k;
  then reconsider m = k-1 as Element of NAT by NAT_1:20;
  pow(x, i*(k-1)) = x|^(i*m) by Def2
    .= (x|^i) |^m by BINOM:11
    .= pow(x|^i,m) by Def2;
  hence thesis;
end;
