
theorem Th22:
  for L being almost_left_invertible associative well-unital
add-associative right_zeroed right_complementable left-distributive commutative
  non degenerated non empty doubleLoopStr, k being Element of NAT, x being
  Element of L st x <> 0.L holds pow(x", k) = pow(x, -k)
proof
  let L be almost_left_invertible associative well-unital add-associative
right_zeroed right_complementable left-distributive commutative non degenerated
  non empty doubleLoopStr;
  let k be Element of NAT;
  let x be Element of L;
  assume
A1: x <> 0.L;
  pow(x", k) = (x") |^ k by Def2
    .= (x |^ k)" by A1,Lm8
    .= (pow(x,k))" by Def2
    .= pow(x,-k) by A1,Th18;
  hence thesis;
end;
