theorem
  for K be add-associative right_zeroed right_complementable
  left-distributive left_unital non empty doubleLoopStr for a be Element of K
  holds (- 1.K) * a = - a
proof
  let K be add-associative right_zeroed right_complementable left-distributive
  left_unital non empty doubleLoopStr;
  let x be Element of K;
  thus (- 1.K) * x = (0.K - 1.K) * x by RLVECT_1:14
    .= 0.K * x - 1.K * x by VECTSP_1:13
    .= 0.K - 1.K * x
    .= - 1.K * x by RLVECT_1:14
    .= - x;
end;
