
theorem Th1:
  for L being add-associative right_zeroed right_complementable
  associative commutative well-unital distributive almost_left_invertible non
  empty doubleLoopStr for x being Element of L st x <> 0.L holds -(x") = (-x)"
proof
  let L be add-associative right_zeroed right_complementable associative
  commutative well-unital distributive almost_left_invertible non empty
  doubleLoopStr;
  let x be Element of L;
  assume
A1: x <> 0.L;
A2: now
    assume -x = 0.L;
    then --x = 0.L by RLVECT_1:12;
    hence contradiction by A1,RLVECT_1:17;
  end;
  (-x) * (-(x")) = -((-x) * x") by VECTSP_1:8
    .= -(-(x * x")) by VECTSP_1:8
    .= -(- 1_L) by A1,VECTSP_1:def 10
    .= 1_L by RLVECT_1:17;
  hence thesis by A2,VECTSP_1:def 10;
end;
