reserve F for Field,
  x for Element of F,
  V for VectSp of F,
  v for Element of V;

theorem
  for V being add-associative right_zeroed right_complementable non
  empty addLoopStr, v,w being Element of V holds v+w=0.V iff -v=w
proof
  let V be add-associative right_zeroed right_complementable non empty
  addLoopStr, v,w be Element of V;
  v+w=0.V implies -v=w
  proof
    assume
A1: v+w=0.V;
    thus w = 0.V + w by RLVECT_1:4
      .= -v + v + w by RLVECT_1:5
      .= -v + 0.V by A1,RLVECT_1:def 3
      .= -v by RLVECT_1:4;
  end;
  hence thesis by RLVECT_1:5;
end;
