
theorem Th49:
  for V being RealUnitarySpace, W being Subspace of V, v being
  VECTOR of V holds v + W = (- v) + W iff v in W
proof
  let V be RealUnitarySpace;
  let W be Subspace of V;
  let v be VECTOR of V;
  thus v + W = (- v) + W implies v in W
  proof
    assume v + W = (- v) + W;
    then v in (- v) + W by Th37;
    then consider u being VECTOR of V such that
A1: v = - v + u and
A2: u in W;
    0.V = v - (- v + u) by A1,RLVECT_1:15
      .= (v - (- v)) - u by RLVECT_1:27
      .= (v + v) - u by RLVECT_1:17
      .= (1 * v + v) - u by RLVECT_1:def 8
      .= (1 * v + 1 * v) - u by RLVECT_1:def 8
      .= ((1 + 1) * v) - u by RLVECT_1:def 6
      .= 2 * v - u;
    then 2" * (2 * v) = 2" * u by RLVECT_1:21;
    then (2" * 2) * v = 2" * u by RLVECT_1:def 7;
    then v = 2" * u by RLVECT_1:def 8;
    hence thesis by A2,Th15;
  end;
  assume
A3: v in W;
  then v + W = the carrier of W by Lm3;
  hence thesis by A3,Th45;
end;
