
theorem
  for V being RealUnitarySpace, W being Subspace of V, u,v being VECTOR
  of V holds u in W iff v + W = (v - u) + W
proof
  let V be RealUnitarySpace;
  let W be Subspace of V;
  let u,v be VECTOR of V;
A1: - u in W implies u in W
  proof
    assume - u in W;
    then - (- u) in W by Th16;
    hence thesis by RLVECT_1:17;
  end;
  - u in W iff v + W = (v + (- u)) + W by Th46;
  hence thesis by A1,Th16;
end;
