
theorem Th55:
  for V being RealUnitarySpace, W being Subspace of V, u,v being
  VECTOR of V holds u + v in v + W iff u in W
proof
  let V be RealUnitarySpace;
  let W be Subspace of V;
  let u,v be VECTOR of V;
  thus u + v in v + W implies u in W
  proof
    assume u + v in v + W;
    then ex v1 being VECTOR of V st u + v = v + v1 & v1 in W;
    hence thesis by RLVECT_1:8;
  end;
  assume u in W;
  hence thesis;
end;
