theorem
  u in W iff v + W = (v - u) + W
proof
A1: - u in W implies u in W
  proof
    assume - u in W;
    then - (- u) in W by Th41;
    hence thesis by RLVECT_1:17;
  end;
  -u in W iff v+W = (v+(-u))+W by Th71;
  hence thesis by A1,Th41;
end;
