theorem Th21:
  V1 is linearly-closed implies for v being VECTOR of V st v in V1
  holds - v in V1
proof
  assume
A1: V1 is linearly-closed;
  let v be VECTOR of V;
  assume v in V1;
  then (- 1r) * v in V1 by A1;
  hence thesis by Th3;
end;
