
theorem
  for V being RealLinearSpace holds Up((0).V) is circled
proof
  let V be RealLinearSpace, u be VECTOR of V, r be Real;
  assume that
  |.r.| <= 1 and
A1: u in Up((0).V);
  u in the carrier of (0).V by A1,RUSUB_4:def 5;
  then u in {0.V} by RLSUB_1:def 3;
  then u = 0.V by TARSKI:def 1;
  then r * u = 0.V;
  then r * u in {0.V} by TARSKI:def 1;
  then r * u in the carrier of (0).V by RLSUB_1:def 3;
  hence thesis by RUSUB_4:def 5;
end;
