
theorem
  for V being RealLinearSpace holds Up((0).V) is convex
proof
  let V be RealLinearSpace;
  let u,v be VECTOR of V;
  let r be Real;
  assume that
  0 < r and
  r < 1 and
A1: u in Up((0).V) and
A2: v in Up((0).V);
  v in the carrier of (0).V by A2,RUSUB_4:def 5;
  then v in {0.V} by RLSUB_1:def 3;
  then
A3: v = 0.V by TARSKI:def 1;
  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 + (1-r) * v = 0.V + (1-r) * 0.V by A3
    .= 0.V + 0.V
    .= 0.V;
  then r * u + (1-r) * v in {0.V} by TARSKI:def 1;
  then r * u + (1-r) * v in the carrier of (0).V by RLSUB_1:def 3;
  hence thesis by RUSUB_4:def 5;
end;
