theorem
  for V being ComplexLinearSpace holds Up((0).V) is convex
proof
  let V be ComplexLinearSpace;
  let u,v be VECTOR of V;
  let z be Complex;
  assume that
  ex r being Real st z=r & 0 < r & r < 1 and
A1: u in Up((0).V) and
A2: v in Up((0).V);
  v in {0.V} by A2,CLVECT_1:def 9;
  then
A3: v = 0.V by TARSKI:def 1;
  u in {0.V} by A1,CLVECT_1:def 9;
  then u = 0.V by TARSKI:def 1;
  then z * u + (1r-z) * v = 0.V + (1r-z) * 0.V by A3,CLVECT_1:1
    .= 0.V + 0.V by CLVECT_1:1
    .= 0.V;
  then z * u + (1r-z) * v in {0.V} by TARSKI:def 1;
  hence thesis by CLVECT_1:def 9;
end;
