
theorem
  for V being RealLinearSpace, W being Subspace of V holds Up(W) is
  Affine & 0.V in the carrier of W
proof
  let V be RealLinearSpace;
  let W be Subspace of V;
  for x,y being VECTOR of V, a being Real st x in Up(W) & y in Up(W)
    holds (1 - a) * x + a * y in Up(W)
  proof
    let x,y be VECTOR of V;
    let a be Real;
    assume that
A1: x in Up(W) and
A2: y in Up(W);
    reconsider aa=a as Real;
    y in W by A2;
    then
A3: aa * y in W by RLSUB_1:21;
    x in W by A1;
    then (1 - aa) * x in W by RLSUB_1:21;
    then (1 - a) * x + a * y in W by A3,RLSUB_1:20;
    hence thesis;
  end;
  hence Up(W) is Affine;
  0.V in W by RLSUB_1:17;
  hence thesis;
end;
