
theorem Th14:
  for V being RealUnitarySpace, W being Subspace of V, u,v being
  VECTOR of V st u in W & v in W holds u + v in W
proof
  let V be RealUnitarySpace;
  let W be Subspace of V;
  reconsider VW = the carrier of W as Subset of V by Def1;
  let u,v be VECTOR of V;
  assume u in W & v in W;
  then
A1: u in the carrier of W & v in the carrier of W;
  VW is linearly-closed by Lm1;
  then u + v in the carrier of W by A1,RLSUB_1:def 1;
  hence thesis;
end;
