
theorem Th15:
  for V being RealUnitarySpace, W being Subspace of V, v being
  VECTOR of V, a being Real st v in W holds a * 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 v be VECTOR of V;
  let a be Real;
  assume v in W;
  then
A1: v in the carrier of W;
  reconsider aa=a as Real;
  VW is linearly-closed by Lm1;
  then aa * v in the carrier of W by A1,RLSUB_1:def 1;
  hence thesis;
end;
