
theorem
  for V being RealUnitarySpace, W being Subspace of V, A being Subset of
  V st A c= the carrier of W holds Lin(A) is Subspace of W
proof
  let V be RealUnitarySpace;
  let W be Subspace of V;
  let A be Subset of V;
  assume
A1: A c= the carrier of W;
  now
    let w be object;
    assume w in the carrier of Lin(A);
    then w in Lin(A) by STRUCT_0:def 5;
    then consider L being Linear_Combination of A such that
A2: w = Sum(L) by Th1;
    Carrier(L) c= A by RLVECT_2:def 6;
    then
    ex K being Linear_Combination of W st Carrier(K) = Carrier(L) & Sum(L)
    = Sum(K) by A1,Th20,XBOOLE_1:1;
    hence w in the carrier of W by A2;
  end;
  then the carrier of Lin(A) c= the carrier of W;
  hence thesis by RUSUB_1:22;
end;
