reserve GF for Field,
  V for VectSp of GF,
  W for Subspace of V,
  x, y, y1, y2 for set,
  i, n, m for Nat;

theorem Th16:
  for A being Subset of V st A c= the carrier of W holds Lin(A) is
  Subspace of W
proof
  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 VECTSP_7:7;
    Carrier(L) c= A by VECTSP_6:def 4;
    then
    ex K being Linear_Combination of W st Carrier(K) = Carrier (L) & Sum(L)
    = Sum(K) by A1,Th9,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 VECTSP_4:27;
end;
