reserve V for RealLinearSpace,
  W for Subspace of V,
  x, y, y1, y2 for set,
  i, n for Element of NAT,
  v for VECTOR of V,
  KL1, KL2 for Linear_Combination of V,
  X for Subset of V;

theorem Th19:
  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 RLVECT_3:14;
    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,Th12,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 RLSUB_1:28;
end;
