reserve x,y for object,
  R for Ring,
  V for LeftMod of R,
  L for Linear_Combination of V,
  a for Scalar of R,
  v,u for Vector of V,
  F,G for FinSequence of the carrier of V,
  C for finite Subset of V;
reserve X,Y,Z for set,
  A,B for Subset of V,
  T for finite Subset of V,
  l for Linear_Combination of A,
  f,g for Function of the carrier of V,the carrier of R;

theorem Th8:
  for R being non degenerated Ring,
      V being LeftMod of R,
      A being Subset of V
  for W being strict Subspace of V st A = the carrier of W
  holds Lin(A) = W
proof
  let R be non degenerated Ring,
      V be LeftMod of R,
      A be Subset of V;
  let W be strict Subspace of V;
  assume that
A2: A = the carrier of W;
  now
    let v be Vector of V;
    thus v in Lin(A) implies v in W
    proof
      assume v in Lin(A); then
A3:   ex l being Linear_Combination of A st v = Sum(l) by Th4;
A1:   0.R <> 1.R;
      A is linearly-closed by A2,VECTSP_4:33;
      then v in the carrier of W by A1,A2,A3,VECTSP_6:14;
      hence thesis by STRUCT_0:def 5;
    end;
    v in W iff v in the carrier of W by STRUCT_0:def 5;
    hence v in W implies v in Lin(A) by A2,Th5;
  end;
  hence thesis by VECTSP_4:30;
end;
