 reserve x,y for object, X,Y,Z for set;
 reserve GF for commutative
     Abelian add-associative right_zeroed right_complementable
     associative well-unital distributive non empty doubleLoopStr;
 reserve a,b for Element of GF;
 reserve V for scalar-distributive vector-distributive
   scalar-associative scalar-unital add-associative right_zeroed
     right_complementable Abelian non empty ModuleStr over GF;
 reserve v,v1,v2,u for Vector of V;
 reserve A,B,C for Subset of V;
 reserve T for finite Subset of V;
 reserve l for Linear_Combination of A;
 reserve f,g for Function of V, GF;
 reserve GF for commutative non degenerated almost_left_invertible
     Abelian add-associative right_zeroed right_complementable
     associative well-unital distributive non empty doubleLoopStr;
 reserve a,b for Element of GF;
 reserve V for scalar-distributive vector-distributive
   scalar-associative scalar-unital add-associative right_zeroed
     right_complementable Abelian non empty ModuleStr over GF;
 reserve v,v1,v2,u for Vector of V;
 reserve A,B,C for Subset of V;
 reserve T for finite Subset of V;
 reserve l for Linear_Combination of A;
 reserve f,g for Function of V, GF;
reserve l0 for Linear_Combination of {}(the carrier of V);

theorem Th11:
  for GF be non degenerated Ring,
      V be LeftMod of GF,
      A be Subset of V holds
  for W being strict Subspace of V st A = the carrier of W holds Lin(A) = W
proof
  let GF be non degenerated Ring,
      V be LeftMod of GF,
      A be Subset of V;
  let W be strict Subspace of V;
  assume
A1: A = the carrier of W;
  now
    let v be Vector of V;
A2: 0.GF <> 1.GF;
    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 Th7;
      A is linearly-closed by A1,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 A1,Th8;
  end;
  hence thesis by VECTSP_4:30;
end;
