 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);
reserve x for set,
  R for Ring,
  V for LeftMod of R,
  v,v1,v2 for Vector of V,
  A, B for Subset of V;
reserve R for domRing,
  V for LeftMod of R,
  A,B for Subset of V,
  l for Linear_Combination of A,
  f,g for Function of the carrier of V, the carrier of R;

theorem Th12:
  for W being strict Subspace of V st
    R is non degenerated & A = the carrier of W holds Lin(A) = W
proof
  let W be strict Subspace of V;
  assume that
  R is non degenerated and
A2: A = the carrier of W;
A1: 0.R <> 1.R;
  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 st v = Sum(l) by Th7;
      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,Th8;
  end;
  hence thesis by VECTSP_4:30;
end;
