 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 Th9:
  for GF be Ring,
      V be LeftMod of GF holds
  Lin({}(the carrier of V)) = (0).V
proof
  let GF be Ring,
      V be LeftMod of GF;
  set A = Lin({}(the carrier of V));
  now
    let v be Vector of V;
    thus v in A implies v in (0).V
    proof
      assume v in A; then
A1:   v in the carrier of A by STRUCT_0:def 5;
      the carrier of A = the set of all Sum(l0) where
        l0 is Linear_Combination of {}the carrier of V by Def2;
      then ex l0 being Linear_Combination of
        {}the carrier of V st v = Sum(l0) by A1;
      then v = 0.V by VECTSP_6:16;
      hence thesis by VECTSP_4:35;
    end;
    assume v in (0).V;
    then v = 0.V by VECTSP_4:35;
    hence v in A by VECTSP_4:17;
  end;
  hence thesis by VECTSP_4:30;
end;
