 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 
  for GF be Ring,
      V be LeftMod of GF,
      A be Subset of V holds
  Lin(A) = (0).V implies A = {} or A = {0.V}
proof
  let GF be Ring,
      V be LeftMod of GF,
      A be Subset of V;
  assume that
A1: Lin(A) = (0).V and
A2: A <> {};
  thus A c= {0.V}
  proof
    let x be object;
    assume x in A;
    then x in Lin(A) by Th8;
    then x = 0.V by A1,VECTSP_4:35;
    hence thesis by TARSKI:def 1;
  end;
  set y = the Element of A;
  let x be object;
  assume x in {0.V};
  then
A3: x = 0.V by TARSKI:def 1;
  y in A & y in Lin(A) by A2,Th8;
  hence thesis by A1,A3,VECTSP_4:35;
end;
