 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;

theorem Th2:
  for R being non degenerated Ring,
     V being LeftMod of R,
     A being Subset of V st
  A is linearly-independent holds not 0.V in A
proof
  let R be non degenerated Ring,
     V be LeftMod of R,
     A be Subset of V;
  assume that
A2: A is linearly-independent and
A3: 0.V in A;
  deffunc U(set) = 0.R;
  consider f be Function of the carrier of V, the carrier of R such that
A4: f.(0.V) = 1.R and
A5: for v be Element of V st v <> 0.V holds f.v = U(v) from FUNCT_2:sch 6;
  reconsider f as Element of Funcs(the carrier of V, the carrier of R) by
FUNCT_2:8;
  ex T be finite Subset of V st for v being Vector of V
    st not v in T holds f.v = 0.R
  proof
    take T = {0.V};
    let v be Vector of V;
    assume not v in T;
    then v <> 0.V by TARSKI:def 1;
    hence thesis by A5;
  end;
  then reconsider f as Linear_Combination of V by VECTSP_6:def 1;
A6: Carrier(f) = {0.V}
  proof
    thus Carrier(f) c= {0.V}
    proof
      let x be object;
      assume x in Carrier(f);
      then consider v being Vector of V such that
A7:   v = x and
A8:   f.v <> 0.R;
      v = 0.V by A5,A8;
      hence thesis by A7,TARSKI:def 1;
    end;
    let x be object;
    assume x in {0.V};
    then x = 0.V by TARSKI:def 1;
    hence thesis by A4;
  end;
  then Carrier(f) c= A by A3,ZFMISC_1:31;
  then reconsider f as Linear_Combination of A by VECTSP_6:def 4;
  Sum(f) = f.(0.V) * 0.V by A6,VECTSP_6:20
    .= 0.V by VECTSP_1:14;
  hence contradiction by A2,A6;
end;
