 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
  for R being commutative non degenerated almost_left_invertible
     Abelian add-associative right_zeroed right_complementable
     associative well-unital distributive non empty doubleLoopStr,
      V being LeftMod of R,
      v being Vector of V holds
  {v} is linearly-independent iff v <> 0.V
proof
  let R be commutative non degenerated almost_left_invertible
     Abelian add-associative right_zeroed right_complementable
     associative well-unital distributive non empty doubleLoopStr;
  let V be LeftMod of R,
      v be Vector of V;
  thus {v} is linearly-independent implies v <> 0.V
  proof
    assume {v} is linearly-independent;
    then not 0.V in {v} by Th2;
    hence thesis by TARSKI:def 1;
  end;
  assume
A1: v <> 0.V;
  let l be Linear_Combination of {v};
A2: Carrier(l) c= {v} by VECTSP_6:def 4;
  assume
A3: Sum(l) = 0.V;
  now
    per cases by A2,ZFMISC_1:33;
    suppose
      Carrier(l) = {};
      hence thesis;
    end;
    suppose
A4:   Carrier(l) = {v};
A5:   0.V = l.v * v by A3,VECTSP_6:17;
      now
        assume v in Carrier(l);
        then ex u being Vector of V st v = u & l.u <> 0.R;
        hence contradiction by A1,A5,VECTSP_1:15;
      end;
      hence thesis by A4,TARSKI:def 1;
    end;
  end;
  hence thesis;
end;
