reserve x,y for object,
  R for Ring,
  V for LeftMod of R,
  L for Linear_Combination of V,
  a for Scalar of R,
  v,u for Vector of V,
  F,G for FinSequence of the carrier of V,
  C for finite Subset of V;
reserve X,Y,Z for set,
  A,B for Subset of V,
  T for finite Subset of V,
  l for Linear_Combination of A,
  f,g for Function of the carrier of V,the carrier of R;
reserve R for Skew-Field;
reserve a,b for Scalar of R;
reserve V for LeftMod of R;
reserve v,v1,v2,u for Vector of V;
reserve f for Function of the carrier of V, the carrier of R;

theorem
  {v} is linearly-independent iff v <> 0.V
proof
  thus {v} is linearly-independent implies v <> 0.V
  proof
    assume {v} is linearly-independent;
    then not 0.V in {v} by VECTSP_7:2;
    hence thesis by TARSKI:def 1;
  end;
  assume
A2: v <> 0.V;
  let l be Linear_Combination of {v};
A3: Carrier(l) c= {v} by VECTSP_6:def 4;
  assume
A4: Sum(l) = 0.V;
  now
    per cases by A3,ZFMISC_1:33;
    suppose
      Carrier(l) = {};
      hence thesis;
    end;
    suppose
A5:   Carrier(l) = {v};
A6:   0.V = l.v * v by A4,VECTSP_6:17;
       not v in Carrier(l) by A2,A6,VECTSP_2:30,VECTSP_6:2;
      hence thesis by A5,TARSKI:def 1;
    end;
  end;
  hence thesis;
end;
