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;
