theorem
  R is non degenerated & V is Mult-cancelable implies
  ({v} is linearly-independent iff v <> 0.V)
  proof
    assume A1: R is non degenerated & V is Mult-cancelable;
    thus {v} is linearly-independent implies v <> 0.V
    proof
      assume {v} is linearly-independent;
      then not 0.V in {v} by Th57,A1;
      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};
        then
        A6: 0.V = l.v * v by A4,Th24;
        now
          assume v in Carrier(l);
          then ex u st v = u & l.u <> 0.R;
          hence contradiction by A2,A6,A1;
        end;
        hence thesis by A5,TARSKI:def 1;
      end;
    end;
    hence thesis;
  end;
