reserve x, y, y1, y2 for set;
reserve R for Ring;
reserve V for LeftMod of R;
reserve u, v, w for VECTOR of V;
reserve F, G, H, I for FinSequence of V;
reserve i, j, k, n for Element of NAT;
reserve f, f9, g for sequence of V;
reserve K,L,L1,L2,L3 for Linear_Combination of V;
reserve a, b for Element of R;
reserve G, H1, H2, F, F1, F2, F3 for FinSequence of V;
reserve A, B for Subset of V,
  v1, v2, v3, u1, u2, u3 for Vector of V,
  f for Function of V, R,
  i for Element of NAT;
reserve l, l1, l2 for Linear_Combination of A;
 reserve e, e1, e2 for Element of LinComb(V);
reserve W, W1, W2, W3 for Submodule of V;
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, L1, L2 for Linear_Combination of V;
reserve l for Linear_Combination of A;
reserve F, G, H for FinSequence of V;
reserve f, g for Function of V, R;

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;
