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 = INT.Ring &
  V is Mult-cancelable implies
  (v1 <> v2 & {v1,v2} is linearly-independent iff
  for a,b st a * v1 + b * v2 = 0.V holds a = 0.R & b = 0.R)
  proof
    assume A1: R = INT.Ring & V is Mult-cancelable;
    thus v1 <> v2 & {v1,v2} is linearly-independent implies
    for a,b st a * v1 + b * v2 = 0.V holds a = 0.R & b = 0.R
    proof
      assume
      A2: v1 <> v2 & {v1,v2} is linearly-independent;
      let a, b;
      assume that
      A3: a * v1 + b * v2 = 0.V and
      A4: a <> 0.R or b <> 0.R;
      now
        per cases by A4;
        suppose
          A5: a <> 0.R;
          a * v1 = - (b * v2) by A3,RLVECT_1:6
          .= (- 1.R) * (b * v2) by ZMODUL01:2
          .=(-1.R)*b*v2 by VECTSP_1:def 16;
          hence thesis by A1,A2,A5,Th62;
        end;
        suppose
          A6: b <> 0.R;
          b * v2 = - (a * v1) by A3,RLVECT_1:6
          .= (- 1.R) * (a * v1) by ZMODUL01:2
          .=(-1.R)*a*v1 by VECTSP_1:def 16;
          hence thesis by A1,A2,A6,Th62;
        end;
      end;
      hence thesis;
    end;
    assume
    A7: for a, b st a * v1 + b * v2 = 0.V holds a = 0.R & b = 0.R;
    A8:
    now
      let a, b;
      assume A9: b <> 0.R;
      assume b*v1 = a * v2;
      then b*v1 = 0.V + a * v2 by RLVECT_1:4;
      then 0.V = b*v1 - a * v2 by RLSUB_2:61
      .= b*v1 + a * (- v2) by A1,ZMODUL01:6
      .= b*v1 + (- a) * v2 by A1,ZMODUL01:5;
      hence contradiction by A9,A7;
    end;
    now
      assume
      A10: v2 = 0.V;
      0.V = 0.V + 0.V by RLVECT_1:4
      .= 0.R * v1 + 0.V by ZMODUL01:1,A1
      .= 0.R * v1 + 1.R * v2 by A10;
      hence contradiction by A7,A1;
    end;
    hence thesis by A8,A1,Th62;
  end;
