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 Th62:
  R = INT.Ring &
:::  R is commutative non degenerated &
  V is Mult-cancelable implies
  (v1 <> v2 & {v1,v2} is linearly-independent iff v2 <> 0.V &
  for a, b being Element of R st b <> 0.R holds
    b * v1 <> a * v2)
  proof
    assume A1: R = INT.Ring & V is Mult-cancelable;
    thus v1 <> v2 & {v1,v2} is linearly-independent implies v2 <> 0.V &
    for a, b being Element of R st b <> 0.R holds
      b * v1 <> a * v2
    proof
      set N0=0.R,N1=-1.R;
      deffunc F(Element of V)=0.R;
      assume that
      A2: v1 <> v2 and
      A3: {v1,v2} is linearly-independent;
      thus v2 <> 0.V by A3,Th60,A1;
      let a, b be Element of R;
      assume A4: b <> 0.R;
      set Na= a;
      set Nb=-b;
      consider f such that
      A5: f.v1 = Nb & f.v2 = Na and
      A6: for v being Element of V st v <> v1 & v <> v2 holds f.v = F(v)
      from FUNCT_2:sch 7(A2);
      reconsider f as Element of Funcs(the carrier of V,
        the carrier of R) by FUNCT_2:8;
      now
        let v;
        assume not v in {v1,v2};
        then v <> v1 & v <> v2 by TARSKI:def 2;
        hence f.v = 0.R by A6;
      end;
      then reconsider f as Linear_Combination of V by VECTSP_6:def 1;
      Carrier(f) c= {v1,v2}
      proof
        let x be object;
        assume x in Carrier(f); then
        A7: ex u st x = u & f.u <> 0.R;
        assume not x in {v1,v2};
        then x <> v1 & x <> v2 by TARSKI:def 2;
        hence thesis by A6,A7;
      end;
      then reconsider f as Linear_Combination of {v1,v2} by VECTSP_6:def 4;
      Nb <> 0.R by A4,VECTSP_1:28; then
      f.v1 <> 0.R by A5;
      then
      A8: v1 in Carrier(f);
      set w = a * v2;
      assume A9: b * v1 = a * v2;
      Sum(f) = Nb*v1 + Na*v2 by A2,A5,Th22
      .= b*(-v1)+ Na*v2 by ZMODUL01:5,A1
      .= (- w) + w by A9,ZMODUL01:6,A1
      .= - (w - w) by RLVECT_1:33
      .= - 0.V by RLVECT_1:15
      .= 0.V by RLVECT_1:12; then
      Carrier f = {} by VECTSP_7:def 1,A3;
      hence thesis by A8;
    end;
    assume
    A10: v2 <> 0.V;
    assume
    A11: for a, b being Element of R st b <> 0.R holds
      b * v1 <> a * v2;
    A12: 1.R * v2 = v2 & 1.R * v1 = v1;
    hence v1 <> v2 by A11,A1;
    let l be Linear_Combination of {v1,v2};
    assume that
    A13: Sum(l) = 0.V and
    A14: Carrier(l) <> {};
    A15: 0.V = l.v1 * v1 + l.v2 * v2 by A11,A12,A13,Th22,A1;
    set x = the Element of Carrier(l);
    Carrier(l) c= {v1,v2} by VECTSP_6:def 4; then
    A16: x in {v1,v2} by A14;
    x in Carrier(l) by A14; then
    A17: ex u st x = u & l.u <> 0.R;
    now
      per cases by A17,A16,TARSKI:def 2;
      suppose
        A18: l.v1 <> 0.R;
        l.v1 * v1 = - (l.v2 * v2) by A15,RLVECT_1:6
        .= (- 1.R) * (l.v2 * v2) by ZMODUL01:2
        .=(-1.R)*(l.v2)*v2 by VECTSP_1:def 16;
        hence thesis by A11,A18;
      end;
      suppose
        A19: l.v2 <> 0.R & l.v1 = 0.R;
        0.V = l.v1 * v1 + l.v2 * v2 by A11,A12,A13,Th22,A1
        .= 0.V + l.v2 * v2 by A19,ZMODUL01:1,A1
        .= l.v2 * v2 by RLVECT_1:4;
        hence thesis by A1,A10,A19;
      end;
    end;
    hence thesis;
  end;
