reserve x,y for object,
  R for Ring,
  V for LeftMod of R,
  L for Linear_Combination of V,
  a for Scalar of R,
  v,u for Vector of V,
  F,G for FinSequence of the carrier of V,
  C for finite Subset of V;
reserve X,Y,Z for set,
  A,B for Subset of V,
  T for finite Subset of V,
  l for Linear_Combination of A,
  f,g for Function of the carrier of V,the carrier of R;
reserve R for Skew-Field;
reserve a,b for Scalar of R;
reserve V for LeftMod of R;
reserve v,v1,v2,u for Vector of V;
reserve f for Function of the carrier of V, the carrier of R;

theorem Th16:
  v1 <> v2 & {v1,v2} is linearly-independent iff v2 <> 0.V &
    for a holds v1 <> a * v2
proof
  thus v1 <> v2 & {v1,v2} is linearly-independent implies v2 <> 0.V & for a
  holds v1 <> a * v2
  proof
    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,VECTSP_7:28;
    let a;
    consider f such that
A4: f.v1 = - 1.R & f.v2 = a and
A5: 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 A5;
    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
A6:   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 A5,A6;
    end;
    then reconsider f as Linear_Combination of {v1,v2} by VECTSP_6:def 4;
A7: now
      assume not v1 in Carrier(f);
      then 0.R = - 1.R by A4;
      hence contradiction by Th1;
    end;
    set w = a * v2;
    assume v1 = a * v2;
    then Sum(f) = (- 1.R) * w + w by A2,A4,VECTSP_6:18
      .= (- w) + w by VECTSP_1:14
      .= 0.V by RLVECT_1:5;
    hence thesis by A3,A7;
  end;
  assume
A8: v2 <> 0.V;
  assume
A9: for a holds v1 <> a * v2;
A10: 1.R * v2 = v2;
  hence v1 <> v2 by A9;
  let l be Linear_Combination of {v1,v2};
  assume that
A11: Sum(l) = 0.V and
A12: Carrier(l) <> {};
A13: 0.V = l.v1 * v1 + l.v2 * v2 by A9,A10,A11,VECTSP_6:18;
  set x = the Element of Carrier(l);
  Carrier(l) c= {v1,v2} by VECTSP_6:def 4; then
A14: x in {v1,v2} by A12;
  x in Carrier(l) by A12; then
A15: ex u st x = u & l.u <> 0.R;
  now
    per cases by A15,A14,TARSKI:def 2;
    suppose
A16:  l.v1 <> 0.R;
      0.V = (l.v1)" * (l.v1 * v1 + l.v2 * v2) by A13,VECTSP_2:30
        .= (l.v1)" * (l.v1 * v1) + (l.v1)" * (l.v2 * v2) by VECTSP_1:def 14
        .= (l.v1)" * l.v1 * v1 + (l.v1)" * (l.v2 * v2) by VECTSP_1:def 16
        .= (l.v1)" * l.v1 * v1 + (l.v1)" * l.v2 * v2 by VECTSP_1:def 16
        .= 1.R * v1 + (l.v1)" * l.v2 * v2 by A16,Lm2
        .= v1 + (l.v1)" * l.v2 * v2;
      then v1 = - ((l.v1)" * l.v2 * v2) by VECTSP_1:16
        .= (- 1.R) * ((l.v1)" * l.v2 * v2) by VECTSP_1:14
        .= ((- 1.R) * ((l.v1)" * l.v2)) * v2 by VECTSP_1:def 16;
      hence thesis by A9;
    end;
    suppose
A17:  l.v2 <> 0.R & l.v1 = 0.R;
      0.V = (l.v2)" * (l.v1 * v1 + l.v2 * v2) by A13,VECTSP_2:30
        .= (l.v2)" * (l.v1 * v1) + (l.v2)" * (l.v2 * v2) by VECTSP_1:def 14
        .= (l.v2)" * l.v1 * v1 + (l.v2)" * (l.v2 * v2) by VECTSP_1:def 16
        .= (l.v2)" * l.v1 * v1 + 1.R * v2 by A17,Lm2
        .= (l.v2)" * l.v1 * v1 + v2
        .= 0.R * v1 + v2 by A17
        .= 0.V + v2 by VECTSP_2:30
        .= v2 by RLVECT_1:def 4;
      hence thesis by A8;
    end;
  end;
  hence thesis;
end;
