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
  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
  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
A1: v1 <> v2 & {v1,v2} is linearly-independent;
    let a,b;
    assume that
A2: a * v1 + b * v2 = 0.V and
A3: a <> 0.R or b <> 0.R;
    now
      per cases by A3;
      suppose
A4:     a <> 0.R;
        0.V = a" * (a * v1 + b * v2) by A2,VECTSP_2:30
          .= a" * (a * v1) + a" * (b * v2) by VECTSP_1:def 14
          .= (a" * a) * v1 + a" * (b * v2) by VECTSP_1:def 16
          .= (a" * a) * v1 + (a" * b) * v2 by VECTSP_1:def 16
          .= 1.R * v1 + (a" * b) * v2 by A4,Lm2
          .= v1 + (a" * b) * v2;
        then v1 = - ((a" * b) * v2) by VECTSP_1:16
          .= (- 1.R) * ((a" * b) * v2) by VECTSP_1:14
          .= (- 1.R) * (a" * b) * v2 by VECTSP_1:def 16;
        hence thesis by A1,Th16;
      end;
      suppose
A5:     b <> 0.R;
        0.V = b" * (a * v1 + b * v2) by A2,VECTSP_2:30
          .= b" * (a * v1) + b" * (b * v2) by VECTSP_1:def 14
          .= (b" * a) * v1 + b" * (b * v2) by VECTSP_1:def 16
          .= (b" * a) * v1 + 1.R* v2 by A5,Lm2
          .= (b" * a) * v1 + v2;
        then v2 = - ((b" * a) * v1) by VECTSP_1:16
          .= (- 1.R) * ((b" * a) * v1) by VECTSP_1:14
          .= (- 1.R) * (b" * a) * v1 by VECTSP_1:def 16;
        hence thesis by A1,Th16;
      end;
    end;
    hence thesis;
  end;
  assume
A6: for a,b st a * v1 + b * v2 = 0.V holds a = 0.R & b = 0.R;
A7: now
    let a;
    assume v1 = a * v2;
    then v1 = 0.V + a * v2 by RLVECT_1:def 4;
    then 0.V = v1 - a * v2 by RLSUB_2:61
      .= v1 + ((- a) * v2) by VECTSP_1:21
      .= 1.R * v1 + (- a) * v2;
    hence contradiction by A6;
  end;
  now
    assume
A8: v2 = 0.V;
    0.V = 0.V + 0.V by RLVECT_1:def 4
      .= 0.R * v1 + 0.V by VECTSP_2:30
      .= 0.R * v1 + 1.R * v2 by A8;
    hence contradiction by A6;
  end;
  hence thesis by A7,Th16;
end;
