 reserve x,y for object, X,Y,Z for set;
 reserve GF for commutative
     Abelian add-associative right_zeroed right_complementable
     associative well-unital distributive non empty doubleLoopStr;
 reserve a,b for Element of GF;
 reserve V for scalar-distributive vector-distributive
   scalar-associative scalar-unital add-associative right_zeroed
     right_complementable Abelian non empty ModuleStr over GF;
 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 for Linear_Combination of A;
 reserve f,g for Function of V, GF;
 reserve GF for commutative non degenerated almost_left_invertible
     Abelian add-associative right_zeroed right_complementable
     associative well-unital distributive non empty doubleLoopStr;
 reserve a,b for Element of GF;
 reserve V for scalar-distributive vector-distributive
   scalar-associative scalar-unital add-associative right_zeroed
     right_complementable Abelian non empty ModuleStr over GF;
 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 for Linear_Combination of A;
 reserve f,g for Function of V, GF;

theorem
  v1 <> v2 & {v1,v2} is linearly-independent iff
  for a,b st a * v1 + b * v2 = 0.V holds a = 0.GF & b = 0.GF
proof
  thus v1 <> v2 & {v1,v2} is linearly-independent implies for a,b st a * v1 +
  b * v2 = 0.V holds a = 0.GF & b = 0.GF
  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.GF or b <> 0.GF;
    now
      per cases by A3;
      suppose
A4:     a <> 0.GF;
        0.V = a" * (a * v1 + b * v2) by A2,VECTSP_1:15
          .= 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_GF * v1 + (a" * b) * v2 by A4,VECTSP_1:def 10
          .= v1 + (a" * b) * v2;
        then v1 = - ((a" * b) * v2) by VECTSP_1:16
          .= (- 1_GF) * ((a" * b) * v2) by VECTSP_1:14
          .= (- 1_GF) * (a" * b) * v2 by VECTSP_1:def 16;
        hence thesis by A1,Th5;
      end;
      suppose
A5:     b <> 0.GF;
        0.V = b" * (a * v1 + b * v2) by A2,VECTSP_1:15
          .= 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 + (b" * b) * v2 by VECTSP_1:def 16
          .= (b" * a) * v1 + 1_GF* v2 by A5,VECTSP_1:def 10
          .= (b" * a) * v1 + v2;
        then v2 = - ((b" * a) * v1) by VECTSP_1:16
          .= (- 1_GF) * ((b" * a) * v1) by VECTSP_1:14
          .= (- 1_GF) * (b" * a) * v1 by VECTSP_1:def 16;
        hence thesis by A1,Th5;
      end;
    end;
    hence thesis;
  end;
  assume
A6: for a,b st a * v1 + b * v2 = 0.V holds a = 0.GF & b = 0.GF;
A7: now
    let a;
    assume v1 = a * v2;
    then v1 = 0.V + a * v2 by RLVECT_1:4;
    then 0.V = v1 - a * v2 by VECTSP_2:2
      .= v1 + ((- a) * v2) by VECTSP_1:21
      .= 1.GF * v1 + (- a) * v2;
    hence contradiction by A6;
  end;
  now
    assume
A8: v2 = 0.V;
    0.V = 0.V + 0.V by RLVECT_1:4
      .= 0.GF * v1 + 0.V by VECTSP_1:15
      .= 0.GF * v1 + 1.GF * v2 by A8;
    hence contradiction by A6;
  end;
  hence thesis by A7,Th5;
end;
