 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 Th5:
  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(set) = 0.GF;
    assume that
A1: v1 <> v2 and
A2: {v1,v2} is linearly-independent;
    thus v2 <> 0.V by A2,Th4;
    let a;
    consider f such that
A3: f.v1 = - 1_GF & f.v2 = a and
A4: for v being Element of V st v <> v1 & v <> v2 holds f.v = F(v)
    from FUNCT_2:sch 7(A1);
    reconsider f as Element of Funcs(the carrier of V, the carrier of GF) 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.GF by A4;
    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
A5:   ex u st x = u & f.u <> 0.GF;
      assume not x in {v1,v2};
      then x <> v1 & x <> v2 by TARSKI:def 2;
      hence thesis by A4,A5;
    end;
    then reconsider f as Linear_Combination of {v1,v2} by VECTSP_6:def 4;
A6: now
      assume not v1 in Carrier(f);
      then 0.GF = - 1_GF by A3;
      hence contradiction by VECTSP_6:49;
    end;
    set w = a * v2;
    assume v1 = a * v2;
    then Sum(f) = (- 1_GF) * w + w by A1,A3,VECTSP_6:18
      .= (- w) + w by VECTSP_1:14
      .= - (w - w) by VECTSP_1:17
      .= - 0.V by VECTSP_1:19
      .= 0.V by RLVECT_1:12;
    hence thesis by A2,A6;
  end;
  assume
A7: v2 <> 0.V;
  assume
A8: for a holds v1 <> a * v2;
A9: 1_GF * v2 = v2;
  hence v1 <> v2 by A8;
  let l be Linear_Combination of {v1,v2};
  assume that
A10: Sum(l) = 0.V and
A11: Carrier(l) <> {};
A12: 0.V = l.v1 * v1 + l.v2 * v2 by A8,A9,A10,VECTSP_6:18;
  set x = the Element of Carrier(l);
  Carrier(l) c= {v1,v2} by VECTSP_6:def 4;
  then
A13: x in {v1,v2} by A11;
  x in Carrier(l) by A11;
  then
A14: ex u st x = u & l.u <> 0.GF;
  now
    per cases by A14,A13,TARSKI:def 2;
    suppose
A15:  l.v1 <> 0.GF;
      0.V = (l.v1)" * (l.v1 * v1 + l.v2 * v2) by A12,VECTSP_1:15
        .= (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_GF * v1 + (l.v1)" * l.v2 * v2 by A15,VECTSP_1:def 10
        .= v1 + (l.v1)" * l.v2 * v2;
      then v1 = - ((l.v1)" * l.v2 * v2) by VECTSP_1:16
        .= (- 1_GF) * ((l.v1)" * l.v2 * v2) by VECTSP_1:14
        .= ((- 1_GF) * ((l.v1)" * l.v2)) * v2 by VECTSP_1:def 16;
      hence thesis by A8;
    end;
    suppose
A16:  l.v2 <> 0.GF & l.v1 = 0.GF;
      0.V = (l.v2)" * (l.v1 * v1 + l.v2 * v2) by A12,VECTSP_1:15
        .= (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 + (l.v2)" * l.v2 * v2 by VECTSP_1:def 16
        .= (l.v2)" * l.v1 * v1 + 1_GF * v2 by A16,VECTSP_1:def 10
        .= (l.v2)" * l.v1 * v1 + v2
        .= 0.GF * v1 + v2 by A16
        .= 0.V + v2 by VECTSP_1:15
        .= v2 by RLVECT_1:4;
      hence thesis by A7;
    end;
  end;
  hence thesis;
end;
