reserve x,y for object, X,Y,Z for set;
reserve a,b for Real;
reserve k for Element of NAT;
reserve V for RealLinearSpace;
reserve W1,W2,W3 for Subspace 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 the carrier of V;
reserve f,g for Function of the carrier of V, REAL;
reserve p,q,r for FinSequence;
reserve M for non empty set;
reserve CF for Choice_Function of M;

theorem
  v1 <> v2 & {v1,v2} is linearly-independent iff for a,b st a * v1 + b *
  v2 = 0.V holds a = 0 & b = 0
proof
  thus v1 <> v2 & {v1,v2} is linearly-independent implies for a,b st a * v1 +
  b * v2 = 0.V holds a = 0 & b = 0
  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 or b <> 0;
    now
      per cases by A3;
      suppose
A4:     a <> 0;
        0.V = a" * (a * v1 + b * v2) by A2
          .= a" * (a * v1) + a" * (b * v2) by RLVECT_1:def 5
          .= (a" * a) * v1 + a" * (b * v2) by RLVECT_1:def 7
          .= (a" * a) * v1 + (a" * b) * v2 by RLVECT_1:def 7
          .= 1 * v1 + (a" * b) * v2 by A4,XCMPLX_0:def 7
          .= v1 + (a" * b) * v2 by RLVECT_1:def 8;
        then v1 = - ((a" * b) * v2) by RLVECT_1:6
          .= (- 1) * ((a" * b) * v2) by RLVECT_1:16
          .= (- 1) * (a" * b) * v2 by RLVECT_1:def 7;
        hence thesis by A1,Th12;
      end;
      suppose
A5:     b <> 0;
        0.V = b" * (a * v1 + b * v2) by A2
          .= b" * (a * v1) + b" * (b * v2) by RLVECT_1:def 5
          .= (b" * a) * v1 + b" * (b * v2) by RLVECT_1:def 7
          .= (b" * a) * v1 + (b" * b) * v2 by RLVECT_1:def 7
          .= (b" * a) * v1 + 1* v2 by A5,XCMPLX_0:def 7
          .= (b" * a) * v1 + v2 by RLVECT_1:def 8;
        then v2 = - ((b" * a) * v1) by RLVECT_1:def 10
          .= (- 1) * ((b" * a) * v1) by RLVECT_1:16
          .= (- 1) * (b" * a) * v1 by RLVECT_1:def 7;
        hence thesis by A1,Th12;
      end;
    end;
    hence thesis;
  end;
  assume
A6: for a,b st a * v1 + b * v2 = 0.V holds a = 0 & b = 0;
A7: now
    let a;
    assume v1 = a * v2;
    then v1 = 0.V + a * v2;
    then 0.V = v1 - a * v2 by RLSUB_2:61
      .= v1 + (- a * v2) by RLVECT_1:def 11
      .= v1 + a * (- v2) by RLVECT_1:25
      .= v1 + ((- a) * v2) by RLVECT_1:24
      .= 1 * v1 + (- a) * v2 by RLVECT_1:def 8;
    hence contradiction by A6;
  end;
  now
    assume
A8: v2 = 0.V;
    0.V = 0.V + 0.V
      .= 0 * v1 + 0.V by RLVECT_1:10
      .= 0 * v1 + 1 * v2 by A8;
    hence contradiction by A6;
  end;
  hence thesis by A7,Th12;
end;
