
theorem
  for V being RealUnitarySpace, v1,v2,w1,w2 being VECTOR of V st v1 <>
  v2 & {v1,v2} is linearly-independent & v1 in Lin{w1,w2} & v2 in Lin{w1,w2}
  holds Lin{w1,w2} = Lin{v1,v2} & {w1,w2} is linearly-independent & w1 <> w2
proof
  let V be RealUnitarySpace;
  let v1,v2,w1,w2 be VECTOR of V;
  assume that
A1: v1 <> v2 and
A2: {v1,v2} is linearly-independent and
A3: v1 in Lin{w1,w2} and
A4: v2 in Lin{w1,w2};
  consider r1,r2 being Real such that
A5: v1 = r1 * w1 + r2 * w2 by A3,Th32;
  consider r3,r4 being Real such that
A6: v2 = r3 * w1 + r4 * w2 by A4,Th32;
  set t = r1 * r4 - r2 * r3;
A7: now
    assume r1 = 0 & r2 = 0;
    then v1 = 0.V + 0 * w2 by A5,RLVECT_1:10
      .= 0.V + 0.V by RLVECT_1:10
      .= 0.V by RLVECT_1:4;
    hence contradiction by A2,RLVECT_3:11;
  end;
  now
    assume
A8: r1 * r4 = r2 * r3;
    now
      per cases by A7;
      suppose
A9:     r1 <> 0;
        r1" * r1 * r4 = r1" * (r2 * r3) by A8,XCMPLX_1:4;
        then 1 * r4 = r1" * (r2 * r3) by A9,XCMPLX_0:def 7;
        then v2 = r3 * w1 + r3 * (r1" * r2) * w2 by A6
          .= r3 * w1 + r3 * (r1" * r2 * w2) by RLVECT_1:def 7
          .= r3 * 1 * (w1 + r1" * r2 * w2) by RLVECT_1:def 5
          .= r3 * (r1" * r1) * (w1 + r1" * r2 * w2) by A9,XCMPLX_0:def 7
          .= r3 * r1" * r1 * (w1 + r1" * r2 * w2)
          .= r3 * r1" * (r1 * (w1 + r1" * r2 * w2)) by RLVECT_1:def 7
          .= r3 * r1" * (r1 * w1 + r1 * (r1" * r2 * w2)) by RLVECT_1:def 5
          .= r3 * r1" * (r1 * w1 + r1 * (r1" * r2) * w2) by RLVECT_1:def 7
          .= r3 * r1" * (r1 * w1 + r1 * r1" * r2 * w2)
          .= r3 * r1" * (r1 * w1 + 1 * r2 * w2) by A9,XCMPLX_0:def 7
          .= r3 * r1" * (r1 * w1 + r2 * w2);
        hence contradiction by A1,A2,A5,RLVECT_3:12;
      end;
      suppose
A10:    r2 <> 0;
        r2" * (r1 * r4) = r2" * (r2 * r3) by A8
          .= r2" * r2 * r3
          .= 1 * r3 by A10,XCMPLX_0:def 7
          .= r3;
        then v2 = r4 * (r2" * r1) * w1 + r4 * w2 by A6
          .= r4 * (r2" * r1 * w1) + r4 * w2 by RLVECT_1:def 7
          .= r4 * 1 * ((r2" * r1 * w1) + w2) by RLVECT_1:def 5
          .= r4 * (r2" * r2) * ((r2" * r1 * w1) + w2) by A10,XCMPLX_0:def 7
          .= r4 * r2" * r2 * ((r2" * r1 * w1) + w2)
          .= r4 * r2" * (r2 * ((r2" * r1 * w1) + w2)) by RLVECT_1:def 7
          .= r4 * r2" * (r2 * (r2" * r1 * w1) + r2 * w2) by RLVECT_1:def 5
          .= r4 * r2" * (r2 * (r2" * r1) * w1 + r2 * w2) by RLVECT_1:def 7
          .= r4 * r2" * (r2 * r2" * r1 * w1 + r2 * w2)
          .= r4 * r2" * (1 * r1 * w1 + r2 * w2) by A10,XCMPLX_0:def 7
          .= r4 * r2" * (r1 * w1 + r2 * w2);
        hence contradiction by A1,A2,A5,RLVECT_3:12;
      end;
    end;
    hence contradiction;
  end;
  then
A11: r1 * r4 - r2 * r3 <> 0;
A12: now
    assume
A13: r2 <> 0;
    r2" * v1 = r2" * (r1 * w1) + r2" * (r2 * w2) by A5,RLVECT_1:def 5
      .= r2" * r1 * w1 + r2" * (r2 * w2) by RLVECT_1:def 7
      .= r2" * r1 * w1 + r2" * r2 * w2 by RLVECT_1:def 7
      .= r2" * r1 * w1 + 1 * w2 by A13,XCMPLX_0:def 7
      .= r2" * r1 * w1 + w2 by RLVECT_1:def 8;
    then
A14: w2 = r2" * v1 - r2" * r1 * w1 by RLSUB_2:61;
    then w2 = r2" * v1 - r2" * (r1 * w1) by RLVECT_1:def 7;
    then v2 = r3 * w1 + r4 * (r2" * (v1 - r1 * w1)) by A6,RLVECT_1:34
      .= r3 * w1 + r4 * r2" * (v1 - r1 * w1) by RLVECT_1:def 7
      .= r3 * w1 + (r4 * r2" * v1 - r4 * r2" * (r1 * w1)) by RLVECT_1:34
      .= r3 * w1 + r4 * r2" * v1 - r4 * r2" * (r1 * w1) by RLVECT_1:def 3
      .= r4 * r2" * v1 + r3 * w1 - (r4 * r2" * r1) * w1 by RLVECT_1:def 7
      .= r4 * r2" * v1 + (r3 * w1 - (r4 * r2" * r1) * w1) by RLVECT_1:def 3
      .= r4 * r2" * v1 + (r3 - (r4 * r2" * r1)) * w1 by RLVECT_1:35;
    then r2 * v2 = r2 * (r4 * r2" * v1) + r2 * ((r3 - (r4 * r2" * r1)) * w1)
    by RLVECT_1:def 5
      .= r2 * (r4 * r2") * v1 + r2 * ((r3 - (r4 * r2" * r1)) * w1) by
RLVECT_1:def 7
      .= r4 * (r2 * r2") * v1 + r2 * ((r3 - (r4 * r2" * r1)) * w1)
      .= r4 * 1 * v1 + r2 * ((r3 - (r4 * r2" * r1)) * w1) by A13,XCMPLX_0:def 7
      .= r4 * v1 + r2 * (r3 - (r4 * r2" * r1)) * w1 by RLVECT_1:def 7
      .= r4 * v1 + (r2 * r3 - r2 * r2" * (r4 * r1)) * w1
      .= r4 * v1 + (r2 * r3 - 1 * (r4 * r1)) * w1 by A13,XCMPLX_0:def 7
      .= r4 * v1 + (- t) * w1
      .= r4 * v1 + - t * w1 by RLVECT_4:3;
    then - t * w1 = r2 * v2 - r4 * v1 by RLSUB_2:61;
    then t * w1 = - (r2 * v2 - r4 * v1) by RLVECT_1:17
      .= r4 * v1 + - r2 * v2 by RLVECT_1:33;
    then t" * t * w1 = t" * (r4 * v1 + - r2 * v2) by RLVECT_1:def 7;
    then 1 * w1 = t" * (r4 * v1 + - r2 * v2) by A11,XCMPLX_0:def 7;
    then w1 = t" * (r4 * v1 + - r2 * v2) by RLVECT_1:def 8
      .= t" * (r4 * v1) + t" * (- r2 * v2) by RLVECT_1:def 5
      .= t" * r4 * v1 + t" * (- r2 * v2) by RLVECT_1:def 7
      .= t" * r4 * v1 + t" * ((- r2) * v2) by RLVECT_4:3
      .= t" * r4 * v1 + t" * (- r2) * v2 by RLVECT_1:def 7
      .= t" * r4 * v1 + (- t") * r2 * v2
      .= t" * r4 * v1 + (- t") * (r2 * v2) by RLVECT_1:def 7
      .= t" * r4 * v1 + - t" * (r2 * v2) by RLVECT_4:3;
    hence w1 = t" * r4 * v1 + - t" * r2 * v2 by RLVECT_1:def 7;
    then
A15: w2 = r2" * v1 - r2" * r1 * (t" * (r4 * v1) - t" * r2 * v2) by A14,
RLVECT_1:def 7
      .= r2" * v1 - r2" * r1 * (t" * (r4 * v1) - t" * (r2 * v2)) by
RLVECT_1:def 7
      .= r2" * v1 - r2" * r1 * (t" * (r4 * v1 - r2 * v2)) by RLVECT_1:34
      .= r2" * v1 - r1 * r2" * t" * (r4 * v1 - r2 * v2) by RLVECT_1:def 7
      .= r2" * v1 - r1 * (t" * r2") * (r4 * v1 - r2 * v2)
      .= r2" * v1 - r1 * ((t" * r2") * (r4 * v1 - r2 * v2)) by RLVECT_1:def 7
      .= r2" * v1 - r1 * (t" * (r2" * (r4 * v1 - r2 * v2))) by RLVECT_1:def 7
      .= r2" * v1 - r1 * (t" * (r2" * (r4 * v1) - r2" * (r2 * v2))) by
RLVECT_1:34
      .= r2" * v1 - r1 * (t" * (r2" * (r4 * v1) - r2" * r2 * v2)) by
RLVECT_1:def 7
      .= r2" * v1 - r1 * (t" * (r2" * r4 * v1 - r2" * r2 * v2)) by
RLVECT_1:def 7
      .= r2" * v1 - r1 * (t" * (r2" * r4 * v1 - 1 * v2)) by A13,XCMPLX_0:def 7
      .= r2" * v1 - r1 * (t" * (r2" * r4 * v1 - v2)) by RLVECT_1:def 8
      .= r2" * v1 - r1 * (t" * (r2" * r4 * v1) - t" * v2) by RLVECT_1:34
      .= r2" * v1 - (r1 * (t" * (r2" * r4 * v1)) - r1 * (t" * v2)) by
RLVECT_1:34
      .= r2" * v1 - r1 * (t" * (r2" * r4 * v1)) + r1 * (t" * v2) by RLVECT_1:29
      .= r2" * v1 - r1 * (t" * (r2" * r4 * v1)) + r1 * t" * v2 by
RLVECT_1:def 7
      .= r2" * v1 - (r1 * t" * (r2" * r4 * v1)) + r1 * t" * v2 by
RLVECT_1:def 7
      .= r2" * v1 - (r1 * t" * (r2" * r4)) * v1 + r1 * t" * v2 by
RLVECT_1:def 7
      .= (r2" - (r1 * t" * (r2" * r4))) * v1 + t" * r1 * v2 by RLVECT_1:35;
    r2" - (r1 * t" * (r2" * r4)) = r2" * (1 - (r1 * (t" * r4)))
      .= r2" * (t" * t - (t" * (r1 * r4))) by A11,XCMPLX_0:def 7
      .= r2" * r2 * t" * (- r3)
      .= 1 * t" * (- r3) by A13,XCMPLX_0:def 7
      .= - (t" * r3);
    hence w2 = - t" * r3 * v1 + t" * r1 * v2 by A15,RLVECT_4:3;
  end;
  set a4 = t" * r1;
  set a3 = - t" * r3;
  set a2 = - t" * r2;
  set a1 = t" * r4;
  now
    assume
A16: r1 <> 0;
A17: r1" + (t" * r2 * r1" * r3) = r1" * (1 + (t" * (r2 * r3)))
      .= r1" * (t" * t + (t" * (r2 * r3))) by A11,XCMPLX_0:def 7
      .= t" * (r1" * r1 * r4)
      .= t" * (1 * r4) by A16,XCMPLX_0:def 7
      .= t" * r4;
    r1" * v1 = r1" * (r1 * w1) + r1" * (r2 * w2) by A5,RLVECT_1:def 5
      .= r1" * r1 * w1 + r1" * (r2 * w2) by RLVECT_1:def 7
      .= 1 * w1 + r1" * (r2 * w2) by A16,XCMPLX_0:def 7
      .= w1 + r1" * (r2 * w2) by RLVECT_1:def 8
      .= w1 + r2 * r1" * w2 by RLVECT_1:def 7;
    then
A18: w1 = r1" * v1 - r2 * r1" * w2 by RLSUB_2:61;
    then v2 = r3 * (r1" * v1) - r3 * (r2 * r1"* w2) + r4 * w2 by A6,RLVECT_1:34
      .= r3 * r1" * v1 - r3 * (r1" * r2 * w2) + r4 * w2 by RLVECT_1:def 7
      .= r3 * r1" * v1 - r3 * (r1" * r2) * w2 + r4 * w2 by RLVECT_1:def 7
      .= r3 * r1" * v1 - r1" * (r3 * r2) * w2 + r4 * w2
      .= r1" * r3 * v1 - r1" * (r3 * r2 * w2) + r4 * w2 by RLVECT_1:def 7
      .= r1" * (r3 * v1) - r1" * (r3 * r2 * w2) + r4 * w2 by RLVECT_1:def 7;
    then r1 * v2 = r1 * (r1" * (r3 * v1) - r1" * (r3 * r2 * w2)) + r1 * (r4 *
    w2) by RLVECT_1:def 5
      .= r1 * (r1" * (r3 * v1) - r1" * (r3 * r2 * w2)) + r1 * r4 * w2 by
RLVECT_1:def 7
      .= r1 * (r1" * (r3 * v1)) - r1 * (r1" * (r3 * r2 * w2)) + r1 * r4 * w2
    by RLVECT_1:34
      .= r1 * r1" * (r3 * v1) - r1 * (r1" * (r3 * r2 * w2)) + r1 * r4 * w2
    by RLVECT_1:def 7
      .= r1 * r1" * (r3 * v1) - r1 * r1" * (r3 * r2 * w2) + r1 * r4 * w2 by
RLVECT_1:def 7
      .= 1 * (r3 * v1) - r1 * r1" * (r3 * r2 * w2) + r1 * r4 * w2 by A16,
XCMPLX_0:def 7
      .= 1 * (r3 * v1) - 1 * (r3 * r2 * w2) + r1 * r4 * w2 by A16,
XCMPLX_0:def 7
      .= r3 * v1 - 1 * (r3 * r2 * w2) + r1 * r4 * w2 by RLVECT_1:def 8
      .= r3 * v1 - r3 * r2 * w2 + r1 * r4 * w2 by RLVECT_1:def 8
      .= r3 * v1 - (r3 * r2 * w2 - r1 * r4 * w2) by RLVECT_1:29
      .= r3 * v1 + - (r3 * r2 - r1 * r4) * w2 by RLVECT_1:35
      .= r3 * v1 + (- (r3 * r2 - r1 * r4)) * w2 by RLVECT_4:3
      .= r3 * v1 + t * w2;
    then t" * (r1 * v2) = t" * (r3 * v1) + t" * (t * w2) by RLVECT_1:def 5
      .= t" * (r3 * v1) + t" * t * w2 by RLVECT_1:def 7
      .= t" * (r3 * v1) + 1 * w2 by A11,XCMPLX_0:def 7
      .= t" * (r3 * v1) + w2 by RLVECT_1:def 8;
    hence w2 = t" * (r1 * v2) - t" * (r3 * v1) by RLSUB_2:61
      .= t" * r1 * v2 - t" * (r3 * v1) by RLVECT_1:def 7
      .= - t" * r3 * v1 + t" * r1 * v2 by RLVECT_1:def 7;
    hence w1 = r1" * v1 - (r2 * r1" * (- t" * r3 * v1) + r2 * r1" * (t" * r1 *
    v2)) by A18,RLVECT_1:def 5
      .= r1" * v1 - (r2 * r1" * (- t" * r3 * v1) + r2 * r1" * (t" * r1) * v2
    ) by RLVECT_1:def 7
      .= r1" * v1 - (r2 * r1" * (- t" * r3 * v1) + r2 * (r1" * r1 * t") * v2
    )
      .= r1" * v1 - (r2 * r1" * (- t" * r3 * v1) + r2 * (1 * t") * v2) by A16,
XCMPLX_0:def 7
      .= r1" * v1 - (r2 * r1" * (- t" * (r3 * v1)) + r2 * t" * v2) by
RLVECT_1:def 7
      .= r1" * v1 - (r2 * r1" * ((- t") * (r3 * v1)) + r2 * t" * v2) by
RLVECT_4:3
      .= r1" * v1 - (r2 * r1" * (- t") * (r3 * v1) + r2 * t" * v2) by
RLVECT_1:def 7
      .= r1" * v1 - (((- jj) * t" * r2) * r1" * (r3 * v1) + r2 * t" * v2)
      .= r1" * v1 - (((- jj) * t" * r2) * (r1" * (r3 * v1)) + r2 * t" * v2)
    by RLVECT_1:def 7
      .= r1" * v1 - (((- jj) * t") * (r2 * (r1" * (r3 * v1))) + r2 * t" * v2)
    by RLVECT_1:def 7
      .= r1" * v1 - ((- jj) * (t" * (r2 * (r1" * (r3 * v1)))) + r2 * t" * v2)
    by RLVECT_1:def 7
      .= r1" * v1 - (- (t" * (r2 * (r1" * (r3 * v1)))) + r2 * t" * v2) by
RLVECT_1:16
      .= r1" * v1 - (- (t" * r2 * (r1" * (r3 * v1))) + r2 * t" * v2) by
RLVECT_1:def 7
      .= r1" * v1 - (- ((t" * r2 * r1") * (r3 * v1)) + r2 * t" * v2) by
RLVECT_1:def 7
      .= r1" * v1 - (- ((t" * r2 * r1" * r3) * v1) + r2 * t" * v2) by
RLVECT_1:def 7
      .= r1" * v1 - - ((t" * r2 * r1" * r3) * v1) - r2 * t" * v2 by RLVECT_1:27
      .= r1" * v1 + (t" * r2 * r1" * r3) * v1 - r2 * t" * v2 by RLVECT_1:17
      .= t" * r4 * v1 + - t" * r2 * v2 by A17,RLVECT_1:def 6;
  end;
  then
A19: w1 = t" * r4 * v1 + (- t" * r2) * v2 & w2 = (- t" * r3) * v1 + t" * r1
  * v2 by A7,A12,RLVECT_4:3;
  now
    let u being VECTOR of V;
    thus u in Lin{w1,w2} implies u in Lin{v1,v2}
    proof
      assume u in Lin{w1,w2};
      then consider r5,r6 being Real such that
A20:  u = r5 * w1 + r6 * w2 by Th32;
      u = r5 * (a1 * v1) + r5 * (a2 * v2) + r6 * (a3 * v1 + a4 * v2) by A19,A20
,RLVECT_1:def 5
        .= r5 * (a1 * v1) + r5 * (a2 * v2) + (r6 * (a3 * v1) + r6 * (a4 * v2
      )) by RLVECT_1:def 5
        .= r5 * a1 * v1 + r5 * (a2 * v2) + (r6 * (a3 * v1) + r6 * (a4 * v2))
      by RLVECT_1:def 7
        .= r5 * a1 * v1 + r5 * a2 * v2 + (r6 * (a3 * v1) + r6 * (a4 * v2))
      by RLVECT_1:def 7
        .= r5 * a1 * v1 + r5 * a2 * v2 + (r6 * a3 * v1 + r6 * (a4 * v2)) by
RLVECT_1:def 7
        .= r5 * a1 * v1 + r5 * a2 * v2 + (r6 * a3 * v1 + r6 * a4 * v2) by
RLVECT_1:def 7
        .= r5 * a1 * v1 + (r5 * a2 * v2 + (r6 * a3 * v1 + r6 * a4 * v2)) by
RLVECT_1:def 3
        .= r5 * a1 * v1 + (r6 * a3 * v1 + (r5 * a2 * v2 + r6 * a4 * v2)) by
RLVECT_1:def 3
        .= r5 * a1 * v1 + r6 * a3 * v1 + (r5 * a2 * v2 + r6 * a4 * v2) by
RLVECT_1:def 3
        .= (r5 * a1 + r6 * a3) * v1 + (r5 * a2 * v2 + r6 * a4 * v2) by
RLVECT_1:def 6
        .= (r5 * a1 + r6 * a3) * v1 + (r5 * a2 + r6 * a4) * v2 by
RLVECT_1:def 6;
      hence thesis by Th32;
    end;
    assume u in Lin{v1,v2};
    then consider r5,r6 being Real such that
A21: u = r5 * v1 + r6 * v2 by Th32;
    u = r5 * (r1 * w1 + r2 * w2) + (r6 * (r3 * w1) + r6 * (r4 * w2)) by A5,A6
,A21,RLVECT_1:def 5
      .= r5 * (r1 * w1) + r5 * (r2 * w2) + (r6 * (r3 * w1) + r6 * (r4 * w2))
    by RLVECT_1:def 5
      .= r5 * (r1 * w1) + r5 * (r2 * w2) + r6 * (r3 * w1) + r6 * (r4 * w2)
    by RLVECT_1:def 3
      .= r5 * (r1 * w1) + r6 * (r3 * w1) + r5 * (r2 * w2) + r6 * (r4 * w2)
    by RLVECT_1:def 3
      .= (r5 * r1) * w1 + r6 * (r3 * w1) + r5 * (r2 * w2) + r6 * (r4 * w2)
    by RLVECT_1:def 7
      .= (r5 * r1) * w1 + (r6 * r3) * w1 + r5 * (r2 * w2) + r6 * (r4 * w2)
    by RLVECT_1:def 7
      .= (r5 * r1) * w1 + (r6 * r3) * w1 + (r5 * r2) * w2 + r6 * (r4 * w2)
    by RLVECT_1:def 7
      .= (r5 * r1) * w1 + (r6 * r3) * w1 + (r5 * r2) * w2 + (r6 * r4) * w2
    by RLVECT_1:def 7
      .= ((r5 * r1) + (r6 * r3)) * w1 + (r5 * r2) * w2 + (r6 * r4) * w2 by
RLVECT_1:def 6
      .= ((r5 * r1) + (r6 * r3)) * w1 + ((r5 * r2) * w2 + (r6 * r4) * w2) by
RLVECT_1:def 3
      .= ((r5 * r1) + (r6 * r3)) * w1 + ((r5 * r2) + (r6 * r4)) * w2 by
RLVECT_1:def 6;
    hence u in Lin{w1,w2} by Th32;
  end;
  hence Lin{w1,w2} = Lin{v1,v2} by RUSUB_1:25;
  now
    let a,b be Real;
A22: t" <> 0 by A11,XCMPLX_1:202;
    assume a * w1 + b * w2 = 0.V;
    then
A23: 0.V = a * (a1 * v1) + a * (a2 * v2) + b * (a3 * v1 + a4 * v2) by A19,
RLVECT_1:def 5
      .= a * (a1 * v1) + a * (a2 * v2) + (b * (a3 * v1) + b * (a4 * v2)) by
RLVECT_1:def 5
      .= a * a1 * v1 + a * (a2 * v2) + (b * (a3 * v1) + b * (a4 * v2)) by
RLVECT_1:def 7
      .= a * a1 * v1 + a * (a2 * v2) + (b * (a3 * v1) + b * a4 * v2) by
RLVECT_1:def 7
      .= a * a1 * v1 + a * (a2 * v2) + (b * a3 * v1 + b * a4 * v2) by
RLVECT_1:def 7
      .= a * a1 * v1 + a * a2 * v2 + (b * a3 * v1 + b * a4 * v2) by
RLVECT_1:def 7
      .= a * a1 * v1 + (a * a2 * v2 + (b * a3 * v1 + b * a4 * v2)) by
RLVECT_1:def 3
      .= a * a1 * v1 + (b * a3 * v1 + (a * a2 * v2 + b * a4 * v2)) by
RLVECT_1:def 3
      .= a * a1 * v1 + b * a3 * v1 + (a * a2 * v2 + b * a4 * v2) by
RLVECT_1:def 3
      .= (a * a1 + b * a3) * v1 + (a * a2 * v2 + b * a4 * v2) by RLVECT_1:def 6
      .= (a * a1 + b * a3) * v1 + (a * a2 + b * a4) * v2 by RLVECT_1:def 6;
    then 0 = t" * (r4 * a + (- r3) * b) by A1,A2,RLVECT_3:13;
    then
A24: r4 * a - r3 * b = 0 by A22,XCMPLX_1:6;
    0 = t" * ((- r2) * a + r1 * b) by A1,A2,A23,RLVECT_3:13;
    then
A25: r1 * b + - r2 * a = 0 by A22,XCMPLX_1:6;
    assume
A26: a <> 0 or b <> 0;
    now
      per cases by A26;
      suppose
A27:    a <> 0;
        a" * (r1 * b) = a" * a * r2 by A25,XCMPLX_1:4;
        then a" * (r1 * b) = 1 * r2 by A27,XCMPLX_0:def 7;
        then r2 = r1 * (a" * b);
        then v1 = r1 * w1 + r1 * ((a" * b) * w2) by A5,RLVECT_1:def 7;
        then
A28:    v1 = r1 * (w1 + a" * b * w2) by RLVECT_1:def 5;
        a" * a * r4 = a" * (r3 * b) by A24,XCMPLX_1:4;
        then 1 * r4 = a" * (r3 * b) by A27,XCMPLX_0:def 7;
        then r4 = r3 * (a" * b);
        then v2 = r3 * w1 + r3 * ((a" * b) * w2) by A6,RLVECT_1:def 7;
        then v2 = r3 * (w1 + a" * b * w2) by RLVECT_1:def 5;
        hence contradiction by A1,A2,A28,RLVECT_4:21;
      end;
      suppose
A29:    b <> 0;
        b" * b * r1 = b" * (r2 * a) by A25,XCMPLX_1:4;
        then 1 * r1 = b" * (r2 * a) by A29,XCMPLX_0:def 7;
        then r1 = r2 * (b" * a);
        then v1 = r2 * ((b" * a) * w1) + r2 * w2 by A5,RLVECT_1:def 7;
        then
A30:    v1 = r2 * ((b" * a) * w1 + w2) by RLVECT_1:def 5;
        b" * b * r3 = b" * (r4 * a) by A24,XCMPLX_1:4;
        then 1 * r3 = b" * (r4 * a) by A29,XCMPLX_0:def 7;
        then r3 = r4 * (b" * a);
        then v2 = r4 * ((b" * a) * w1) + r4 * w2 by A6,RLVECT_1:def 7;
        then v2 = r4 * ((b" * a) * w1 + w2) by RLVECT_1:def 5;
        hence contradiction by A1,A2,A30,RLVECT_4:21;
      end;
    end;
    hence contradiction;
  end;
  hence thesis by RLVECT_3:13;
end;
