
theorem Th26:
  for M being Matrix of 3,REAL
  for P being Element of absolute
  for Q being Element of real_projective_plane
  for u,v being non zero Element of TOP-REAL 3
  for fp,fq being FinSequence of REAL st
  M = symmetric_3(1,1,-1,0,0,0) &
  P = Dir u & Q = Dir v  &
  u = fp & v = fq &
  Q in tangent P holds SumAll QuadraticForm(fq,M,fp) = 0
  proof
    let M being Matrix of 3,REAL;
    let P being Element of absolute;
    let Q being Element of real_projective_plane;
    let u,v being non zero Element of TOP-REAL 3;
    let fp,fq being FinSequence of REAL;
    assume that
A1: M = symmetric_3(1,1,-1,0,0,0) and
A2: P = Dir u and
A3: Q = Dir v and
A4: u = fp and
A5: v = fq and
A6: Q in tangent P;
    consider p be Element of real_projective_plane such that
A7: p = P and
A8: tangent P = Line(p,pole_infty P) by Def04;
A9: p,pole_infty P,Q are_collinear by A6,A8,COLLSP:11;
    consider w be non zero Element of TOP-REAL 3 such that
A10: P = Dir w & w.3 = 1 & (w.1)^2 + (w.2)^2 = 1 &
      pole_infty P = Dir |[- w.2,w.1,0]| by Def03;
    are_Prop w,u by A2,A10,ANPROJ_1:22;
    then consider aa be Real such that
A11: aa <> 0 and
A12: w = aa * u by ANPROJ_1:1;
A13: w.1 = aa * u`1 & w.2 = aa * u`2 & w.3 = aa * u`3
    proof
      thus w.1 = aa * u.1 by A12,RVSUM_1:44
              .= aa * u`1 by EUCLID_5:def 1;
      thus w.2 = aa * u.2 by A12,RVSUM_1:44
              .= aa * u`2 by EUCLID_5:def 2;
      thus w.3 = aa * u.3 by A12,RVSUM_1:44
              .= aa * u`3 by EUCLID_5:def 3;
    end;
    then 1 * w.1 = aa * u`1 & 1 * w.2 = aa * u`2 & 1 = aa * u`3 by A10;
    then
A14: (aa * (1/aa)) * w.1 = aa * u`1 & (aa * (1/aa)) * w.2 = aa * u`2 &
      (aa * (1/aa)) * 1 = aa * u`3 by A11,XCMPLX_1:106;
A16: 1 = aa^2 * (u`1 * u`1 + u`2 * u`2) by A13,A10;
A17: |[ -w.2,w.1,0]|`1 = - aa * u`2 & |[ -w.2,w.1,0]|`2 = aa * u`1 &
      |[ -w.2,w.1,0]|`3 = 0 by A13,EUCLID_5:2;
    |[-w.2,w.1,0]| is non zero by BKMODEL1:91,A10;
    then 0 = |{u, |[-w.2,w.1,0]| ,v}| by A7,A10,A2,A3,A9,BKMODEL1:1
          .= u`1 * |[-w.2,w.1,0]|`2 * v`3 - u`3*|[-w.2,w.1,0]|`2*v`1
                - u`1*|[-w.2,w.1,0]|`3*v`2 + u`2*|[-w.2,w.1,0]|`3*v`1
                - u`2*|[-w.2,w.1,0]|`1*v`3 + u`3*|[-w.2,w.1,0]|`1*v`2
                  by ANPROJ_8:27
          .= aa * (u`1 * u`1 * v`3 - u`1 * u`3 * v`1 + u`2 * u`2 *v`3
                - u`2 *u`3 * v`2) by A17;
    then 0 = v`3 * (u`1 * u`1 + u`2 * u`2) - u`3 * (u`1 * v`1 + u`2 * v`2)
              by A11
          .= v`3 * (1/(aa^2)) - u`3 * (u`1 * v`1 + u`2 * v`2)
              by A16,XCMPLX_1:73
          .= v`3 * (1/(aa^2)) - (1/aa) * (u`1 * v`1 + u`2 * v`2)
              by A14,A11,XCMPLX_1:5
          .= v`3 * ((1/aa) * (1/aa)) - (1/aa) * (u`1 * v`1 + u`2 * v`2)
              by XCMPLX_1:102
          .= (1/aa) * (v`3 * (1/aa) - (u`1 * v`1 + u`2 * v`2));
    then
A18: v`3 * (1/aa) - (u`1 * v`1 + u`2 * v`2) = 0 by A11;
A19: len fp = width M & len fq = len M & len fp = len M &
    len fq = width M & len fp > 0 & len fq > 0
    proof
      len M = 3 & width M = 3 by MATRIX_0:24;
      hence thesis by A5,FINSEQ_3:153,A4;
    end;
    then
A20: SumAll QuadraticForm(fq,M,fp) = |(fq * M,fp)| by MATRPROB:46
                                  .= |(fq,M * fp)| by A19,MATRPROB:47;
A21: M * fp = Col(M * (ColVec2Mx fp),1) by MATRIXR1:def 11;
A22: fp is Element of REAL 3 by A4,EUCLID:22; then
A23: len fp = 3 by EUCLID_8:50;
    reconsider fa = 1, fb = -1 ,z = 0 as Element of F_Real by XREAL_0:def 1;
A24: M = <* <* fa,z,z *>,
            <* z,fa,z *>,
            <* z,z,fb *> *> by A1,PASCAL:def 3;
    reconsider fp1 = fp.1,fp2 = fp.2, fp3 = fp.3 as Element of F_Real
      by XREAL_0:def 1;
A25: ColVec2Mx fp = MXR2MXF ColVec2Mx fp by MATRIXR1:def 1
                .= <*fp*>@ by A22,ANPROJ_8:72
                .= F2M fp by A22,ANPROJ_8:88,EUCLID_8:50
                .= <* <* fp1 *>,<* fp2 *>, <* fp3 *> *> by A23,ANPROJ_8:def 1;
    reconsider M1 = M as Matrix of 3,3,F_Real;
    reconsider M2 =  <* <* fp1 *>,<* fp2 *>, <* fp3 *> *>
      as Matrix of 3,1,F_Real
      by ANPROJ_8:4;
A26: for n,k,m be Nat for A be Matrix of n,k,F_Real
    for B be Matrix of width A,m,F_Real
    holds A*B is Matrix of len A,width B,F_Real;
A27: len M1 = 3 & width M2 = 1 by MATRIX_0:23;
    width M1 = 3 by MATRIX_0:23; then
A28: M1 * M2 is Matrix of 3,1,F_Real by A26,A27;
A29: M * ColVec2Mx fp = M1 * M2 by A25,ANPROJ_8:17;
    M * ColVec2Mx fp = M1 * M2 by A25,ANPROJ_8:17
                    .= <* <* fa * fp1 + z * fp2 + z * fp3 *> ,
                          <* z * fp1 + fa * fp2 + z * fp3 *>,
                          <* z * fp1 + z * fp2 + fb * fp3 *> *>
                        by A24,ANPROJ_9:7;
    then SumAll QuadraticForm(fq,M,fp)
      = |(v, |[ fp1,fp2,-fp3 ]| )| by A5,A20,A21,A28,A29,ANPROJ_8:5
     .= v`1 * |[ fp1,fp2,-fp3 ]|`1 + v`2 * |[ fp1,fp2,-fp3 ]|`2
         + v`3 * |[ fp1,fp2,-fp3 ]|`3 by EUCLID_5:29
     .= v`1 * fp1 + v`2 * |[ fp1,fp2,-fp3 ]|`2 + v`3 * |[ fp1,fp2,-fp3 ]|`3
        by EUCLID_5:2
     .= v`1 * fp1 + v`2 * fp2 + v`3 * |[ fp1,fp2,-fp3 ]|`3 by EUCLID_5:2
     .= v`1 * fp1 + v`2 * fp2 + v`3 * (-fp3) by EUCLID_5:2
     .= v`1 * u.1 + v`2 * u.2 - v`3 * u.3 by A4
     .= v`1 * u`1 + v`2 * u.2 - v`3 * u.3 by EUCLID_5:def 1
     .= v`1 * u`1 + v`2 * u`2 - v`3 * u.3 by EUCLID_5:def 2
     .= v`1 * u`1 + v`2 * u`2 - v`3 * u`3 by EUCLID_5:def 3
     .= 0 by A18,A14,A11,XCMPLX_1:5;
    hence thesis;
  end;
