
theorem
  for P,Q being Element of absolute
  for R being Element of real_projective_plane
  for u,v,w being non zero Element of TOP-REAL 3 st P = Dir u & Q = Dir v &
  R = Dir w & R in tangent P & R in tangent Q & u.3 = 1 & v.3 = 1 &
  w.3 = 0 holds P = Q or (u.1 = -v.1 & u.2 = -v.2)
  proof
    let P,Q being Element of absolute;
    let R being Element of real_projective_plane;
    let u,v,w being non zero Element of TOP-REAL 3;
    assume that
A1: P = Dir u & Q = Dir v & R = Dir w & R in tangent P &
    R in tangent Q & u.3 = 1 & v.3 = 1 & w.3 = 0;
    assume
A2: P <> Q;
    |[u.1,u.2]| in circle(0,0,1) & |[v.1,v.2]| in circle(0,0,1)
      by A1,BKMODEL1:84; then
A3: (u.1)^2 + (u.2)^2 = 1 & (v.1)^2 + (v.2)^2 = 1 by BKMODEL1:13;
    reconsider M = symmetric_3(1,1,-1,0,0,0) as Matrix of 3,REAL;
    reconsider fp = u, fq = v,fr = w as FinSequence of REAL by EUCLID:24;
    reconsider fr1 = w`1,fr2 = w`2, fr3 = w`3 as Element of REAL
      by XREAL_0:def 1;
A4: fr = <* fr1,fr2,fr3 *> by EUCLID_5:3;
A5: SumAll QuadraticForm(fr,M,fp) = 0 &
    SumAll QuadraticForm(fr,M,fq) = 0 by A1,Th26;
    u is Element of REAL 3 & v is Element of REAL 3 &
    w is Element of REAL 3 by EUCLID:22; then
A6: len fp = 3 & len fq = 3 & len fr = 3 by EUCLID_8:50;
    len fr = len M & len fp = width M & len fp > 0 &
    len fq = width M & len fq > 0 by A6,MATRIX_0:24;
    then
A7: |( fr, M * fp )| = 0 & |( fr, M * fq )| = 0 by A5,MATRPROB:44;
    reconsider m1 = 1,m2 = 0,m3 = 0,m4 = 0,m5 = 1,m6 = 0,m7 = 0,
    m8 = 0,m9 = -1 as Element of REAL by XREAL_0:def 1;
A8: M = <* <* m1,m2,m3 *>,
    <* m4,m5,m6 *>,
    <* m7,m8,m9 *> *> by PASCAL:def 3;
    reconsider fp1 = u`1,fp2 = u`2,fp3 = u`3,
    fq1 = v`1,fq2 = v`2, fq3 = v`3 as Element of REAL by XREAL_0:def 1;
A9: u.1 = fp1 & u.2 = fp2 & v.1 = fq1 & v.2 = fq2 &
    fp3 = 1 & fq3 = 1 & fr3 = 0 by A1,EUCLID_5:def 1,def 2,def 3;
A10: fp = <*fp1,fp2,fp3*> & fq = <*fq1,fq2,fq3*> &
    fr = <* fr1,fr2,fr3 *> by EUCLID_5:3;
    then M * fp = <* 1 * fp1 + 0 * fp2 + 0 * fp3,
                     0 * fp1 + 1 * fp2 + 0 * fp3,
                     0 * fp1 + 0 * fp2 +(-1) * fp3 *> by A8,PASCAL:9
               .= <* fp1,fp2,-fp3 *>;
    then
A11: fr1 * fp1 + fr2 * fp2 + fr3 * (-fp3) = 0 by A7,A4,EUCLID_5:30;
    M * fq = <* 1 * fq1 + 0 * fq2 + 0 * fq3,
                0 * fq1 + 1 * fq2 + 0 * fq3,
                0 * fq1 + 0 * fq2 +(-1) * fq3 *> by A10,A8,PASCAL:9
          .= <* fq1,fq2,-fq3 *>;
    then
A12: fr1 * fq1 + fr2 * fq2 + fr3 * (-fq3) = 0 by A7,A4,EUCLID_5:30;
A13: fr3 = 0 by A1,EUCLID_5:def 3;
    per cases;
    suppose
A14:  fr2 = 0;
      then fr1 <> 0 by A10,EUCLID_5:4,A1;
      then
A15:  fp1 = 0 & fq1 = 0 by A14,A13,A11,A12;
      then (fp2 = 1 or fp2 = -1) & (fq2 = 1 or fq2 = -1) by SQUARE_1:41,A9,A3;
      hence thesis by A1,A2,A15,A10;
    end;
    suppose
A16:  fr2 <> 0;
      fq1 * (fr1 * fp1 + fr2 * fp2) = 0 & fp1 * (fr1 * fq1 + fr2 * fq2) = 0
        by A13,A11,A12;
      then fr2 * (fq1 * fp2 - fp1 * fq2) = 0;
      then
A17:  fq1 * fp2 - fp1 * fq2 = 0 by A16;
      per cases;
      suppose
A18:    fp2 = 0; then
        fp1 = 0 or fq2 = 0 by A17; then
        (fq1 = 1 or fq1 = -1) & (fp1 = 1 or fp1 = -1) by SQUARE_1:41,
          A3,A9,A18;
        hence thesis by A1,A2,A10,A17;
      end;
      suppose
A20:    fp2 <> 0;
        per cases;
        suppose fp1 = 0;
          hence thesis by A9,A11,A16,A20;
        end;
        suppose
A21:      fp1 <> 0;
          reconsider l = fq1 / fp1 as Real;
A22:      l = fq2 / fp2
          proof
            fq1 = fq1 * (fp2 / fp2) by XCMPLX_1:88,A20
               .= fp1 * fq2 / fp2 by A17,XCMPLX_1:74
               .= fp1 * (fq2 / fp2) by XCMPLX_1:74;
            hence thesis by A21,XCMPLX_1:89;
          end;
          then
A23:      fq1 = l * fp1 & fq2 = l * fp2 by A21,A20,XCMPLX_1:87;
          v.1 = l * fp1 & v.2 = l * fp2 by A22,A9,A21,A20,XCMPLX_1:87;
          then l = 1 or l = -1 by A9,A3,BKMODEL1:3;
          hence thesis by A1,A2,A10,A23;
        end;
      end;
    end;
  end;
