
theorem Th28:
  for P being Element of absolute
  for R being Element of real_projective_plane
  for u being non zero Element of TOP-REAL 3 st
  R in tangent P & R = Dir u & u.3 = 0 holds R = pole_infty P
  proof
    let P be Element of absolute;
    let R be Element of real_projective_plane;
    let u be non zero Element of TOP-REAL 3;
    assume that
A1: R in tangent P and
A2: R = Dir u and
A3: u.3 = 0;
    consider w be non zero Element of TOP-REAL 3 such that
A4: P = Dir w & w.3 = 1 & (w.1)^2 + (w.2)^2 = 1 &
      pole_infty P = Dir |[- w.2,w.1,0]| by Def03;
    consider v be non zero Element of TOP-REAL 3 such that
A5: (v.1)^2 + (v.2)^2 = 1 & v.3 = 1 & P = Dir v by BKMODEL1:89;
    reconsider M = symmetric_3(1,1,-1,0,0,0) as Matrix of 3,REAL;
    reconsider fp = v, fr = u as FinSequence of REAL by EUCLID:24;
    reconsider fr1 = u`1,fr2 = u`2, fr3 = u`3 as Element of REAL
      by XREAL_0:def 1;
A6: fr = <* fr1,fr2,fr3 *> by EUCLID_5:3;
A7: SumAll QuadraticForm(fr,M,fp) = 0 by A2,A1,A5,Th26;
    u is Element of REAL 3 & v is Element of REAL 3 by EUCLID:22;
    then
A8: len fp = 3 & len fr = 3 by EUCLID_8:50;
    len M = 3 & width M = 3 by MATRIX_0:24; then
A9: |( fr, M * fp )| = 0 by A7,A8,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;
A10: M = <* <* m1,m2,m3 *>,
            <* m4,m5,m6 *>,
            <* m7,m8,m9 *> *> by PASCAL:def 3;
    reconsider fp1 = v`1,fp2 = v`2,fp3 = v`3 as Element of REAL
      by XREAL_0:def 1;
A11: v.1 = fp1 & v.2 = fp2 & fp3 = 1 & fr3 = 0
      by A3,A5,EUCLID_5:def 1,def 2,def 3;
A12: fp = <*fp1,fp2,fp3*> & 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 A10,PASCAL:9
               .= <* fp1,fp2,-fp3 *>;
    then
A13: fr1 * fp1 + fr2 * fp2 + fr3 * (-fp3) = 0
      by A9,A6,EUCLID_5:30;
    per cases;
    suppose
A14:  fr1 = 0;
      then
A15:  fr2 <> 0 by A12,EUCLID_5:4,A3;
      then
A16:  fp2 = 0 by A14,A11,A13;
A17:  w.1 = v.1 by A4,A5,BKMODEL1:43
         .= fp1 by EUCLID_5:def 1;
A18:  w.2 = v.2 by A4,A5,BKMODEL1:43
         .= 0 by A15,A14,A11,A13;
      now
A19:    fp1 <> 0 by A5,A16,A11;
        thus |[0,fp1,0]| is non zero
          by A5,A16,A11,FINSEQ_1:78,EUCLID_5:4;
        thus are_Prop u, |[0,fp1,0]|
        proof
          u = |[ (fr2/fp1) * 0, (fr2/fp1) * fp1, (fr2/fp1) * 0 ]|
             by A16,A11,A5,XCMPLX_1:87,A12,A14
           .= (fr2/fp1) * |[ 0,fp1,0 ]| by EUCLID_5:8;
          hence thesis by A15,A19,ANPROJ_1:1;
        end;
      end;
      hence thesis by A2,ANPROJ_1:22,A4,A17,A18;
    end;
    suppose
A20:  fr1 <> 0;
A21:  fp2 <> 0
      proof
        assume
A22:    fp2 = 0;
        then fp1 = 0 by A13,A11,A20;
        hence contradiction by A22,A11,A5;
      end;
      then
A23:  fr2 = (fp1 * (-fr1)) / fp2 by A13,A11,XCMPLX_1:89
         .= fp1 * ((-fr1) / fp2) by XCMPLX_1:74;
      reconsider l = (-fr1)/fp2 as non zero Real by A20,A21;
A24:  now
        thus |[-fp2,fp1,0]| is non zero by A21,FINSEQ_1:78,EUCLID_5:4;
        fr1 =  - (-fr1)
           .= - (l * fp2) by A21,XCMPLX_1:87;
        then fr = |[ l  * (- fp2), l * fp1, l * 0 ]|
                   by A23,A12,A3
               .= l * |[ - fp2,fp1,0]| by EUCLID_5:8;
        hence are_Prop u, |[ -fp2,fp1,0]| by ANPROJ_1:1;
      end;
      w = v by BKMODEL1:43,A4,A5;
      hence thesis by A24,A2,ANPROJ_1:22,A4,A11;
    end;
  end;
