
theorem Th54:
  for P being Point of real_projective_plane
  for l being Element of ProjectiveLines real_projective_plane st P in l holds
  dual l in dual P
  proof
    let P be Point of real_projective_plane;
    let l be Element of ProjectiveLines real_projective_plane;
    assume
A1: P in l;
    consider u be Element of TOP-REAL 3 such that
A2: u is not zero and
A3: P = Dir u by ANPROJ_1:26;
    reconsider u as non zero Element of TOP-REAL 3 by A2;
    reconsider P9 = P as Element of ProjectiveSpace TOP-REAL 3;
    reconsider dl = dual l as Point of ProjectiveSpace TOP-REAL 3;
    consider Pl,Ql be Point of real_projective_plane such that
A4: Pl <> Ql and
A5: l = Line(Pl,Ql) and
A6: dual l = L2P(Pl,Ql) by Def25;
    consider ul,vl be non zero Element of TOP-REAL 3 such that
A7: Pl = Dir ul and
A8: Ql = Dir vl and
A9: L2P(Pl,Ql) = Dir(ul <X> vl) by A4,BKMODEL1:def 5;
    reconsider ulvl = ul <X> vl as non zero Element of TOP-REAL 3
      by A4,A7,A8,BKMODEL1:78;
    consider S be Point of real_projective_plane such that
A10: P = S and
A11: Pl,Ql,S are_collinear by A1,A5;
    P,Pl,Ql are_collinear by A10,A11,ANPROJ_2:24;
    then
A12: |{u,ul,vl}| = 0 by A3,A7,A8,BKMODEL1:1;
    per cases by Th37;
    suppose P9 is non zero_proj1;
      then reconsider P9 as non zero_proj1 Point of ProjectiveSpace TOP-REAL 3;
      consider P99 be non zero_proj1 Point of ProjectiveSpace TOP-REAL 3
      such that
A13:  P9 = P99 and
A14:  dual P = dual1 P99 by Th38;
      consider S be Point of real_projective_plane such that
A15:  P = S and
A16:  Pl,Ql,S are_collinear by A1,A5;
      P,Pl,Ql are_collinear by A15,A16,ANPROJ_2:24;
      then
A17:  |{u,ul,vl}| = 0 by A3,A7,A8,BKMODEL1:1;
      Dir normalize_proj1 P9 = Dir u by A3,Def2;
      then
A18:  are_Prop normalize_proj1 P9,u by ANPROJ_1:22;
      |{dir1a P9,dir1b P9, ulvl }| = |(normalize_proj1 P9, ulvl)| by Th21
                                  .= 0 by A17,A18,Th7;
      then Pdir1a P9,Pdir1b P9, dl are_collinear by A6,A9,BKMODEL1:1;
      hence thesis by A13,A14;
    end;
    suppose P9 is non zero_proj2;
      then reconsider P9 as non zero_proj2 Point of ProjectiveSpace TOP-REAL 3;
      consider P99 be non zero_proj2 Point of ProjectiveSpace TOP-REAL 3
      such that
A19:  P9 = P99 and
A20:  dual P = dual2 P99 by Th39;
      Dir normalize_proj2 P9 = Dir u by A3,Def4;
      then
A21:  are_Prop normalize_proj2 P9,u by ANPROJ_1:22;
      |{dir2a P9,dir2b P9, ulvl }| = - |(normalize_proj2 P9, ulvl)| by Th25
                                  .= - 0 by A12,A21,Th7;
      then Pdir2a P9,Pdir2b P9, dl are_collinear by A6,A9,BKMODEL1:1;
      hence thesis by A19,A20;
    end;
    suppose P9 is non zero_proj3;
      then reconsider P9 as non zero_proj3 Point of ProjectiveSpace TOP-REAL 3;
      consider P99 be non zero_proj3 Point of ProjectiveSpace TOP-REAL 3
      such that
A22:  P9 = P99 and
A23:  dual P = dual3 P99 by Th40;
      Dir normalize_proj3 P9 = Dir u by A3,Def6;
      then
A24:  are_Prop normalize_proj3 P9,u by ANPROJ_1:22;
      |{dir3a P9,dir3b P9, ulvl }| = |(normalize_proj3 P9, ulvl)| by Th29
                                  .= 0 by A12,A24,Th7;
      then Pdir3a P9,Pdir3b P9, dl are_collinear by A6,A9,BKMODEL1:1;
      hence thesis by A22,A23;
    end;
  end;
