
theorem Th46:
  for l being Element of ProjectiveLines real_projective_plane holds
  dual dual l = l
  proof
    let l be Element of ProjectiveLines real_projective_plane;
    consider P,Q be Point of real_projective_plane such that
A1: P <> Q and
A2: l = Line(P,Q) and
A3: dual l = L2P(P,Q) by Def25;
    reconsider P9 = P,Q9 = Q as Point of ProjectiveSpace TOP-REAL 3;
    consider u,v be non zero Element of TOP-REAL 3 such that
A4: P = Dir u and
A5: Q = Dir v and
A6: L2P(P,Q) = Dir(u <X> v) by A1,BKMODEL1:def 5;
    reconsider l2 = Line(P,Q) as LINE of real_projective_plane
      by A1,COLLSP:def 7;
    not are_Prop u,v by A1,A4,A5,ANPROJ_1:22;
    then u <X> v is non zero by ANPROJ_8:51;
    then reconsider uv = u <X> v as non zero Element of TOP-REAL 3;
    reconsider R = Dir uv as Point of ProjectiveSpace TOP-REAL 3
      by ANPROJ_1:26;
    reconsider R9 = R as Element of real_projective_plane;
    per cases by Th37;
    suppose
A9:   R is non zero_proj1;
      then reconsider R as non zero_proj1 Point of ProjectiveSpace TOP-REAL 3;
      #R9 = R;
      then consider P99 be non zero_proj1 Point of ProjectiveSpace TOP-REAL 3
      such that
A10:  R9 = P99 and
A11:  dual R9 = dual1 P99 by Th38;
A12:  uv.1 / uv.1 = 1 by A9,Th10,XCMPLX_1:60;
      normalize_proj1 P99 = |[1, uv.2/uv.1,uv.3/uv.1]| by A10,Th11;
      then
A13:  (normalize_proj1 P99)`2 = uv.2/uv.1 &
        (normalize_proj1 P99)`3 = uv.3/uv.1;
      reconsider l1 = Line(Pdir1a P99,Pdir1b P99) as
        LINE of real_projective_plane by Th20,COLLSP:def 7;
      now
        |{ |[- uv.2/uv.1, 1,  0]|,
           |[- uv.3/uv.1, 0,  1]|,
           |[u`1,         u`2,u`3]| }|
          = u`1 - (-uv.2/uv.1) * u`2 - u`3 * (-uv.3/uv.1) by Th2
         .= (uv.1 / uv.1) * u`1  + (uv.2/uv.1) * u`2 + u`3 * (uv.3/uv.1)
           by A12
         .= (1/uv.1) * (uv.1 * u`1 + uv.2 * u`2 + uv.3 * u`3)
         .= 0;
        then|{ dir1a P99,dir1b P99,u }| = 0 by A13;
        then Pdir1a P99,Pdir1b P99, P9 are_collinear by A4,BKMODEL1:1;
        hence P in l1;
        |{ |[- uv.2/uv.1, 1,  0]|,
           |[- uv.3/uv.1, 0,  1]|,
           |[v`1,         v`2,v`3]| }|
          = (uv.1 / uv.1) * v`1 - (-uv.2/uv.1) * v`2 - v`3 * (-uv.3/uv.1)
           by A12,Th2
         .= (1/uv.1) * (uv.1 * v`1 + uv.2 * v`2 + uv.3 * v`3)
         .= 0;
        then|{ dir1a P99,dir1b P99,v }| = 0 by A13;
        then Pdir1a P99,Pdir1b P99, Q9 are_collinear by A5,BKMODEL1:1;
        hence Q in l1;
        thus P in l2 & Q in l2 by COLLSP:10;
      end;
      hence thesis by A1,A3,A6,A11,A2,COLLSP:20;
    end;
    suppose
A14:  R is non zero_proj2;
      then reconsider R as non zero_proj2 Point of ProjectiveSpace TOP-REAL 3;
      #R9 = R;
      then consider P99 be non zero_proj2 Point of ProjectiveSpace TOP-REAL 3
      such that
A15:  R9 = P99 and
A16:  dual R9 = dual2 P99 by Th39;
A17:  uv.2 / uv.2 = 1 by A14,Th13,XCMPLX_1:60;
      normalize_proj2 P99 = |[uv.1/uv.2,1,uv.3/uv.2]| by A15,Th14;
      then
A18:  (normalize_proj2 P99)`1 = uv.1/uv.2 &
        (normalize_proj2 P99)`3 = uv.3/uv.2;
      reconsider l1 = Line(Pdir2a P99,Pdir2b P99) as
        LINE of real_projective_plane by Th24,COLLSP:def 7;
      now
        |{ |[1  , - uv.1/uv.2, 0]|,
           |[0  , - uv.3/uv.2, 1]|,
           |[u`1, u`2,         u`3]| }|
          = (-uv.3/uv.2) * u`3 + (-uv.1/uv.2) * u`1 - u`2 by Th3
         .= -((uv.3/uv.2) * u`3 + (uv.1/uv.2) * u`1 + (uv.2/uv.2) * u`2) by A17
         .= - ((1 / uv.2) * (uv.1 * u`1 + uv.2 * u`2 + uv.3 * u`3))
         .= 0;
        then|{ dir2a P99,dir2b P99,u }| = 0 by A18;
        then Pdir2a P99,Pdir2b P99, P9 are_collinear by A4,BKMODEL1:1;
        hence P in l1;
        |{ |[ 1 , - uv.1/uv.2, 0  ]|,
           |[ 0 , - uv.3/uv.2, 1  ]|,
           |[v`1, v`2,         v`3]| }|
          = (-uv.3/uv.2) * v`3 + (-uv.1/uv.2) * v`1 - v`2 by Th3
         .= -((uv.3/uv.2) * v`3 + (uv.1/uv.2) * v`1 + (uv.2/uv.2) * v`2) by A17
         .= - ((1 / uv.2) * (uv.1 * v`1 + uv.2 * v`2 + uv.3 * v`3))
         .= 0;
        then|{ dir2a P99,dir2b P99,v }| = 0 by A18;
        then Pdir2a P99,Pdir2b P99, Q9 are_collinear by A5,BKMODEL1:1;
        hence Q in l1;
        thus P in l2 & Q in l2 by COLLSP:10;
      end;
      hence thesis by A3,A6,A16,A2,A1,COLLSP:20;
    end;
    suppose
A19:  R is non zero_proj3;
      then reconsider R as non zero_proj3 Point of ProjectiveSpace TOP-REAL 3;
      #R9 = R;
      then consider P99 be non zero_proj3 Point of ProjectiveSpace TOP-REAL 3
      such that
A20:  R9 = P99 and
A21:  dual R9 = dual3 P99 by Th40;
A22:  uv.3 / uv.3 = 1 by A19,Th16,XCMPLX_1:60;
      normalize_proj3 P99 = |[uv.1/uv.3,uv.2/uv.3,1]| by A20,Th17;
      then
A23:  (normalize_proj3 P99)`1 = uv.1/uv.3 &
        (normalize_proj3 P99)`2 = uv.2/uv.3;
      reconsider l1 = Line(Pdir3a P99,Pdir3b P99) as
        LINE of real_projective_plane by Th28,COLLSP:def 7;
      now
        |{ |[1  ,0  , - uv.1/uv.3]|,
           |[0  ,1  , - uv.2/uv.3]|,
           |[u`1,u`2, u`3        ]| }|
          = u`3 - u`1 * (-uv.1/uv.3) - u`2 * (-uv.2/uv.3) by Th4
         .= u`1 * (uv.1/uv.3) + u`2 * (uv.2/uv.3) + u`3 * (uv.3 / uv.3) by A22
         .= (1 / uv.3) * (uv.1 * u`1 + uv.2 * u`2 + uv.3 * u`3)
         .= 0;
        then|{ dir3a P99,dir3b P99,u }| = 0 by A23;
        then Pdir3a P99,Pdir3b P99, P9 are_collinear by A4,BKMODEL1:1;
        hence P in l1;
        |{ |[1  ,0  ,- uv.1/uv.3]|,
           |[0  ,1  ,- uv.2/uv.3]|,
           |[v`1,v`2,v`3]| }|
          = v`3 - v`1 * (-uv.1/uv.3) - v`2 * (-uv.2/uv.3) by Th4
         .= v`1 * (uv.1/uv.3) + v`2 * (uv.2/uv.3) + v`3 * (uv.3 / uv.3) by A22
         .= (1 / uv.3) * (uv.1 * v`1 + uv.2 * v`2 + uv.3 * v`3)
         .= 0;
        then|{ dir3a P99,dir3b P99,v }| = 0 by A23;
        then Pdir3a P99,Pdir3b P99, Q9 are_collinear by A5,BKMODEL1:1;
        hence Q in l1;
        thus P in l2 & Q in l2 by COLLSP:10;
      end;
      hence thesis by A3,A6,A21,A2,A1,COLLSP:20;
    end;
  end;
