
theorem Th35:
  for P being non zero_proj1 non zero_proj3 Point of ProjectiveSpace
  TOP-REAL 3 holds dual1 P = dual3 P
  proof
    let P be non zero_proj1 non zero_proj3 Point of ProjectiveSpace TOP-REAL 3;
    consider u be Element of TOP-REAL 3 such that
A1: u is not zero and
A2: P = Dir u by ANPROJ_1:26;
    reconsider u as non zero Element of TOP-REAL 3 by A1;
A3: normalize_proj1 P = |[1, u.2/u.1,u.3/u.1]| &
      normalize_proj3 P = |[u.1/u.3,u.2/u.3,1]| by A2,Th11,Th17;
    now
      now
        let x be object;
        assume x in Line(Pdir1a P,Pdir1b P);
        then consider P9 be Point of ProjectiveSpace TOP-REAL 3 such that
A4:     x = P9 and
A5:     Pdir1a P,Pdir1b P,P9 are_collinear;
        consider u9 be Element of TOP-REAL 3 such that
A6:     u9 is non zero and
A7:     P9 = Dir u9 by ANPROJ_1:26;
        set a2 = - (normalize_proj1(P)).2,
            a3 = - (normalize_proj1(P)).3,
            b1 = u9`1, b2 = u9`2, b3 = u9`3;
A8:     a2 = - (normalize_proj1(P))`2
          .= - u.2/u.1 by A3;
A9:     a3 = - (normalize_proj1(P))`3
          .= - u.3/u.1 by A3;
        0 = |{ dir1a P,dir1b P,u9 }| by A5,A6,A7,BKMODEL1:1
         .= |{ |[a2, 1 , 0]| ,
               |[a3, 0 , 1]|,
               |[b1, b2, b3]| }|
         .= b1 - a2 * b2 - a3 * b3 by Th2
         .= b1 + u.2/u.1 * b2 + u.3/u.1 * b3 by A8,A9;
        then
A10:    0 = u.1 * (b1 + u.2/u.1 * b2 + u.3/u.1 * b3)
         .= u.1 * b1 + u.1 * (u.2 / u.1) * b2 + u.1 * (u.3/u.1) * b3
         .= u.1 * b1 + u.2 * b2 + u.1 * (u.3/u.1) * b3
           by A2,Th10,XCMPLX_1:87
         .= u.1 * b1 + u.2 * b2 + u.3 * b3 by A2,Th10,XCMPLX_1:87;
        set c2 = - (normalize_proj3(P)).1,
            c3 = - (normalize_proj3(P)).2;
A11:    c2 = - (normalize_proj3(P))`1
          .= - u.1/u.3 by A3;
A12:    c3 = - (normalize_proj3(P))`2
          .= - u.2/u.3 by A3;
        |{ |[1,   0, c2 ]|,
           |[0,   1, c3]|,
           |[u9`1,u9`2,u9`3]| }| = b3 - c2 * b1 - c3 * b2 by Th4;
        then |{dir3a P,dir3b P,u9}|
          = (u.1/u.3) * b1 + (u.2/u.3) * b2 + 1 * b3 by A11,A12
         .= (u.1/u.3) *  b1 + (u.2/u.3) * b2 + (u.3/u.3) * b3
           by XCMPLX_1:60,A2,Th16
         .= (1 / u.3) * (u.1 * b1 + u.2 * b2 + u.3 * b3)
         .= 0 by A10;
        then Pdir3a P,Pdir3b P,P9 are_collinear by A6,A7,BKMODEL1:1;
        hence x in Line(Pdir3a P,Pdir3b P) by A4;
      end;
      hence Line(Pdir1a P,Pdir1b P) c= Line(Pdir3a P,Pdir3b P);
      now
        let x be object;
        assume x in Line(Pdir3a P,Pdir3b P);
        then consider P9 be Point of ProjectiveSpace TOP-REAL 3 such that
A13:    x = P9 and
A14:    Pdir3a P,Pdir3b P,P9 are_collinear;
        consider u9 be Element of TOP-REAL 3 such that
A15:    u9 is non zero and
A16:    P9 = Dir u9 by ANPROJ_1:26;
        set a2 = - (normalize_proj1(P)).2,
            a3 = - (normalize_proj1(P)).3,
            b1 = u9`1, b2 = u9`2, b3 = u9`3;
        set c2 = - (normalize_proj3(P)).1,
            c3 = - (normalize_proj3(P)).2;
A17:    a2 = - (normalize_proj1(P))`2
          .= - u.2/u.1 by A3;
A18:    a3 = - (normalize_proj1(P))`3
          .= - u.3/u.1 by A3;
A19:    c2 = - (normalize_proj3(P))`1
          .= - u.1/u.3 by A3;
A20:    c3 = - (normalize_proj3(P))`2
          .= - u.2/u.3 by A3;
A21:    0 = |{ dir3a P,dir3b P,u9 }| by A14,A15,A16,BKMODEL1:1
         .= |{ |[1, 0,c2]| ,
               |[0, 1,c3]|,
               |[b1, b2, b3]| }|
         .= b3 - c2 * b1 - c3 * b2 by Th4
         .= (u.1/u.3) * b1 + (u.2/u.3) * b2 + 1 * b3 by A19,A20
         .= (u.1/u.3) * b1 + (u.2/u.3) * b2 + (u.3/u.3) * b3
           by XCMPLX_1:60,A2,Th16
         .= (1 / u.3) * (u.1 * b1 + u.2 * b2 + u.3 * b3);
A22:    u.3 <> 0 by A2,Th16;
A23:    u.1/u.1 = 1 by XCMPLX_1:60,A2,Th10;
        |{dir1a P,dir1b P,u9}| = |{ |[a2, 1 , 0]| ,
                                    |[a3, 0 , 1]|,
                                    |[b1, b2, b3]| }|
         .= b1 - a2 * b2 - a3 * b3 by Th2
         .= (u.1/u.1) * b1 + u.2/u.1 * b2 + u.3/u.1 * b3 by A17,A18,A23
         .= (1/u.1) * (u.1 * b1 + u.2 * b2 + u.3 * b3)
         .= (1 / u.1) * 0 by A22,A21,XCMPLX_1:6
         .= 0;
        then Pdir1a P,Pdir1b P,P9 are_collinear by A15,A16,BKMODEL1:1;
        hence x in Line(Pdir1a P,Pdir1b P) by A13;
      end;
      hence Line(Pdir3a P,Pdir3b P) c= Line(Pdir1a P,Pdir1b P);
    end;
    hence thesis;
  end;
