
theorem Th33:
  for P being non zero_proj1 non zero_proj2 Point of ProjectiveSpace
  TOP-REAL 3 holds dual1 P = dual2 P
  proof
    let P be non zero_proj1 non zero_proj2 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_proj2 P = |[u.1/u.2,1,u.3/u.2]| by A2,Th11,Th14;
    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_proj2(P)).1,
            c3 = - (normalize_proj2(P)).3;
A11:    c2 = - (normalize_proj2(P))`1
          .= - u.1/u.2 by A3;
A12:    c3 = - (normalize_proj2(P))`3
          .= - u.3/u.2 by A3;
        |{ |[1,   c2,  0]|,
           |[0,   c3,  1]|,
           |[u9`1,u9`2,u9`3]| }| = (- u.1/u.2) *  b1 + (-u.3/u.2) * b3 - b2
             by A11,A12,Th3;
        then |{dir2a P,dir2b P,u9}|
          = (- u.1/u.2) *  b1 + (-u.3/u.2) * b3 + (-1) * b2
         .= (- u.1/u.2) *  b1 + (-u.3/u.2) * b3 + (-u.2/u.2) * b2
           by XCMPLX_1:60,A2,Th13
         .= (u.1/(-u.2)) *  b1 + (-u.3/u.2) * b3 + (-u.2/u.2) * b2
           by XCMPLX_1:188
         .= (u.1/(-u.2)) *  b1 + (u.3/(-u.2)) * b3 + (-u.2/u.2) * b2
           by XCMPLX_1:188
         .= (u.1/(-u.2)) *  b1 + (u.3/(-u.2)) * b3 + (u.2/(-u.2)) * b2
           by XCMPLX_1:188
         .= (1 / -u.2) * (u.1 * b1 + u.2 * b2 + u.3 * b3)
         .= 0 by A10;
        then Pdir2a P,Pdir2b P,P9 are_collinear by A6,A7,BKMODEL1:1;
        hence x in Line(Pdir2a P,Pdir2b P) by A4;
      end;
      hence Line(Pdir1a P,Pdir1b P) c= Line(Pdir2a P,Pdir2b P);
      now
        let x be object;
        assume x in Line(Pdir2a P,Pdir2b P);
        then consider P9 be Point of ProjectiveSpace TOP-REAL 3 such that
A13:    x = P9 and
A14:    Pdir2a P,Pdir2b 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_proj2(P)).1,
            c3 = - (normalize_proj2(P)).3;
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_proj2(P))`1
          .= - u.1/u.2 by A3;
A20:    c3 = - (normalize_proj2(P))`3
          .= - u.3/u.2 by A3;
A21:    - u.2 <> 0 by A2,Th13;
A22:    0 = |{ dir2a P,dir2b P,u9 }| by A14,A15,A16,BKMODEL1:1
         .= |{ |[1, c2 , 0]| ,
               |[0, c3 , 1]|,
               |[b1, b2, b3]| }|
         .= c3 * b3 + c2 * b1 - b2 by Th3
         .= (- u.1/u.2) *  b1 + (-u.3/u.2) * b3 + (-1) * b2 by A19,A20
         .= (- u.1/u.2) *  b1 + (-u.3/u.2) * b3 + (-u.2/u.2) * b2
           by XCMPLX_1:60,A2,Th13
         .= (u.1/(-u.2)) *  b1 + (-u.3/u.2) * b3 + (-u.2/u.2) * b2
           by XCMPLX_1:188
         .= (u.1/(-u.2)) *  b1 + (u.3/(-u.2)) * b3 + (-u.2/u.2) * b2
           by XCMPLX_1:188
         .= (u.1/(-u.2)) *  b1 + (u.3/(-u.2)) * b3 + (u.2/(-u.2)) * b2
           by XCMPLX_1:188
         .= (1 / -u.2) * (u.1 * b1 + u.2 * b2 + u.3 * b3);
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(Pdir2a P,Pdir2b P) c= Line(Pdir1a P,Pdir1b P);
    end;
    hence thesis;
  end;
