
theorem Th34:
  for P being non zero_proj2 non zero_proj3 Point of ProjectiveSpace
  TOP-REAL 3 holds dual2 P = dual3 P
  proof
    let P be non zero_proj2 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_proj3 P = |[u.1/u.3,u.2/u.3,1]| &
      normalize_proj2 P = |[u.1/u.2,1,u.3/u.2]| by A2,Th17,Th14;
    now
      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
A4:     x = P9 and
A5:     Pdir2a P,Pdir2b 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_proj2(P)).1,
            a3 = - (normalize_proj2(P)).3,
            b1 = u9`1, b2 = u9`2, b3 = u9`3;
A8:     a2 = - (normalize_proj2(P))`1
          .= - u.1/u.2 by A3;
A9:     a3 = - (normalize_proj2(P))`3
          .= - u.3/u.2 by A3;
        0 = |{ dir2a P,dir2b P,u9 }| by A5,A6,A7,BKMODEL1:1
         .= |{ |[1, a2, 0]| ,
               |[0, a3, 1]|,
               |[b1, b2, b3]| }|
         .= a2 * b1 + a3 * b3 - b2 by Th3
         .= -(u.1/u.2 * b1 + b2 + u.3/u.2 * b3) by A8,A9;
        then
A10:    0 = u.2 * (u.1/u.2 *b1 + b2 + u.3/u.2 * b3)
         .= u.2 * b2 + u.2 * (u.1 / u.2) * b1 + u.2 * (u.3/u.2) * b3
         .= u.2 * b2 + u.1 * b1 + u.2 * (u.3/u.2) * b3
           by A2,Th13,XCMPLX_1:87
         .= u.2 * b2 + u.1 * b1 + u.3 * b3 by A2,Th13,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;
A13:    u.3 / u.3 = 1 by A2,Th16,XCMPLX_1:60;
        |{ |[1,   0,c2]|,
           |[0,   1,c3]|,
           |[u9`1,u9`2,u9`3]| }| = b3 - b1 * (-u.1/u.3) - b2 * (-u.2/u.3)
             by A11,A12,Th4
                                .= (u.1/u.3) * b1 + (u.2/u.3) * b2 + b3;
        then |{dir3a P,dir3b P,u9}|
          = (u.1 * (1/u.3)) *  b1 + (u.2/(u.3)) * b2 + (u.3/u.3) * b3
            by A13

         .= (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(Pdir2a P,Pdir2b 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
A14:    x = P9 and
A15:    Pdir3a P,Pdir3b P,P9 are_collinear;
        consider u9 be Element of TOP-REAL 3 such that
A16:    u9 is non zero and
A17:    P9 = Dir u9 by ANPROJ_1:26;
        set a2 = - (normalize_proj3(P)).1,
            a3 = - (normalize_proj3(P)).2,
            b1 = u9`1, b2 = u9`2, b3 = u9`3;
        set c2 = - (normalize_proj2(P)).1,
            c3 = - (normalize_proj2(P)).3;
A18:    a2 = - (normalize_proj3(P))`1
          .= - u.1/u.3 by A3;
A19:    a3 = - (normalize_proj3(P))`2
          .= - u.2/u.3 by A3;
A20:    c2 = - (normalize_proj2(P))`1
          .= - u.1/u.2 by A3;
A21:    c3 = - (normalize_proj2(P))`3
          .= - u.3/u.2 by A3;
A22:    0 = |{ dir3a P,dir3b P,u9 }| by A15,A16,A17,BKMODEL1:1
         .= |{ |[1, 0, a2]| ,
               |[0, 1, a3]|,
               |[b1, b2, b3]| }|
         .= b3 - a2 * b1 - a3 * b2 by Th4
         .= (u.1/u.3) *  b1 + (u.2/u.3) * b2 + 1 * b3 by A18,A19
         .= (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);
A23:    u.3 <> 0 by A2,Th16;
        |{dir2a P,dir2b P,u9}| = |{ |[1 ,c2, 0]| ,
                                    |[0 ,c3, 1]|,
                                    |[b1, b2, b3]| }|
          .= c3 * b3 + c2 * b1 - b2 by Th3
          .= (-u.1/u.2) * b1 + (-1) * b2 + (-u.3/u.2) * b3 by A20,A21
          .= (-u.1/u.2) * b1 + (-u.2/u.2) * b2 + (-u.3/u.2) * b3
            by XCMPLX_1:60,A2,Th13
          .= (u.1/-u.2) * b1 + (-u.2/u.2) * b2 + (-u.3/u.2) * b3
            by XCMPLX_1:188
          .= (u.1/-u.2) * b1 + (u.2/-u.2) * b2 + (-u.3/u.2) * b3
            by XCMPLX_1:188
          .= (u.1/-u.2) * b1 + (u.2/-u.2) * b2 + (u.3/-u.2) * b3
            by XCMPLX_1:188
          .= (1/-u.2) * (u.1 * b1 + u.2 * b2 + u.3 * b3)
          .= (1/-u.2) * 0 by A23,XCMPLX_1:6,A22
          .= 0;
        then Pdir2a P,Pdir2b P,P9 are_collinear by A16,A17,BKMODEL1:1;
        hence x in Line(Pdir2a P,Pdir2b P) by A14;
      end;
      hence Line(Pdir3a P,Pdir3b P) c= Line(Pdir2a P,Pdir2b P);
    end;
    hence thesis;
  end;
