
theorem Th45:
  for P being Point of real_projective_plane holds dual dual P = P
  proof
    let P be Point of real_projective_plane;
    reconsider P9 = P as Point of ProjectiveSpace TOP-REAL 3;
    per cases by Th37;
    suppose P9 is non zero_proj1;
      then reconsider P9 = P as
        non zero_proj1 Point of ProjectiveSpace TOP-REAL 3;
      reconsider P = P9 as Point of real_projective_plane;
      consider P99 be non zero_proj1 Point of ProjectiveSpace TOP-REAL 3
      such that
A1:   P = P99 & dual P= dual1 P99 by Th38;
      reconsider l = Line(Pdir1a P9,Pdir1b P9) as
        Element of ProjectiveLines real_projective_plane by A1;
      consider P1,P2 be Point of real_projective_plane such that
A2:   P1 <> P2 and
A3:   l = Line(P1,P2) and
A4:   dual l = L2P(P1,P2) by Def25;
A5:   Line(P1,P2) = Line(Pdir1a P9,Pdir1b P9) & P1 in Line(P1,P2) &
        P2 in Line(P1,P2) by A3,COLLSP:10;
      then consider Q1 be Point of ProjectiveSpace TOP-REAL 3 such that
A6:   P1 = Q1 and
A7:   Pdir1a P9,Pdir1b P9,Q1 are_collinear;
      consider Q2 be Point of ProjectiveSpace TOP-REAL 3 such that
A8:   P2 = Q2 and
A9:   Pdir1a P9,Pdir1b P9,Q2 are_collinear by A5;
      consider u,v be non zero Element of TOP-REAL 3 such that
A10:  P1 = Dir u and
A11:  P2 = Dir v and
A12:  L2P(P1,P2) = Dir(u <X> v) by A2,BKMODEL1:def 5;
      consider w be Element of TOP-REAL 3 such that
A13:  w is not zero and
A14:  P9 = Dir w by ANPROJ_1:26;
      reconsider w as non zero Element of TOP-REAL 3 by A13;
      normalize_proj1 P9 = |[1, w.2/w.1,w.3/w.1]| by A14,Th11;
      then (normalize_proj1(P9))`2 = w.2/w.1 &
        (normalize_proj1(P9))`3 = w.3/w.1;
      then
A15:  dir1a P9 <X> dir1b P9
        = |[ (1 * 1) - (0 * 0),
             (0 * (-w.3/w.1)) - ((-w.2/w.1) * 1),
             ((-w.2/w.1) * 0) - ((-w.3/w.1) * 1) ]|
       .= |[ w`1/w.1, w`2/w.1,w`3/w.1 ]| by A14,Th10,XCMPLX_1:60
       .= 1/w.1 * w by EUCLID_5:7;
      w.1 <> 0 by A14,Th10;
      then reconsider a = 1/w.1 * w as non zero Element of TOP-REAL 3
        by ANPROJ_9:3;
      now
        assume u <X> v = 0.TOP-REAL 3;
        then are_Prop u,v by ANPROJ_8:51;
        hence contradiction by A2,A10,A11,ANPROJ_1:22;
      end;
      then
A16:  u <X> v is non zero;
      now
        now
          thus not are_Prop u,v by A2,A10,A11,ANPROJ_1:22;
          thus 0 = |{ dir1a P9,dir1b P9,u }| by A10,A6,A7,BKMODEL1:1
                .= |{ u, dir1a P9,dir1b P9 }| by EUCLID_5:34
                .= |( a, u )| by A15;
          thus 0 = |{ dir1a P9,dir1b P9,v }| by A11,A8,A9,BKMODEL1:1
                .= |{ v, dir1a P9,dir1b P9 }| by EUCLID_5:34
                .= |( a, v )| by A15;
        end;
        then are_Prop 1/w.1 * w, u <X> v by Th8;
        hence are_Prop w.1 * a,u <X> v by A14,Th10,A16,Th9;
        thus w.1 * a = (w.1 * (1/w.1)) * w by RVSUM_1:49
                    .= 1 * w by A14,Th10,XCMPLX_1:106
                    .= w by RVSUM_1:52;
      end;
      hence thesis by A14,A1,A4,A12,A16,ANPROJ_1:22;
    end;
    suppose P9 is non zero_proj2;
      then reconsider P9 = P as
        non zero_proj2 Point of ProjectiveSpace TOP-REAL 3;
      reconsider P = P9 as Point of real_projective_plane;
      consider P99 be non zero_proj2 Point of ProjectiveSpace TOP-REAL 3
      such that
A17:  P = P99 & dual P= dual2 P99 by Th39;
      reconsider l = Line(Pdir2a P9,Pdir2b P9) as
        Element of ProjectiveLines real_projective_plane by A17;
      consider P1,P2 be Point of real_projective_plane such that
A18:  P1 <> P2 and
A19:  l = Line(P1,P2) and
A20:  dual l = L2P(P1,P2) by Def25;
A21:  Line(P1,P2) = Line(Pdir2a P9,Pdir2b P9) & P1 in Line(P1,P2) &
        P2 in Line(P1,P2) by A19,COLLSP:10;
      then consider Q1 be Point of ProjectiveSpace TOP-REAL 3 such that
A22:  P1 = Q1 and
A23:  Pdir2a P9,Pdir2b P9,Q1 are_collinear;

      consider Q2 be Point of ProjectiveSpace TOP-REAL 3 such that
A24:  P2 = Q2 and
A25:  Pdir2a P9,Pdir2b P9,Q2 are_collinear by A21;
      consider u,v be non zero Element of TOP-REAL 3 such that
A26:  P1 = Dir u and
A27:  P2 = Dir v and
A28:  L2P(P1,P2) = Dir(u <X> v) by A18,BKMODEL1:def 5;
      consider w be Element of TOP-REAL 3 such that
A29:  w is not zero and
A30:  P9 = Dir w by ANPROJ_1:26;
      reconsider w as non zero Element of TOP-REAL 3 by A29;
      normalize_proj2 P9 = |[w.1/w.2, 1, w.3/w.2]| by A30,Th14;
      then (normalize_proj2(P9))`1 = w.1/w.2 &
        (normalize_proj2(P9))`3 = w.3/w.2;
      then
A31:  dir2a P9 <X> dir2b P9
        = |[ ((-w.1/w.2) * 1) - (0 * (-w.3/w.2)),
             (0 * 0) - (1 * 1),
             (1 * (-w.3/w.2)) - (0 * (-w.1/w.2)) ]|
       .= |[ -w.1/w.2, -w.2/w.2,-w.3/w.2 ]| by A30,Th13,XCMPLX_1:60
       .= |[ w.1/(-w.2), -w.2/w.2,-w.3/w.2 ]| by XCMPLX_1:188
       .= |[ w.1/(-w.2), w.2/(-w.2),-w.3/w.2 ]| by XCMPLX_1:188
       .= |[ w`1/(-w.2), w`2/(-w.2),w`3/(-w.2) ]| by XCMPLX_1:188
       .= 1/(-w.2) * w by EUCLID_5:7;
A32:  w.2 <> 0 by A30,Th13;
      then reconsider a = 1/(-w.2) * w as non zero Element of TOP-REAL 3
        by ANPROJ_9:3;
      now
        assume u <X> v = 0.TOP-REAL 3;
        then are_Prop u,v by ANPROJ_8:51;
        hence contradiction by A18,A26,A27,ANPROJ_1:22;
      end;
      then
A33:  u <X> v is non zero;
      now
        now
          thus not are_Prop u,v by A18,A26,A27,ANPROJ_1:22;
          thus 0 = |{ dir2a P9,dir2b P9,u }| by A26,A22,A23,BKMODEL1:1
                .= |{ u, dir2a P9,dir2b P9 }| by EUCLID_5:34
                .= |( a, u )| by A31;
          thus 0 = |{ dir2a P9,dir2b P9,v }| by A27,A24,A25,BKMODEL1:1
                .= |{ v, dir2a P9,dir2b P9 }| by EUCLID_5:34
                .= |( a, v )| by A31;
        end;
        then are_Prop 1/(-w.2) * w, u <X> v by Th8;
        hence are_Prop (-w.2) * a,u <X> v by A32,A33,Th9;
        thus (-w.2) * a = ((-w.2) * (1/(-w.2))) * w by RVSUM_1:49
                       .= 1 * w by A32,XCMPLX_1:106
                       .= w by RVSUM_1:52;
      end;
      hence thesis by A30,A17,A20,A28,A33,ANPROJ_1:22;
    end;
    suppose P9 is non zero_proj3;
      then reconsider P9 = P as
        non zero_proj3 Point of ProjectiveSpace TOP-REAL 3;
      reconsider P = P9 as Point of real_projective_plane;
      consider P99 be non zero_proj3 Point of ProjectiveSpace TOP-REAL 3
      such that
A34:  P = P99 & dual P= dual3 P99 by Th40;
      reconsider l = Line(Pdir3a P9,Pdir3b P9) as
        Element of ProjectiveLines real_projective_plane by A34;
      consider P1,P2 be Point of real_projective_plane such that
A35:  P1 <> P2 and
A36:  l = Line(P1,P2) and
A37:  dual l = L2P(P1,P2) by Def25;
A38:  Line(P1,P2) = Line(Pdir3a P9,Pdir3b P9) & P1 in Line(P1,P2) &
        P2 in Line(P1,P2) by A36,COLLSP:10;
      then consider Q1 be Point of ProjectiveSpace TOP-REAL 3 such that
A39:  P1 = Q1 and
A40:  Pdir3a P9,Pdir3b P9,Q1 are_collinear;
      consider Q2 be Point of ProjectiveSpace TOP-REAL 3 such that
A41:  P2 = Q2 and
A42:  Pdir3a P9,Pdir3b P9,Q2 are_collinear by A38;
      consider u,v be non zero Element of TOP-REAL 3 such that
A43:  P1 = Dir u and
A44:  P2 = Dir v and
A45:  L2P(P1,P2) = Dir(u <X> v) by A35,BKMODEL1:def 5;
      consider w be Element of TOP-REAL 3 such that
A46:  w is not zero and
A47:  P9 = Dir w by ANPROJ_1:26;
      reconsider w as non zero Element of TOP-REAL 3 by A46;
      normalize_proj3 P9 = |[w.1/w.3, w.2/w.3, 1]| by A47,Th17;
      then (normalize_proj3(P9))`1 = w.1/w.3 &
        (normalize_proj3(P9))`2 = w.2/w.3;
      then
A48:  dir3a P9 <X> dir3b P9
        = |[ (0 * (-w.2/w.3)) - ((-w.1/w.3) * 1),
             ((-w.1/w.3) * 0) - (1 * (-w.2/w.3)),
             (1 * 1) - (0 * 0) ]|
       .= |[ w`1/w.3, w`2/w.3,w`3/w.3 ]| by A47,Th16,XCMPLX_1:60
       .= 1/(w.3) * w by EUCLID_5:7;
      w.3 <> 0 by A47,Th16;
      then reconsider a = 1/(w.3) * w as non zero Element of TOP-REAL 3
        by ANPROJ_9:3;
      now
        assume u <X> v = 0.TOP-REAL 3;
        then are_Prop u,v by ANPROJ_8:51;
        hence contradiction by A35,A43,A44,ANPROJ_1:22;
      end;
      then
A49:  u <X> v is non zero;
      now
        now
          thus not are_Prop u,v by A35,A43,A44,ANPROJ_1:22;
          thus 0 = |{ dir3a P9,dir3b P9,u }| by A43,A39,A40,BKMODEL1:1
                .= |{ u, dir3a P9,dir3b P9 }| by EUCLID_5:34
                .= |( a, u )| by A48;
          thus 0 = |{ dir3a P9,dir3b P9,v }| by A44,A41,A42,BKMODEL1:1
          .= |{ v, dir3a P9,dir3b P9 }| by EUCLID_5:34
          .= |( a, v )| by A48;
        end;
        then are_Prop 1/(w.3) * w, u <X> v by Th8;
        hence are_Prop (w.3) * a,u <X> v by A47,Th16,A49,Th9;
        thus w.3 * a = (w.3 * (1/w.3)) * w by RVSUM_1:49
                    .= 1 * w by A47,Th16,XCMPLX_1:106
                    .= w by RVSUM_1:52;
      end;
      hence thesis by A47,A34,A37,A45,A49,ANPROJ_1:22;
    end;
  end;
