
theorem
  for P being Point of real_projective_plane holds not P in dual P
  proof
    let P be Point of real_projective_plane;
    reconsider P9 = P as Element of ProjectiveSpace TOP-REAL 3;
    per cases by Th37;
    suppose P9 is non zero_proj1;
      then reconsider P9 = P as
        non zero_proj1 Element of ProjectiveSpace TOP-REAL 3;
      #P = P9;
      then consider P99 be non zero_proj1 Point of ProjectiveSpace TOP-REAL 3
      such that
A1:   P = P99 and
A2:   dual P = dual1 P99 by Th38;
      assume P in dual P;
      hence contradiction by A1,A2,Th41;
    end;
    suppose P9 is non zero_proj2;
      then reconsider P9 = P as
        non zero_proj2 Element of ProjectiveSpace TOP-REAL 3;
      #P = P9;
      then consider P99 be non zero_proj2 Point of ProjectiveSpace TOP-REAL 3
      such that
A3:   P = P99 and
A4:   dual P = dual2 P99 by Th39;
      assume P in dual P;
      hence contradiction by Th42,A3,A4;
    end;
    suppose P9 is non zero_proj3;
      then reconsider P9 = P as
        non zero_proj3 Element of ProjectiveSpace TOP-REAL 3;
      #P = P9;
      then consider P99 be non zero_proj3 Point of ProjectiveSpace TOP-REAL 3
      such that
A5:   P = P99 and
A6:   dual P = dual3 P99 by Th40;
      assume P in dual P;
      hence contradiction by Th43,A5,A6;
    end;
  end;
