
theorem Th39:
  for P being Element of real_projective_plane st #P is non zero_proj2 holds
  ex P9 being non zero_proj2 Point of ProjectiveSpace TOP-REAL 3 st
  P = P9 & dual P = dual2 P9
  proof
    let P be Element of real_projective_plane;
    assume
A1: #P is non zero_proj2;
    reconsider P1 = #P as non zero_proj2 Point of ProjectiveSpace TOP-REAL 3
      by A1;
    per cases;
    suppose
      P1 is non zero_proj1 & P1 is zero_proj3;
      then reconsider P9 = P1 as
        non zero_proj1 non zero_proj2 Element of ProjectiveSpace TOP-REAL 3;
      dual P1 = dual1 P9 & dual1 P9 = dual2 P9 by Def22,Th33;
      hence thesis;
    end;
    suppose P1 is zero_proj1 & P1 is non zero_proj3;
      hence thesis by Def22;
    end;
    suppose P1 is zero_proj1 & P1 is zero_proj3;
      hence thesis by Def22;
    end;
    suppose P1 is non zero_proj1 & P1 is non zero_proj3;
      then reconsider P9 = P as non zero_proj1 non zero_proj2
        non zero_proj3 Element of ProjectiveSpace TOP-REAL 3;
      dual P1 = dual1 P9 & dual1 P9 = dual2 P9 by Def22,Th33;
      hence thesis;
    end;
  end;
