
theorem Th48:
  for l,m being Element of ProjectiveLines real_projective_plane holds
  (l <> m iff dual l <> dual m)
  proof
    let l,m be Element of ProjectiveLines real_projective_plane;
    now
      assume
A1:   l <> m;
      assume dual l = dual m;
      then l = dual dual m by Th46;
      hence contradiction by A1,Th46;
    end;
    hence thesis;
  end;
