
theorem
  for l,m,n being Element of ProjectiveLines real_projective_plane holds
  l,m,l are_concurrent & l,l,m are_concurrent & l,m,m are_concurrent
  proof
    let l1,l2,l3 be Element of ProjectiveLines real_projective_plane;
    dual l1,dual l2,dual l1 are_collinear &
      dual l1,dual l1,dual l2 are_collinear &
      dual l1,dual l2,dual l2 are_collinear by ANPROJ_2:def 7;
    then dual dual l1,dual dual l2,dual dual l1 are_concurrent &
      dual dual l1,dual dual l1,dual dual l2 are_concurrent &
      dual dual l1,dual dual l2,dual dual l2 are_concurrent by Th59;
    then l1,dual dual l2,dual dual l1 are_concurrent &
      l1,dual dual l1,dual dual l2 are_concurrent &
      l1,dual dual l2,dual dual l2 are_concurrent by Th46;
    then l1,l2,dual dual l1 are_concurrent &
      l1,l1,dual dual l2 are_concurrent &
      l1,l2,dual dual l2 are_concurrent by Th46;
    hence thesis;
  end;
