
theorem
  for k,l1,l2,l3,m1,m2,m3,n1,n2,n3 being Element of ProjectiveLines
  real_projective_plane st
  k <> m1 & l1 <> m1 & k <> m2 & l2 <> m2 & k <> m3 & l3 <> m3 &
  not k,l1,l2 are_concurrent & not k,l1,l3 are_concurrent &
  not k,l2,l3 are_concurrent &
  l1,l2,n3 are_concurrent & m1,m2,n3 are_concurrent &
  l2,l3,n1 are_concurrent & m2,m3,n1 are_concurrent &
  l1,l3,n2 are_concurrent & m1,m3,n2 are_concurrent &
  k,l1,m1 are_concurrent & k,l2,m2 are_concurrent &
  k,l3,m3 are_concurrent holds n1,n2,n3 are_concurrent
  proof
    let k,l1,l2,l3,m1,m2,m3,n1,n2,n3 be Element of ProjectiveLines
      real_projective_plane;
    assume that
A1: k <> m1 and
A2: l1 <> m1 and
A3: k <> m2 and
A4: l2 <> m2 and
A5: k <> m3 and
A6: l3 <> m3 and
A7: not k,l1,l2 are_concurrent and
A8: not k,l1,l3 are_concurrent and
A9: not k,l2,l3 are_concurrent and
A10: l1,l2,n3 are_concurrent and
A11: m1,m2,n3 are_concurrent and
A12: l2,l3,n1 are_concurrent and
A13: m2,m3,n1 are_concurrent and
A14: l1,l3,n2 are_concurrent and
A15: m1,m3,n2 are_concurrent and
A16: k,l1,m1 are_concurrent and
A17: k,l2,m2 are_concurrent and
A18: k,l3,m3 are_concurrent;
    dual k <> dual m1 & dual l1 <> dual m1 & dual k <> dual m2 &
      dual l2 <> dual m2 & dual k <> dual m3 & dual l3 <> dual m3 &
      not dual k,dual l1,dual l2 are_collinear &
      not dual k,dual l1,dual l3 are_collinear &
      not dual k,dual l2,dual l3 are_collinear &
      dual l1,dual l2,dual n3 are_collinear &
      dual m1,dual m2,dual n3 are_collinear &
      dual l2,dual l3,dual n1 are_collinear &
      dual m2,dual m3,dual n1 are_collinear &
      dual l1,dual l3,dual n2 are_collinear &
      dual m1,dual m3,dual n2 are_collinear &
      dual k,dual l1,dual m1 are_collinear &
      dual k,dual l2,dual m2 are_collinear &
      dual k,dual l3,dual m3 are_collinear
      by A1,A2,A3,A4,A5,A6,Th48,
         A7,A8,A9,A10,A11,A12,A13,A14,A15,A16,A17,A18,Th60;
    then dual n1,dual n2, dual n3 are_collinear by ANPROJ_2:def 12;
    hence thesis by Th60;
  end;
