
theorem
  for k,l1,l2,l3,m1,m2,m3,n1,n2,n3 being Element of ProjectiveLines
    real_projective_plane st
  k <> l2 & k <> l3 & l2 <> l3 & l1 <> l2 & l1 <> l3 & k <> m2 &
  k <> m3 & m2 <> m3 & m1 <> m2 & m1 <> m3 &
  not k,l1,m1 are_concurrent & k,l1,l2 are_concurrent &
  k,l1,l3 are_concurrent & k,m1,m2 are_concurrent &
  k,m1,m3 are_concurrent & l1,m2,n3 are_concurrent &
  m1,l2,n3 are_concurrent & l1,m3,n2 are_concurrent &
  l3,m1,n2 are_concurrent & l2,m3,n1 are_concurrent &
  l3,m2,n1 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 <> l2 and
A2: k <> l3 and
A3: l2 <> l3 and
A4: l1 <> l2 and
A5: l1 <> l3 and
A6: k <> m2 and
A7: k <> m3 and
A8: m2 <> m3 and
A9: m1 <> m2 and
A10: m1 <> m3 and
A11: not k,l1,m1 are_concurrent and
A12: k,l1,l2 are_concurrent and
A13: k,l1,l3 are_concurrent and
A14: k,m1,m2 are_concurrent and
A15: k,m1,m3 are_concurrent and
A16: l1,m2,n3 are_concurrent and
A17: m1,l2,n3 are_concurrent and
A18: l1,m3,n2 are_concurrent and
A19: l3,m1,n2 are_concurrent and
A20: l2,m3,n1 are_concurrent and
A21: l3,m2,n1 are_concurrent;
    now
      thus dual k <> dual l2 & dual k <> dual l3 & dual l2 <> dual l3
        by A1,A2,A3,Th48;
      thus dual l1 <> dual l2 & dual l1 <> dual l3 & dual k <> dual m2
        by A4,A5,A6,Th48;
      thus dual k <> dual m3 & dual m2 <> dual m3 & dual m1 <> dual m2 &
        dual m1 <> dual m3 by A7,A8,A9,A10,Th48;
      thus not dual k,dual l1,dual m1 are_collinear &
        dual k,dual l1,dual l2 are_collinear &
        dual k,dual l1,dual l3 are_collinear &
        dual k,dual m1,dual m2 are_collinear &
        dual k,dual m1,dual m3 are_collinear &
        dual l1,dual m2,dual n3 are_collinear &
        dual m1,dual l2,dual n3 are_collinear &
        dual l1,dual m3,dual n2 are_collinear &
        dual l3,dual m1,dual n2 are_collinear &
        dual l2,dual m3,dual n1 are_collinear &
        dual l3,dual m2,dual n1 are_collinear
        by A11,A12,A13,A14,A15,A16,A17,A18,A19,A20,A21,Th60;
    end;
    then dual n1,dual n2, dual n3 are_collinear by ANPROJ_2:def 13;
    hence thesis by Th60;
  end;
