
theorem
  for l,l1,l2,n,n1 being Element of ProjectiveLines real_projective_plane st
  l,l1,n are_concurrent & l1,l2,n1 are_concurrent
  ex n2 being Element of ProjectiveLines real_projective_plane st
  l,l2,n2 are_concurrent & n,n1,n2 are_concurrent
  proof
    let l,l1,l2,n,n1 be Element of ProjectiveLines real_projective_plane;
    assume that
A1: l,l1,n are_concurrent and
A2: l1,l2,n1 are_concurrent;
    dual l, dual l1, dual n are_collinear &
      dual l1,dual l2, dual n1 are_collinear by A1,A2,Th60;
    then consider P be Point of real_projective_plane such that
A3: dual l, dual l2, P are_collinear and
A4: dual n, dual n1, P are_collinear by ANPROJ_2:def 9;
    take dual P;
    dual dual l, dual dual l2, dual P are_concurrent &
      dual dual n, dual dual n1, dual P are_concurrent
      by A3,A4,Th59;
    then l, dual dual l2, dual P are_concurrent &
      n, dual dual n1, dual P are_concurrent by Th46;
    hence thesis by Th46;
  end;
