
theorem
  for l,m being Element of ProjectiveLines real_projective_plane holds
  ex n being Element of ProjectiveLines real_projective_plane st
  l <> n & m <> n & l,m,n are_concurrent
  proof
    let l,m be Element of ProjectiveLines real_projective_plane;
    consider r be Point of real_projective_plane such that
A1: dual l <> r and
A2: dual m <> r and
A3: dual l, dual m, r are_collinear by ANPROJ_2:def 10;
    now
      thus l <> dual r & m <> dual r by Th45,A1,A2;
      dual dual l, dual dual m, dual r are_concurrent by A3,Th59;
      then l, dual dual m, dual r are_concurrent by Th46;
      hence l, m, dual r are_concurrent by Th46;
    end;
    hence thesis;
  end;
