
theorem Th43:
  for N being invertible Matrix of 3,F_Real
  for p,q,r,s,t,u,np,nq,nr,ns being Element of real_projective_plane st
  p <> q & r <> s & np <> nq & nr <> ns &
  p,q,t are_collinear & r,s,t are_collinear &
  np = homography(N).p & nq = homography(N).q &
  nr = homography(N).r & ns = homography(N).s &
  np,nq,u are_collinear & nr,ns,u are_collinear &
  not p,q,r are_collinear holds u = homography(N).t
  proof
    let N be invertible Matrix of 3,F_Real;
    let p,q,r,s,t,u,np,nq,nr,ns being Element of real_projective_plane;
    assume that
A1: p <> q & r <> s & np <> nq & nr <> ns &
      p,q,t are_collinear & r,s,t are_collinear &
      np = homography(N).p & nq = homography(N).q &
      nr = homography(N).r & ns = homography(N).s &
      np,nq,u are_collinear & nr,ns,u are_collinear and
A2: not p,q,r are_collinear;
    u = homography(N).t or Line(np,nq) = Line(nr,ns) by A1,Th42;
    hence thesis by A1,A2,Th41;
  end;
