reserve P for Element of BK_model;
reserve N,N1,N2 for invertible Matrix of 3,F_Real;
reserve l,l1,l2 for Element of the Lines of IncProjSp_of real_projective_plane;

theorem Th16:
  for h1,h2 being Element of EnsLineHomography3 st
  h1 = line_homography(N1) & h2 = line_homography(N2) holds
  line_homography(N1 * N2) = h1 (*) h2
  proof
    let h1,h2 be Element of EnsLineHomography3;
    assume that
A1: h1 = line_homography(N1) and
A2: h2 = line_homography(N2);
    consider M1,M2 be invertible Matrix of 3,F_Real such that
A3: h1 = line_homography(M1) and
A4: h2 = line_homography(M2) and
A5: h1 (*) h2 = line_homography(M1 * M2) by Def03;
    reconsider h12 = h1 (*) h2 as
    Function of the Lines of IncProjSp_of real_projective_plane,
      the Lines of IncProjSp_of real_projective_plane by A5;
    now
      dom line_homography(N1 * N2)
        = the Lines of IncProjSp_of real_projective_plane
        by FUNCT_2:def 1;
      hence dom line_homography(N1 * N2) = dom h12 by FUNCT_2:def 1;
      thus for x be object st x in dom line_homography(N1 * N2) holds
      (line_homography(N1 * N2)).x = h12.x
      proof
        let x be object;
        assume x in dom line_homography(N1 * N2);
        then reconsider xf = x as
          Element of the Lines of IncProjSp_of real_projective_plane;
        (line_homography(N1 * N2)).xf
          = (line_homography(N1)).((line_homography(N2)).xf)
            by Th13
         .= h12.xf by A3,A4,A5,A1,A2,Th13;
        hence thesis;
      end;
    end;
    hence thesis by FUNCT_1:def 11;
  end;
