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 Th17:
  for x,y,z being Element of EnsLineHomography3 holds
  (x (*) y) (*) z = x (*) (y (*) z)
  proof
    let x,y,z be Element of EnsLineHomography3;
    x in EnsLineHomography3;
    then consider Nx be invertible Matrix of 3,F_Real such that
A2: x = line_homography(Nx);
    y in EnsLineHomography3;
    then consider Ny be invertible Matrix of 3,F_Real such that
A3: y = line_homography(Ny);
    z in EnsLineHomography3;
    then consider Nz be invertible Matrix of 3,F_Real such that
A4: z = line_homography(Nz);
A5: width Nx = 3 & len Ny = 3 & width Ny = 3 & len Nz = 3 by MATRIX_0:24;
    y (*) z = line_homography(Ny * Nz) by A3,A4,Th16; then
A6: (x (*) (y (*) z)) = line_homography(Nx * (Ny * Nz)) by A2,Th16
    .= line_homography((Nx * Ny) * Nz) by A5,MATRIX_3:33;
    x (*) y = line_homography(Nx * Ny) by A2,A3,Th16;
    hence thesis by A6,A4,Th16;
  end;
