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 Th13:
  (line_homography(N1)).((line_homography(N2)).l)
    = (line_homography(N1 * N2)).l
  proof
    reconsider l2 = (line_homography(N2)).l as
      LINE of IncProjSp_of real_projective_plane;
A1: l2 = {homography(N2).P where
       P is POINT of IncProjSp_of real_projective_plane : P on l} by Def02;
A2: (line_homography(N1)).((line_homography(N2)).l) = {homography(N1).P where
      P is POINT of IncProjSp_of real_projective_plane : P on l2} by Def02;
    set X = {homography(N1).P where
      P is POINT of IncProjSp_of real_projective_plane : P on l2},
        Y = {homography(N1 * N2).P where
      P is POINT of IncProjSp_of real_projective_plane : P on l};
    {homography(N1).P where
        P is POINT of IncProjSp_of real_projective_plane : P on l2}
      = {homography(N1 * N2).P where
        P is POINT of IncProjSp_of real_projective_plane : P on l}
    proof
A3:  X c= Y
      proof
        let x be object;
        assume x in X;
        then consider P be POINT of IncProjSp_of real_projective_plane
        such that
A4:     x = homography(N1).P and
A5:     P on l2;
A6:     P is Point of real_projective_plane by INCPROJ:3;
        l2 is LINE of real_projective_plane by INCPROJ:4;
        then P in {homography(N2).P where
          P is POINT of IncProjSp_of real_projective_plane : P on l}
          by A6,A1,A5,INCPROJ:5;
        then consider P2 be POINT of IncProjSp_of real_projective_plane
        such that
A7:     P = homography(N2).P2 and
A8:     P2 on l;
        P2 is Point of real_projective_plane by INCPROJ:3;
        then x = (homography(N1*N2)).P2 by A4,A7,ANPROJ_9:13;
        hence thesis by A8;
      end;
      Y c= X
      proof
        let x be object;
        assume x in Y;
        then consider P be POINT of IncProjSp_of real_projective_plane
        such that
A9:     x = homography(N1*N2).P and
A10:    P on l;
A11:    P is Point of real_projective_plane by INCPROJ:3;
        P is Element of ProjectiveSpace TOP-REAL 3 by INCPROJ:3;
        then homography(N2).P is Point of real_projective_plane by FUNCT_2:5;
        then reconsider P2 = homography(N2).P as
          POINT of IncProjSp_of real_projective_plane by INCPROJ:3;
        now
          thus x = homography(N1).P2 by A11,A9,ANPROJ_9:13;
A12:      P2 in l2 by A10,A1;
          l2 is LINE of real_projective_plane by INCPROJ:4;
          hence P2 on l2 by A12,INCPROJ:5;
        end;
        hence thesis;
      end;
      hence thesis by A3;
    end;
    hence thesis by A2,Def02;
  end;
