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 Th14:
  (line_homography(1.(F_Real,3))).l = l
  proof
    set X = {homography(1.(F_Real,3)).P where
      P is POINT of IncProjSp_of real_projective_plane : P on l};
A1: X c= l
    proof
      let x be object;
      assume x in X;
      then consider P be POINT of IncProjSp_of real_projective_plane such that
A2:   x = homography(1.(F_Real,3)).P and
A3:   P on l;
A4:   P is Point of real_projective_plane by INCPROJ:2;
      then
A5:   x = P by A2,ANPROJ_9:14;
      l is LINE of real_projective_plane by INCPROJ:4;
      hence thesis by A4,A3,A5,INCPROJ:5;
    end;
    l c= X
    proof
      let x be object;
      assume
A6:   x in l;
A7:   l is LINE of real_projective_plane by INCPROJ:4;
      l is Subset of real_projective_plane by INCPROJ:4;
      then reconsider x9 = x as Point of real_projective_plane by A6;
      reconsider l9 = l as LINE of IncProjSp_of real_projective_plane;
      reconsider x99 = x9 as POINT of IncProjSp_of real_projective_plane
        by INCPROJ:3;
A8:   x99 on l by A7,A6,INCPROJ:5;
      homography(1.(F_Real,3)).x99 = x99 by ANPROJ_9:14;
      hence thesis by A8;
    end;
    hence thesis by A1,Def02;
  end;
