 reserve i,n for Nat;
 reserve r for Real;
 reserve ra for Element of F_Real;
 reserve a,b,c for non zero Element of F_Real;
 reserve u,v for Element of TOP-REAL 3;
 reserve p1 for FinSequence of (1-tuples_on REAL);
 reserve pf,uf for FinSequence of F_Real;
 reserve N for Matrix of 3,F_Real;
 reserve K for Field;
 reserve k for Element of K;
 reserve N,N1,N2 for invertible Matrix of 3,F_Real;
 reserve P,P1,P2,P3 for Point of ProjectiveSpace TOP-REAL 3;

theorem Th16:
  (homography(N)).((homography(N~)).P) = P &
  (homography(N~)).((homography(N)).P) = P
  proof
A1: N~ is_reverse_of N by MATRIX_6:def 4;
    thus (homography(N)).((homography(N~)).P) = (homography(N * N~)).P
                                               by Th14
                                             .= (homography(1.(F_Real,3))).P
                                               by A1,MATRIX_6:def 2
                                             .= P by Th15;
    thus (homography(N~)).((homography(N)).P) = (homography(N~ * N)).P
                                               by Th14
                                             .= (homography(1.(F_Real,3))).P
                                               by A1,MATRIX_6:def 2
                                             .= P by Th15;
  end;
