 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
  for P1,P2,P3,P4,Q1,Q2,Q3,Q4 being Point of ProjectiveSpace TOP-REAL 3 st
  not (P1,P2,P3 are_collinear) & not (P1,P2,P4 are_collinear) &
  not (P1,P3,P4 are_collinear) & not (P2,P3,P4 are_collinear) &
  not (Q1,Q2,Q3 are_collinear) & not (Q1,Q2,Q4 are_collinear) &
  not (Q1,Q3,Q4 are_collinear) & not (Q2,Q3,Q4 are_collinear) holds
  ex N being invertible Matrix of 3,F_Real st
  (homography(N)).P1 = Q1 & (homography(N)).P2 = Q2 &
  (homography(N)).P3 = Q3 & (homography(N)).P4 = Q4
  proof
    let P1,P2,P3,P4,Q1,Q2,Q3,Q4 be Point of ProjectiveSpace TOP-REAL 3;
    assume that
A1: not (P1,P2,P3 are_collinear) & not (P1,P2,P4 are_collinear) &
    not (P1,P3,P4 are_collinear) & not (P2,P3,P4 are_collinear) and
A2: not (Q1,Q2,Q3 are_collinear) & not (Q1,Q2,Q4 are_collinear) &
    not (Q1,Q3,Q4 are_collinear) & not (Q2,Q3,Q4 are_collinear);
    consider N1 being invertible Matrix of 3,F_Real such that
A3: (homography(N1)).P1 = Dir100 & (homography(N1)).P2 = Dir010 &
    (homography(N1)).P3 = Dir001 & (homography(N1)).P4 = Dir111 by A1,Th26;
    consider N2 being invertible Matrix of 3,F_Real such that
A4: (homography(N2)).Q1 = Dir100 & (homography(N2)).Q2 = Dir010 &
    (homography(N2)).Q3 = Dir001 & (homography(N2)).Q4 = Dir111 by A2,Th26;
    reconsider N = (N2~) * N1 as invertible Matrix of 3,F_Real;
    take N;
    thus Q1 = (homography(N2~)).((homography(N1)).P1) by A3,A4,Th16
           .= (homography(N)).P1 by Th14;
    thus Q2 = (homography(N2~)).((homography(N1)).P2) by A3,A4,Th16
           .= (homography(N)).P2 by Th14;
    thus Q3 = (homography(N2~)).((homography(N1)).P3) by A3,A4,Th16
           .= (homography(N)).P3 by Th14;
    thus Q4 = (homography(N2~)).((homography(N1)).P4) by A3,A4,Th16
           .= (homography(N)).P4 by Th14;
  end;
