 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 Th25:
  for P being Point of ProjectiveSpace TOP-REAL 3 st
  not (Dir100,Dir010,P are_collinear) &
  not (Dir100,Dir001,P are_collinear) &
  not (Dir010,Dir001,P are_collinear) holds
  ex a,b,c being non zero Element of F_Real st
  for N being invertible Matrix of 3,F_Real st
  N = <* <* a,0,0 *>,
         <* 0,b,0 *>,
         <* 0,0,c *> *> holds
  (homography(N)).P = Dir111
  proof
    let P be Point of ProjectiveSpace TOP-REAL 3;
    assume that
A1: not (Dir100,Dir010,P are_collinear) and
A2: not (Dir100,Dir001,P are_collinear) and
A3: not (Dir010,Dir001,P are_collinear);
    consider a,b,c being non zero Element of F_Real such that
A4: P = Dir |[a,b,c]| by Th23,A1,A2,A3;
    reconsider ia = 1/a, ib = 1/b, ic = 1/c as Element of F_Real
      by XREAL_0:def 1;
    ia is non zero & ib is non zero & ic is non zero by XCMPLX_1:50;
    then reconsider ia,ib,ic as non zero Element of F_Real;
    take ia,ib,ic;
    thus thesis by A4,Th24;
  end;
