reserve a,b,c,d,e,f for Real,
        g           for positive Real,
        x,y         for Complex,
        S,T         for Element of REAL 2,
        u,v,w       for Element of TOP-REAL 3;
reserve a,b,c for Element of F_Real,
          M,N for Matrix of 3,F_Real;
reserve D        for non empty set;
reserve d1,d2,d3 for Element of D;
reserve A        for Matrix of 1,3,D;
reserve B        for Matrix of 3,1,D;

theorem
  for N being invertible Matrix of 3,F_Real
  for NR being Matrix of 3,REAL
  for P,Q being Element of ProjectiveSpace TOP-REAL 3
  for u,v being non zero Element of TOP-REAL 3
  for vfr,ufr being FinSequence of REAL
  st P = Dir u & Q = Dir v & u = ufr & v = vfr & N = NR & NR * ufr = vfr holds
  homography(N).P = Q
  proof
    let N be invertible Matrix of 3,F_Real;
    let NR be Matrix of 3,REAL;
    let P,Q be Element of ProjectiveSpace TOP-REAL 3;
    let u,v be non zero Element of TOP-REAL 3;
    let vfr,ufr be FinSequence of REAL;
    assume
A1: P = Dir u & Q = Dir v & u = ufr & v = vfr & N = NR & NR * ufr = vfr;
    consider u1,v1 being Element of TOP-REAL 3,
               u1f being FinSequence of F_Real,
               p1 being FinSequence of 1-tuples_on REAL such that
A2: P = Dir u1 & u1 is not zero & u1 = u1f & p1 = N * u1f & v1 = M2F p1 &
      v1 is not zero & (homography(N)).P = Dir v1 by ANPROJ_8:def 4;
    reconsider u1fr = u1f as FinSequence of REAL;
    u1 in TOP-REAL 3;
    then u1 in REAL 3 by EUCLID:22; then
A3: len u1fr = 3 by A2,EUCLID_8:50; then
A4: v1 = NR * u1fr by A1,A2,Th30;
    are_Prop u,u1 by A1,A2,ANPROJ_1:22;
    then consider a be Real such that
A5: a <> 0 and
A6: u = a * u1 by ANPROJ_1:1;
A7: width NR = 3 by MATRIX_0:23;
A8: width NR = len u1fr & len u1fr > 0 by A3,MATRIX_0:23;
A9: len NR = 3 by MATRIX_0:24;
    u in TOP-REAL 3;
    then u in REAL 3 by EUCLID:22;
    then width NR = len ufr & len ufr > 0 by A7,A1,EUCLID_8:50;
    then len (NR * ufr) = 3 & len (NR * u1fr) = 3
      by A9,A3,MATRIX_0:23,MATRIXR1:61;
    then NR * ufr is Element of REAL 3 & NR * u1fr is Element of REAL 3
      by EUCLID_8:2;
    then reconsider w1 = NR * ufr,w2 = NR * u1fr as Element of TOP-REAL 3
      by EUCLID:22;
    w1 = a * w2 by A8,A1,A2,A6,MATRIXR1:59;
    then are_Prop w1,w2 by A5,ANPROJ_1:1;
    hence thesis by A1,A2,A4,ANPROJ_1:22;
  end;
