
theorem Th18:
  for N being invertible Matrix of 3,F_Real
  for n11,n12,n13,n21,n22,n23,n31,n32,n33 being Element of F_Real
  for P,Q being Point of ProjectiveSpace TOP-REAL 3
  for u,v being non zero Element of TOP-REAL 3 st
  N = <* <* n11,n12,n13 *>,
         <* n21,n22,n23 *>,
         <* n31,n32,n33 *> *> &
  P = Dir u & Q = Dir v & Q = homography(N).P & u.3 = 1
  holds ex a being non zero Real st
  v.1 = a * (n11 * u.1 + n12 * u.2 + n13) &
  v.2 = a * (n21 * u.1 + n22 * u.2 + n23) &
  v.3 = a * (n31 * u.1 + n32 * u.2 + n33)
  proof
    let N be invertible Matrix of 3,F_Real;
    let n11,n12,n13,n21,n22,n23,n31,n32,n33 be Element of F_Real;
    let P,Q be Point of ProjectiveSpace TOP-REAL 3;
    let u,v be non zero Element of TOP-REAL 3;
    assume
A1: N = <* <* n11,n12,n13 *>, <* n21,n22,n23 *>, <* n31,n32,n33 *> *> &
    P = Dir u & Q = Dir v & Q = homography(N).P & u.3 = 1;
    consider u9,v9 be Element of TOP-REAL 3,
                uf be FinSequence of F_Real,
                p be FinSequence of 1-tuples_on REAL
    such that
A2: P = Dir u9 & u9 is not zero & u9 = uf & p = N * uf & v9 = M2F p &
      v9 is not zero & homography(N).P = Dir v9 by ANPROJ_8:def 4;
    are_Prop u,u9 by A1,A2,ANPROJ_1:22;
    then consider a be Real such that
A3: a <> 0 & u9 = a * u by ANPROJ_1:1;
A4: |[u9`1,u9`2,u9`3]| = u9 by EUCLID_5:3
                      .= |[ a * u`1,a * u`2, a * u`3 ]|
                        by A3,EUCLID_5:7
                      .= |[ a * u`1,a * u`2, a * u.3 ]| by EUCLID_5:def 3
                      .= |[ a * u`1,a * u`2, a]| by A1;
    reconsider uf1=a*u`1,uf2=a*u`2,uf3=a as Element of F_Real by XREAL_0:def 1;
    reconsider x1 = n11*u`1+n12*u`2+n13, x2 = n21*u`1+n22*u`2+n23,
    x3 = n31*u`1+n32*u`2+n33 as Element of REAL by XREAL_0:def 1;
    uf = <* uf1,uf2,uf3 *> by A2,A4,EUCLID_5:3;
    then
A5: v9 = <* n11*uf1+n12*uf2+n13*uf3,
            n21*uf1+n22*uf2+n23*uf3,
            n31*uf1+n32*uf2+n33*uf3 *> by A1,A2,PASCAL:8
      .= <* a*(n11*u`1+n12*u`2+n13),
            a*(n21*u`1+n22*u`2+n23),
            a*(n31*u`1+n32*u`2+n33) *>
      .= a * |[x1,x2,x3]| by EUCLID_5:8;
    are_Prop v,v9 by A1,A2,ANPROJ_1:22;
    then consider b be Real such that
A6: b <> 0 and
A7: v = b * v9 by ANPROJ_1:1;
A8: v = b * |[a * x1,a * x2, a * x3]| by A7,A5,EUCLID_5:8
     .= |[ b * (a * x1),b * (a * x2),b * (a * x3)]| by EUCLID_5:8
     .= |[ (b * a) * (n11 * u`1 + n12 * u`2 + n13),
           (b * a) * (n21 * u`1 + n22 * u`2 + n23),
           (b * a) * (n31 * u`1 + n32 * u`2 + n33) ]|;
    reconsider c = b * a as non zero Real by A3,A6;
    take c;
    v`1 = c * (n11 * u`1 + n12 * u`2 + n13) &
      v`2 = c * (n21 * u`1 + n22 * u`2 + n23) &
      v`3 = c * (n31 * u`1 + n32 * u`2 + n33)
      by A8,EUCLID_5:2;
    then v.1 = c * (n11 * u`1 + n12 * u`2 + n13) &
      v.2 = c * (n21 * u`1 + n22 * u`2 + n23) &
      v.3 = c * (n31 * u`1 + n32 * u`2 + n33)
      by EUCLID_5:def 1,def 2,def 3;
    then v.1 = c * (n11 * u.1 + n12 * u`2 + n13) &
      v.2 = c * (n21 * u.1 + n22 * u`2 + n23) &
      v.3 = c * (n31 * u.1 + n32 * u`2 + n33)
      by EUCLID_5:def 1;
    hence thesis by EUCLID_5:def 2;
  end;
