reserve a,b,r for non unit non zero Real;
reserve X for non empty set,
        x for Tuple of 4,X;
reserve V             for RealLinearSpace,
        A,B,C,P,Q,R,S for Element of V;
reserve x           for Tuple of 4,the carrier of V,
        P9,Q9,R9,S9 for Element of V;

theorem Th42:
  x = <* P,Q,R,S *> & P,Q,R,S are_mutually_distinct &
  P,Q,R,S are_collinear implies
  cross-ratio-tuple(pi_1423(x))
    = (cross-ratio-tuple(x) - 1) / cross-ratio-tuple(x) &
  cross-ratio-tuple(pi_2314(x))
    = (cross-ratio-tuple(x) - 1) / cross-ratio-tuple(x) &
  cross-ratio-tuple(pi_4132(x))
    = (cross-ratio-tuple(x) - 1) / cross-ratio-tuple(x) &
  cross-ratio-tuple(pi_3241(x))
    = (cross-ratio-tuple(x) - 1) / cross-ratio-tuple(x)
  proof
    assume that
A1: x = <* P,Q,R,S *> and
A2: P,Q,R,S are_mutually_distinct and
A3: P,Q,R,S are_collinear;
A4: P <> R & P <> S & R <> Q & S <> Q & S <> R & P <> Q
     by A2,ZFMISC_1:def 6;
    cross-ratio(P,Q,R,S) <> 0 by A3,A4,Th31;
    then reconsider cr = cross-ratio-tuple(x) as non zero Complex by A1,Th36;
    pi_1243(x) = <* P,Q,S,R *> & P,Q,S,R are_collinear &
      P,Q,S,R are_mutually_distinct by A4,A1,A3,ZFMISC_1:def 6;
    then A5: cross-ratio-tuple(pi_1324(pi_1243(x)))
      = 1 - cross-ratio-tuple(pi_1243(x)) by Th41
     .= 1 - (1 / cr) by A1,A3,A4,Th40
     .= cr / cr - 1 / cr by XCMPLX_1:60
     .= (cr - 1) / cr;
    hence cross-ratio-tuple(pi_1423(x))
      = (cross-ratio-tuple(x) - 1) / cross-ratio-tuple(x);
    pi_1423(x) = <* P,S,Q,R *> & P,S,Q,R are_collinear by A1,A3;
    then cross-ratio-tuple(pi_3412(pi_1423(x)))=cross-ratio-tuple(pi_1423(x)) &
      cross-ratio-tuple(pi_2143(pi_1423(x)))=cross-ratio-tuple(pi_1423(x))  &
      cross-ratio-tuple(pi_4321(pi_1423(x)))=cross-ratio-tuple(pi_1423(x))
      by A4,Th37,Th38;
    hence thesis by A5;
  end;
