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 Th41:
  x = <* P,Q,R,S *> & P,Q,R,S are_mutually_distinct &
  P,Q,R,S are_collinear implies
  cross-ratio-tuple(pi_1324(x)) = 1 - cross-ratio-tuple(x) &
  cross-ratio-tuple(pi_2413(x)) = 1 - cross-ratio-tuple(x) &
  cross-ratio-tuple(pi_3142(x)) = 1 - cross-ratio-tuple(x) &
  cross-ratio-tuple(pi_4231(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;
A6: pi_1324(x) = <*P,R,Q,S*> & P,R,Q,S are_collinear by A3,A1;
    now
      x.1 = P & x.2 = Q & x.3 = R & x.4 = S & (pi_1324(x)).1 = P &
       (pi_1324(x)).2 = R & (pi_1324(x)).3 = Q & (pi_1324(x)).4 = S by A1;
      hence cross-ratio-tuple(pi_1324(x)) = cross-ratio(P,R,Q,S) &
        cross-ratio-tuple(x) = cross-ratio(P,Q,R,S) by Def03;
      thus cross-ratio-tuple(pi_2413(x))
        = cross-ratio-tuple(pi_3412(pi_1324(x)))
       .= cross-ratio-tuple(pi_1324(x)) by A6,A4,Th37;
      thus cross-ratio-tuple(pi_3142(x))
        = cross-ratio-tuple(pi_2143(pi_1324(x)))
       .= cross-ratio-tuple(pi_1324(x)) by A4,Th38,A6;
      thus cross-ratio-tuple(pi_4231(x))
        = cross-ratio-tuple(pi_4321(pi_1324(x)))
       .= cross-ratio-tuple(pi_1324(x)) by A4,Th38,A6;
    end;
    hence thesis by A2,A3,Th35;
  end;
