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
  x = <* P,Q,R,S *> & P,Q,R,S are_mutually_distinct &
  P,Q,R,S are_collinear implies
  cross-ratio-tuple(pi_1432(x))
    = cross-ratio-tuple(x) / (cross-ratio-tuple(x) - 1) &
  cross-ratio-tuple(pi_2341(x))
    = cross-ratio-tuple(x) / (cross-ratio-tuple(x) - 1) &
  cross-ratio-tuple(pi_3214(x))
    = cross-ratio-tuple(x) / (cross-ratio-tuple(x) - 1) &
  cross-ratio-tuple(pi_4123(x))
    = cross-ratio-tuple(x) / (cross-ratio-tuple(x) - 1)
  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;
A5: P,S,R,Q are_collinear & pi_1432(x) = <*P,S,R,Q*> by A1,A3;
A6: cross-ratio-tuple(pi_1432(x))
      = cross-ratio-tuple(pi_1243(pi_1423(x)))
     .= 1 / (cross-ratio-tuple(pi_1423(x))) by Th39
     .= 1 / ((cross-ratio-tuple(x) - 1) / cross-ratio-tuple(x))
       by A1,A3,A2,Th42;
    hence cross-ratio-tuple(pi_1432(x))
      = cross-ratio-tuple(x) / (cross-ratio-tuple(x) - 1) by XCMPLX_1:57;
    now
      thus cross-ratio-tuple(pi_2341(x))
        = cross-ratio-tuple(pi_4321(pi_1432(x)))
       .= cross-ratio-tuple(pi_1432(x)) by A5,A4,Th38;
      thus cross-ratio-tuple(pi_3214(x))
        = cross-ratio-tuple(pi_3412(pi_1432(x)))
       .= cross-ratio-tuple(pi_1432(x)) by A4,A5,Th37;
      thus cross-ratio-tuple(pi_4123(x))
        = cross-ratio-tuple(pi_2143(pi_1432(x)))
       .= cross-ratio-tuple(pi_1432(x)) by A4,A5,Th38;
    end;
    hence thesis by A6,XCMPLX_1:57;
  end;
