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 Th40:
  x = <*P,Q,R,S*> & P,Q,R,S are_collinear & P <> R & P <> S &
  Q <> R & Q <> S implies
  cross-ratio-tuple(pi_1243(x)) = 1 / cross-ratio-tuple(x) &
  cross-ratio-tuple(pi_2134(x)) = 1 / cross-ratio-tuple(x) &
  cross-ratio-tuple(pi_3421(x)) = 1 / cross-ratio-tuple(x) &
  cross-ratio-tuple(pi_4312(x)) = 1 / cross-ratio-tuple(x)
  proof
    assume that
A1: x = <*P,Q,R,S*> and
A2: P,Q,R,S are_collinear and
A3: P <> R and
A4: P <> S and
A5: Q <> R and
A6: Q <> S;
A7: pi_1243(x) = <*P,Q,S,R*> & P,Q,S,R are_collinear by A2,A1;
    consider P9,Q9,R9,S9 be Element of V such that
A8: P9 = x.1 & Q9 = x.2 & R9 = x.3 & S9 = x.4 &
      cross-ratio-tuple(x) = cross-ratio(P9,Q9,R9,S9) by Def03;
    consider P99,Q99,R99,S99 be Element of V such that
A9: P99 = (pi_1243(x)).1 & Q99 = (pi_1243(x)).2 &
     R99 = (pi_1243(x)).3 & S99 = (pi_1243(x)).4 &
     cross-ratio-tuple(pi_1243(x)) = cross-ratio(P99,Q99,R99,S99) by Def03;
    now
      thus cross-ratio-tuple(pi_2134(x))
        = cross-ratio-tuple(pi_2143(pi_1243(x)))
        .= cross-ratio-tuple(pi_1243(x)) by A7,A3,A4,A5,A6,Th38;
      thus cross-ratio-tuple(pi_3421(x))
        = cross-ratio-tuple(pi_4321(pi_1243(x)))
        .= cross-ratio-tuple(pi_1243(x)) by A7,A3,A4,A5,A6,Th38;
      thus cross-ratio-tuple(pi_4312(x))
        = cross-ratio-tuple(pi_3412(pi_1243(x)))
        .= cross-ratio-tuple(pi_1243(x)) by A7,A3,A5,A6,Th37;
    end;
    hence thesis by A8,A9,XCMPLX_1:57;
  end;
