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 Th39:
  cross-ratio-tuple(pi_1243(x)) = 1 / cross-ratio-tuple(x)
   proof
     consider P9,Q9,R9,S9 be Element of V such that
A1:  P9 = x.1 & Q9 = x.2 & R9 = x.3 & S9 = x.4 &
       cross-ratio-tuple(x) = cross-ratio(P9,Q9,R9,S9) by Def03;
     ex P99,Q99,R99,S99 be Element of V st
     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;
     hence thesis by A1,XCMPLX_1:57;
   end;
