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
  (ex r being non unit non zero Real st r = cross-ratio-tuple(x) &
  cross-ratio-tuple(pi_1243(x)) = op1(r))
  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;
    consider P9,Q9,R9,S9 be Element of V such that
A5: P9 = x.1 & Q9 = x.2 & R9 = x.3 & S9 = x.4 &
      cross-ratio-tuple(x) = cross-ratio(P9,Q9,R9,S9) by Def03;
    reconsider r=cross-ratio-tuple(x) as non unit non zero Real
      by Def01,A1,A3,A5,A4,Th32,Th31;
    take r;
    thus thesis by Th39;
  end;
