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
  P,Q,R,S are_collinear & P,Q,R,S are_mutually_distinct implies
  cross-ratio(R,P,S,Q) = 1 - cross-ratio(P,Q,R,S)
  proof
    assume that
A1: P,Q,R,S are_collinear and
A2: P,Q,R,S are_mutually_distinct;
A3: P <> R & P <> S & R <> Q & S <> Q & S <> R & P <> Q  by A2,ZFMISC_1:def 6;
    P,R,Q,S are_collinear by A1;
    then cross-ratio(P,R,Q,S) = cross-ratio(R,P,S,Q) by A3,Th34;
    hence thesis by A1,A2,Th35;
  end;
