theorem
  P,Q,R,S are_collinear & P,Q,R,S are_mutually_distinct implies
  cross-ratio(S,Q,R,P) = 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(S,Q,R,P) = cross-ratio(P,R,Q,S)
      by A3,Th34bis;
    hence thesis by A1,A2,Th35;
  end;
