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 <> R & P <> S & R <> Q & S <> Q implies
  cross-ratio(Q,P,R,S) = 1 / cross-ratio(P,Q,R,S)
  proof
    assume that
A1: P,Q,R,S are_collinear and
A2: P <> R and
A3: P <> S and
A4: R <> Q and
A5: S <> Q;
A6: cross-ratio(P,Q,S,R) = 1 / cross-ratio(P,Q,R,S) by XCMPLX_1:57;
    Q,P,R,S are_collinear & P <> R & P <> S & R <> Q & S <> Q
      by A2,A3,A4,A5,A1;
    hence thesis by A6,Th34;
  end;
