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 & P9,Q9,R9,S9 are_collinear & S <> P & S <> Q &
  S9 <> P9 & S9 <> Q9 implies
  (cross-ratio(P,Q,R,S) = cross-ratio(P9,Q9,R9,S9) iff
  affine-ratio(R,P,Q) * affine-ratio(S9,P9,Q9) =
  affine-ratio(R9,P9,Q9) * affine-ratio(S,P,Q))
  proof
    assume that
A1: P,Q,R,S are_collinear and
A2: P9,Q9,R9,S9 are_collinear and
A3: S <> P and
A4: S <> Q and
A5: S9 <> P9 and
A6: S9 <> Q9;
    set r  = affine-ratio(R,P,Q),     s = affine-ratio(S,P,Q),
        r9 = affine-ratio(R9,P9,Q9), s9 = affine-ratio(S9,P9,Q9);
    S,P,Q are_collinear & S9,P9,Q9 are_collinear by A1,A2;
    then s <> 0 & s9 <> 0 by A3,A4,A5,A6,Th06;
    hence thesis by XCMPLX_1:94,95;
  end;
