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 <> R & P <> S implies cross-ratio(P,P,R,S) = 1
  proof
    assume that
A1: P <> R and
A2: P <> S;
    R,P,P are_collinear & S,P,P are_collinear by Th05;
    then affine-ratio(R,P,P) = 1 & affine-ratio(S,P,P) = 1
      by A1,A2,Th07;
    hence thesis;
  end;
