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 Th36:
  x = <* P, Q, R, S *> implies
  cross-ratio(P,Q,R,S) = cross-ratio-tuple(x)
  proof
    assume x = <* P, Q, R, S *>;
    then x.1 = P & x.2 = Q & x.3 = R & x.4 = S;
    hence thesis by Def03;
  end;
