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;

theorem
  for r being Real st P,Q,R are_collinear &
  P <> R & Q <> R & P <> Q &
  r = affine-ratio(P,Q,R) holds
  affine-ratio(P,R,Q) = 1 / r &
  affine-ratio(Q,P,R) = r / (r - 1) &
  affine-ratio(Q,R,P) = (r - 1) / r &
  affine-ratio(R,P,Q) = 1 / (1 - r) &
  affine-ratio(R,Q,P) = 1 - r by Th09,Th10,Th11,Th12,Th13;
