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 Th09:
  P,Q,R are_collinear & P <> R & P <> Q implies
  affine-ratio(P,R,Q) = 1 / affine-ratio(P,Q,R)
  proof
    assume that
A1: P,Q,R are_collinear and
A2: P <> R and
A3: P <> Q;
    set r = affine-ratio(P,Q,R),
        s = affine-ratio(P,R,Q);
    P,R,Q are_collinear by A1;
    then R - P = s * (Q - P) by A3,Def02
              .= s * (r * (R - P)) by A1,A2,Def02
              .= (s * r) * (R - P) by RLVECT_1:def 7;
    then 1 * (R - P) = (s * r) * (R - P) by RLVECT_1:def 8;
    hence thesis by A2,Th08,XCMPLX_1:73;
  end;
