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 Th13:
  P,Q,R are_collinear & P <> R & Q <> R implies
  affine-ratio(R,P,Q) = 1 / (1 - affine-ratio(P,Q,R))
  proof
    assume that
A1: P,Q,R are_collinear and
A2: P <> R and
A3: Q <> R;
    set r = affine-ratio(P,Q,R),
        s = affine-ratio(R,P,Q);
A4: 1 - r <> 0 by A1,A2,A3,Th07;
A5: r * (R - P) = Q + 0.V - P by A1,A2,Def02
                .= Q + (-R + R) - P by RLVECT_1:5
                .= Q - R + R - P by RLVECT_1:def 3
                .= Q - R + --(R - P) by RLVECT_1:def 3
                .= Q - R - (P - R) by RLVECT_1:33;
A6: R,P,Q are_collinear by A1;
    then - (s * (Q - R)) = - (P - R) by A3,Def02
                        .= R - P by RLVECT_1:33;
    then
A7: R - P = (-1) * (s * (Q - R)) by RLVECT_1:16
          .= ((-1) * s) * (Q - R) by RLVECT_1:def 7
          .= (-s) * (Q - R);
    r * (R - P) = (Q - R) + -(s * (Q - R)) by A5,A6,A3,Def02
               .= (Q - R) + (-1) * (s * (Q - R)) by RLVECT_1:16
               .= (Q - R) + ((-1) * s) * (Q - R) by RLVECT_1:def 7
               .= 1 * (Q - R) + (-s) * (Q - R) by RLVECT_1:def 8
               .= (1 - s) * (Q - R) by RLVECT_1:def 6;
    then (r * (-s)) * (Q - R) = (1 - s) * (Q - R) by A7,RLVECT_1:def 7;
    then - r * s = (1 - s) by Th08,A3;
    then s * (1 - r) = 1;
    hence thesis by A4,XCMPLX_1:89;
  end;
