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 Th12:
  P,Q,R are_collinear & P <> R & P <> Q implies
  affine-ratio(Q,R,P) =
  (affine-ratio(P,Q,R) - 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(Q,R,P);
A4: r <> 0 by A1,A2,A3,Th06;
A5: Q - P = r * (R + 0.V - P) by A1,A2,Def02
         .= r * (R + (-Q + Q) - P) by RLVECT_1:5
         .= r * ((R + -Q) + Q - P) by RLVECT_1:def 3
         .= r * (R - Q + (Q - P)) by RLVECT_1:def 3
         .= r * (R - Q) + r * (Q - P) by RLVECT_1:def 5;
    Q,R,P are_collinear by A1;
    then Q - P = r * (s * (P - Q)) + r * (Q - P) by A5,A3,Def02
              .= (r * s) * (P - Q) + r * (Q - P) by RLVECT_1:def 7
              .= (r * s) * (-(Q - P)) + r * (Q - P) by RLVECT_1:33
              .= (r * s) * ((-1) * (Q - P)) + r * (Q - P) by RLVECT_1:16
              .= (r * s * (-1)) * (Q - P) + r * (Q - P) by RLVECT_1:def 7
              .= (-r * s + r) * (Q - P) by RLVECT_1:def 6;
    then 1 * (Q - P) = (r - r * s) * (Q - P) by RLVECT_1:def 8;
    then 1 = r - r * s by A3,Th08;
    hence thesis by A4,XCMPLX_1:89;
  end;
