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 Th10:
  P,Q,R are_collinear & P <> R & Q <> R & P <> Q implies
  affine-ratio(Q,P,R) =
    affine-ratio(P,Q,R) / (affine-ratio(P,Q,R) - 1)
  proof
    assume that
A1: P,Q,R are_collinear and
A2: P <> R and
A3: Q <> R and
A4: P <> Q;
    set r = affine-ratio(P,Q,R),
        s = affine-ratio(Q,P,R);
A5: Q - P = r * (R - P) by A1,A2,Def02;
    Q,P,R are_collinear by A1;
    then P - Q = s * (R - Q) by A3,Def02;
    then r * (R - P) = -s * (R + 0.V - Q) by A5,RLVECT_1:33
                    .= -s * (R + (-P + P) - Q) by RLVECT_1:5
                    .= -s * (R - P + P - Q) by RLVECT_1:def 3
                    .= -(s * ((R - P) + (P - Q))) by RLVECT_1:def 3
                    .= (-1) * (s * ((R - P) + (P - Q))) by RLVECT_1:16
                    .= ((-1) * s) * ((R - P) + (P - Q)) by RLVECT_1:def 7
                    .= (-s) * (R - P) + (-s) * (P - Q) by RLVECT_1:def 5;
    then
A6: r * (R - P) + s * (R - P)
       = (-s) * (R - P) + ((-s) * (P - Q) + s * (R - P)) by RLVECT_1:def 3
       .= (-s) * (R - P) + (s * (R - P) + (-s) * (P - Q)) by RLVECT_1:def 2
       .= ((-s) * (R - P) + s * (R - P)) + (-s) * (P - Q) by RLVECT_1:def 3
       .= (-s + s) * (R - P) + (-s) * (P - Q) by RLVECT_1:def 6
       .= 0.V + (-s) * (P - Q) by RLVECT_1:10
       .= (-s) * (P - Q);
    Q,P,R are_collinear by A1; then
A7: s <> 0 by A3,A4,Th06;
    then reconsider s9 = 1 / s as non zero Real;
A8: s9 * s = 1 by A7,XCMPLX_1:106;
A9: r - 1 <> 0 by A1,A2,A3,Th07;
    (r + s) * (R - P) = (-s) * (P - Q) by A6,RLVECT_1:def 6
                     .= s * (-(P - Q)) by RLVECT_1:24
                     .= s * (Q - P) by RLVECT_1:33;
    then (s9 * (r + s)) * (R - P) = s9 * (s * (Q - P)) by RLVECT_1:def 7
                                 .= (s9 * s) * (Q - P) by RLVECT_1:def 7
                                 .= 1 * (Q - P) by A7,XCMPLX_1:106
                                 .= Q - P by RLVECT_1:def 8;
    then s * r  = s * ((1/s) * (r + s)) by A1,Def02,A2;
    then r * s  = (s * (1/s)) * (r + s);
    then r * s = r + s by A8;
    then s * (r - 1) / (r - 1) = r / (r - 1);
    then s * ((r - 1) / (r - 1)) = r / (r - 1);
    then s * 1 = r / (r - 1) by A9,XCMPLX_1:60;
    hence thesis;
  end;
