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;
reserve x           for Tuple of 4,the carrier of V,
        P9,Q9,R9,S9 for Element of V;

theorem Th32:
  P,Q,R,S are_collinear & R <> Q & S <> Q & R <> S &
  cross-ratio(P,Q,R,S) = 1 implies P = Q
  proof
    assume that
A1: P,Q,R,S are_collinear and
A2: R <> Q and
A3: S <> Q and
A4: R <> S and
A5: cross-ratio(P,Q,R,S) = 1;
A6: affine-ratio(R,P,Q) = affine-ratio(S,P,Q) by A5,XCMPLX_1:58;
    set r = affine-ratio(R,P,Q);
A7: R,P,Q are_collinear & S,P,Q are_collinear by A1;
    then P + 0.V - R = r * (Q + 0.V - R) by A2,Def02;
    then P + (-S + S) - R = r * (Q + 0.V - R) by RLVECT_1:5
                         .= r * (Q + (-S + S) - R) by RLVECT_1:5
                         .= r * (Q - S + S - R) by RLVECT_1:def 3;
    then P - S + S - R = r * (Q - S + S - R) by RLVECT_1:def 3
                      .= r * ((Q - S) + (S - R)) by RLVECT_1:def 3
                      .= r * (Q - S) + r * (S - R) by RLVECT_1:def 5
                      .= (P - S) + r * (S - R) by A7,A6,A3,Def02;
    then -(P - S) + ((P - S) + (S - R)) = -(P - S) + ((P - S) + r * (S - R))
      by RLVECT_1:def 3;
    then (-(P - S) + (P - S)) + (S - R) = -(P - S) + ((P - S) + r * (S - R))
                                           by RLVECT_1:def 3
                                       .= (-(P - S) + (P - S)) + r * (S - R)
                                           by RLVECT_1:def 3;
    then r * (S - R) = S - R by RLVECT_1:8
                    .= 1 * (S - R) by RLVECT_1:def 8;
    then r = 1 by A4,Th08;
    hence P = Q by A2,Th07,A7;
  end;
