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 Th33:
  P,Q,R,S are_collinear & P <> S & R <> Q & S <> Q implies
  cross-ratio(P,Q,R,S) = cross-ratio(R,S,P,Q)
  proof
    assume that
A1: P,Q,R,S are_collinear and
A2: P <> S and
A3: R <> Q and
A4: S <> Q;
A5: R,P,Q are_collinear & P,R,S are_collinear & S,P,Q are_collinear &
      Q,R,S are_collinear by A1;
    set r1 = affine-ratio(R,P,Q), r2 = affine-ratio(S,P,Q),
        s1 = affine-ratio(P,R,S), s2 = affine-ratio(Q,R,S);
A6: r2 <> 0 & s2 <> 0 by A2,A3,A4,A5,Th06;
A7: Q - S <> 0.V by A4,RLVECT_1:21;
A8: P - R = r1 * (Q - R) by A5,A3,Def02;
A9: P - S = r2 * (Q - S) by A5,A4,Def02;
    R - Q = s2 * (S - Q) by A5,A4,Def02; then
A10: Q - R = s2 * (Q - S) by Lm02;
    R - P = s1 * (S - P) by A5,A2,Def02;
    then P - R = s1 * (P - S) by Lm02
              .= s1 * r2 * (Q - S) by A9,RLVECT_1:def 7;
    then r1 * s2 * (Q - S) = s1 * r2 * (Q - S) by A8,A10,RLVECT_1:def 7;
    hence thesis by A7,RLVECT_1:37,A6,XCMPLX_1:94;
  end;
