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 Th34:
  for V being RealLinearSpace for P,Q,R,S being Element of V st
  P,Q,R,S are_collinear & P <> R & P <> S & R <> Q & S <> Q holds
  cross-ratio(P,Q,R,S) = cross-ratio(Q,P,S,R)
  proof
    let V be RealLinearSpace;
    let P,Q,R,S be Element of V;
    assume that
A1: P,Q,R,S are_collinear and
A2: P <> R and
A3: P <> S and
A4: R <> Q and
A5: S <> Q;
    set r1 = affine-ratio(R,P,Q), r2 = affine-ratio(S,P,Q),
        s1 = affine-ratio(S,Q,P), s2 = affine-ratio(R,Q,P);
    per cases;
    suppose
A6:   P = Q;
      R,P,P are_collinear & S,P,P are_collinear by Th05;
      then r1 = 1 & r2 = 1 & s1 = 1 & s2 = 1 by A2,A3,A6,Th07;
      hence thesis;
    end;
    suppose
A7:   P <> Q;
      P,Q,R are_collinear by A1;
      then consider r9 be non unit non zero Real such that
A8:   r9 = affine-ratio(P,Q,R) &
      affine-ratio(P,R,Q) = op1(r9) &
      affine-ratio(Q,P,R) = op1(op2(op1(r9))) &
      affine-ratio(Q,R,P) = op2(op1(r9)) &
      affine-ratio(R,P,Q) = op1(op2(r9)) &
      affine-ratio(R,Q,P) = op2(r9) by A2,A4,A7,Th28;
      P,Q,S are_collinear & P <> S & Q <> S by A1,A3,A5;
      then ex s9 be non unit non zero Real st
        s9 = affine-ratio(P,Q,S) &
        affine-ratio(P,S,Q) = op1(s9) &
        affine-ratio(Q,P,S) = op1(op2(op1(s9))) &
        affine-ratio(Q,S,P) = op2(op1(s9)) &
        affine-ratio(S,P,Q) = op1(op2(s9)) &
        affine-ratio(S,Q,P) = op2(s9) by A7,Th28;
      hence thesis by A8;
    end;
  end;
