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 Th28:
  for V being RealLinearSpace for P,Q,R being Element of V st
  P,Q,R are_collinear & P <> R & Q <> R & P <> Q holds
  ex r being non unit non zero Real st
  r = affine-ratio(P,Q,R) &
  affine-ratio(P,R,Q) = op1(r) &
  affine-ratio(Q,P,R) = op1(op2(op1(r))) &
  affine-ratio(Q,R,P) = op2(op1(r)) &
  affine-ratio(R,P,Q) = op1(op2(r)) &
  affine-ratio(R,Q,P) = op2(r)
  proof
    let V be RealLinearSpace;
    let P,Q,R be Element of V;
    assume that
A1: P,Q,R are_collinear and
A2: P <> R and
A3: Q <> R and
A4: P <> Q;
A5: affine-ratio(P,Q,R) <> 0 & affine-ratio(P,Q,R) <> 1
      by A1,A2,A3,A4,Th06,Th07;
    reconsider r = affine-ratio(P,Q,R) as Element of REAL
      by XREAL_0:def 1;
    reconsider r9 = r as non unit non zero Real by A5,Def01;
    take r9;
    affine-ratio(P,R,Q) = 1 / r &
      affine-ratio(Q,P,R) = r / (r - 1) &
      affine-ratio(Q,R,P) = (r - 1) / r &
      affine-ratio(R,P,Q) = 1 / (1 - r) &
      affine-ratio(R,Q,P) = 1 - r
      by A1,A2,A3,A4,Th09,Th10,Th11,Th12,Th13;
    hence thesis by Th01;
  end;
