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 Th31:
  P,Q,R,S are_collinear & R <> Q & S <> Q & S <> P implies
  (R = P iff cross-ratio(P,Q,R,S) = 0)
  proof
    assume that
A1: P,Q,R,S are_collinear and
A2: R <> Q and
A3: S <> Q and
A4: S <> P;
    consider L be line of V such that
A5: P in L & Q in L & R in L & S in L by A1;
A6: R,P,Q are_collinear & S,P,Q are_collinear by A5;
    hereby
      assume R = P;
      then affine-ratio(R,P,Q) = 0 by A6,A2,Th06;
      hence cross-ratio(P,Q,R,S) = 0;
    end;
    assume
A7: cross-ratio(P,Q,R,S) = 0;
    per cases;
    suppose affine-ratio(S,P,Q) = 0;
      hence R = P by A3,A4,A6,Th06;
    end;
    suppose affine-ratio(S,P,Q) <> 0;
      then affine-ratio(R,P,Q) = 0 by A7;
      hence R = P by A2,A6,Th06;
    end;
  end;
