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 Th37:
  x = <* P, Q, R, S *> & P,Q,R,S are_collinear &
  P <> S & Q <> R & Q <> S implies
  cross-ratio-tuple(x) = cross-ratio-tuple(pi_3412(x))
  proof
    assume that
A1: x = <* P, Q, R, S *> and
A2: P,Q,R,S are_collinear and
A3: P <> S and
A4: Q <> R and
A5: Q <> S;
    consider P9,Q9,R9,S9 be Element of V such that
A7: P9 = x.1 & Q9 = x.2 & R9 = x.3 & S9 = x.4 &
      cross-ratio-tuple(x) = cross-ratio(P9,Q9,R9,S9) by Def03;
    ex P99,Q99,R99,S99 be Element of V st
    P99 = (pi_3412(x)).1 & Q99 = (pi_3412(x)).2 &
      R99 = (pi_3412(x)).3 & S99 = (pi_3412(x)).4 &
      cross-ratio-tuple(pi_3412(x)) = cross-ratio(P99,Q99,R99,S99) by Def03;
    hence thesis by A1,A7,A2,A3,A4,A5,Th33;
  end;
