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 Th38:
  x = <* P, Q, R, S *> & P,Q,R,S are_collinear & P <> R & P <> S &
  Q <> R & Q <> S implies
  cross-ratio-tuple(x) = cross-ratio-tuple(pi_2143(x)) &
  cross-ratio-tuple(x) = cross-ratio-tuple(pi_4321(x))
  proof
    assume that
A1: x = <* P, Q, R, S *> and
A2: P,Q,R,S are_collinear and
A3: P <> R and
A4: P <> S and
A5: Q <> R and
A6: Q <> S;
A8: ex P9,Q9,R9,S9 be Element of V st P9 = x.1 & Q9 = x.2 & R9 = x.3 &
    S9 = x.4 & cross-ratio-tuple(x) = cross-ratio(P9,Q9,R9,S9) by Def03;
    H1: ex P99,Q99,R99,S99 be Element of V st P99 = (pi_2143(x)).1 &
        Q99 = (pi_2143(x)).2 & R99 = (pi_2143(x)).3 & S99 = (pi_2143(x)).4 &
        cross-ratio-tuple(pi_2143(x)) = cross-ratio(P99,Q99,R99,S99) by Def03;
    ex P99,Q99,R99,S99 be Element of V st P99 = (pi_4321(x)).1 &
        Q99 = (pi_4321(x)).2 & R99 = (pi_4321(x)).3 & S99 = (pi_4321(x)).4 &
        cross-ratio-tuple(pi_4321(x)) = cross-ratio(P99,Q99,R99,S99) by Def03;
    hence thesis by H1,A6,A1,A8,A2,A3,A4,A5,Th34,Th34bis;
  end;
