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
  for x1,x2,x3,x4 being Element of TOP-REAL 1 st
  x2 <> x3 & x3 <> x1 & x2 <> x4 & x1 <> x4 holds
  ex a,b,c,d being Real st x1 = <* a *> & x2 =  <* b *> & x3 = <* c *> &
  x4 = <* d *> & cross-ratio-tuple(<*x1,x2,x3,x4*>)
    = ((c - a) / (c - b)) * ((d - b) / (d - a))
  proof
    let x1,x2,x3,x4 be Element of TOP-REAL 1;
    assume that
A1: x2 <> x3 and
A2: x3 <> x1 and
A3: x2 <> x4 and
A4: x1 <> x4;
    reconsider x = <*x1,x2,x3,x4*> as Tuple of 4,the carrier of TOP-REAL 1;
    consider P,Q,R,S be Element of TOP-REAL 1 such that
A5: P = x.1 & Q = x.2 & R = x.3 & S = x.4 &
      cross-ratio-tuple(x) = cross-ratio(P,Q,R,S) by Def03;
    consider a1,b1,c1 be Real such that
A7: x3 = <* a1 *> & x1 = <* b1 *> & x2 = <* c1 *> &
    affine-ratio(x3,x1,x2) = (b1 - a1) / (c1 - a1) by A1,A2,Th26;
    consider a2,b2,c2 be Real such that
A8: x4 = <* a2 *> & x1 = <* b2 *> & x2 = <* c2 *> &
    affine-ratio(x4,x1,x2) = (b2 - a2) / (c2 - a2) by A3,A4,Th26;
    take b1,c1,a1,a2;
    b1 = b2 & c1 = c2 by A7,A8,FINSEQ_1:76;
    hence thesis by A7,A8,Lm03,A5;
  end;
