reserve x for object,
  a,b for Real,
  k,k1,i1,j1,w for Nat,
  m,m1,n,n1 for Integer;
reserve p,q for Rational;

theorem
  p < q iff numerator(p)*denominator(q) < numerator(q)*denominator(p)
proof
  p<q iff numerator(p)/denominator(p)<q by Th12;
  then
  p<q iff numerator(p)/denominator(p)<numerator(q)/denominator(q) by Th12;
  hence thesis by XREAL_1:102,106;
end;
