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 < 0 iff numerator(p) < 0
proof
  now
    assume p<0;
    then numerator(p)/denominator(p)<0 by Th12;
    hence numerator(p)<0;
  end;
  hence thesis;
end;
