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 Th32:
  p < 1 iff numerator(p) < denominator(p)
proof
A3: now
    assume numerator(p)<denominator(p);
    then numerator(p)*denominator(p)"<denominator(p)*denominator(p)"
    by XREAL_1:68;
    then numerator(p)*denominator(p)"<1 by XCMPLX_0:def 7;
    hence p<1 by Th12;
  end;
  now
    assume p<1;
    then numerator(p)/denominator(p)<1 by Th12;
    then numerator(p)/denominator(p)*denominator(p)<1*denominator(p)
    by XREAL_1:68;
    hence numerator(p)<denominator(p) by XCMPLX_1:87;
  end;
  hence thesis by A3;
end;
