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 Th36:
  a < p iff a*denominator(p) < numerator(p)
proof
A3: now
    assume a*denominator(p)<numerator(p);
    then
    (a*denominator(p))*denominator(p)"<numerator(p)*denominator(p)"
    by XREAL_1:68;
    then a*(denominator(p)*denominator(p)")<numerator(p)*denominator(p)";
    then a*1<numerator(p)*denominator(p)" by XCMPLX_0:def 7;
    hence a<p by Th12;
  end;
  thus thesis by A3,XREAL_1:68;
end;
