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
  a <= p iff a*denominator(p) <= numerator(p)
proof
A2: now
    assume
A3: a*denominator(p)<=numerator(p);
    per cases by A3,XXREAL_0:1;
    suppose
      a*denominator(p)=numerator(p);
      then p=(a*denominator(p))/(1*denominator(p)) by Th12
        .=(a/1)*(denominator(p)/denominator(p))
        .=a by XCMPLX_1:60;
      hence a<=p;
    end;
    suppose
      a*denominator(p)<numerator(p);
      hence a<=p by Th36;
    end;
  end;
  now
    assume
A4: a<=p;
    per cases by A4,XXREAL_0:1;
    suppose
      a=p;
      hence numerator(p)>=a*denominator(p);
    end;
    suppose
      a<p;
      hence a*denominator(p)<=numerator(p) by Th36;
    end;
  end;
  hence thesis by A2;
end;
