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 <= 1 iff numerator(p) <= denominator(p)
proof
A1: now
    assume
A2: p<=1;
    per cases by A2,XXREAL_0:1;
    suppose
      p=1;
      hence numerator(p)<=denominator(p);
    end;
    suppose
      p<1;
      hence numerator(p)<=denominator(p) by Th32;
    end;
  end;
  now
    assume
A3: numerator(p)<=denominator(p);
    per cases by A3,XXREAL_0:1;
    suppose
      numerator(p)=denominator(p);
      hence p<=1 by Th20;
    end;
    suppose
      numerator(p)<denominator(p);
      hence p<=1 by Th32;
    end;
  end;
  hence thesis by A1;
end;
