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 Th29:
  p <= -1 iff numerator(p) <= -denominator(p)
proof
  hereby
    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 Th28;
    end;
  end;
  assume
A3: numerator(p)<=-denominator(p);
    per cases by A3,XXREAL_0:1;
    suppose
      numerator(p)=-denominator(p);
      hence p<=-1 by Th21;
    end;
    suppose
      numerator(p)<-denominator(p);
      hence p<=-1 by Th28;
    end;
end;
