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 Th28:
  p < -1 iff numerator(p) < -denominator(p)
proof
  hereby
    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;
  assume numerator(p)<-denominator(p);
  then
  numerator(p)*denominator(p)"<((-1)*denominator(p))*denominator(p)"
  by XREAL_1:68;
  then numerator(p)*denominator(p)"<(-1)*(denominator(p)*denominator(p)");
  then numerator(p)*denominator(p)"<(-1)*1 by XCMPLX_0:def 7;
  hence p<-1 by Th12;
end;
