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 Th35:
  p <= 0 iff numerator(p) <= 0
proof
  now
    assume
A1: numerator(p)<=0;
    per cases by A1;
    suppose
      numerator(p)=0;
      hence p<=0 by Th11;
    end;
    suppose
      numerator(p)<0;
      hence p<=0;
    end;
  end;
  hence thesis;
end;
