question about function object

I have a code as following:

Code:

#include <iostream>
#include <algorithm>
#include <list>
using namespace std;

//the class Nth is a predicates
class Nth{
    private:
         int nth;
         int count;
    public:
         Nth(int n):nth(n),count(0){}
         bool operator()(int){
             return ++count == nth;
        }
};

int main()
{
       list<int> lst;
 
       for(int i=1;i<10;i++)
           lst.push_back(i);

       copy(lst.begin(),lst.end(),ostream_iterator<int>(cout," "));
       cout<<endl;

       list<int>::iterator pos = remove_if(lst.begin(),lst.end(),Nth(3));
       lst.erase(pos,lst.end());

       copy(lst.begin(),lst.end(),ostream_iterator<int>(cout," "));
       cout<<endl;
} 

the output is:
1 2 3 4 5 6 7 8 9
1 2 4 5 7 8 9
Not as I expect:
1 2 3 4 5 6 7 8 9
1 2 4 5 6 7 8 9
The book explained:
This happens because the usual implementation of the algorithm copies the predicate internally during the algorithm:

Code:

template<class ForwIter,class Predicate>
ForwIter std::remove_if(ForwIter beg,ForeIter end,Predicate [B]op[/B])
{
       ForwIter pos = find_if(beg,end,[B]op[/B]);
       if(pos == end)
            return beg;
       else{
            ForwIter next = beg;
            return remove_copy_if(++next,end,beg,[B]op[/B]);
       }
}

and it says it is possible to avoid this surprise behavior like this:

Code:

template<class ForwIter,class Predicate>
ForwIter std::remove_if(ForwIter beg,ForeIter end,Predicate [B]op[/B])
{
       while(beg != end && !op(*beg))
            beg++;
       if(pos == end)
            return beg;
       else{
            ForwIter next = beg;
            return remove_copy_if(++next,end,beg,[B]op[/B]);
       }
}

I don't know what the difference between the two implementations is.I think i was able to understand the first one,but why the second one works fine?I thought they were the same.Any help will be appreciated.