# Strict inclusion for facility location formula and aggregate facility location formula

I am trying to prove that $$P_{FL} \subset P_{AFL}$$ where \begin{align}P_{FL}&=\left\{({\bf x},{\bf y})\,\,\middle\vert\,\,\forall i,j:\sum_{j=1}^nx_{ij}=1,x_{ij}\le y_j,0\le x_{ij},y_j\le1\right\}\\P_{AFL}&=\left\{({\bf x},{\bf y})\,\,\middle\vert\,\,\forall i,j:\sum_{j=1}^nx_{ij}=1,\sum_{i=1}^mx_{ij}\le my_j,0\le x_{ij},y_j\le1\right\}.\end{align} I could get to a point that given $$x_{ij} \leq y_j$$ from $$P_{FL}$$ summing over $$m$$ to get $$\sum\limits_{i=1}^m x_{ij} \leq my_j$$. Hence, $$P_{FL}$$ is at least as strong as $$P_{AFL}$$. However, I couldn't provide a point where $$P_{FL} \setminus P_{AFL}$$ to prove the inclusion is strict. How can I prove it?

Without loss of generality take $$m=2$$. Then
$$x_i \leq y\implies\sum_{i=1}^m x_i \leq my$$
On the other hand $$\sum_{i=1}^m x_i \leq my\quad\not\!\!\!\!\implies x_i \leq y$$ is proven by carefully choosing a counterexample. I shall take $$x_1 = y+\frac{\epsilon}{2}> y , x_2 = y-\epsilon$$. The FL constraint is clearly violated for $$i=1$$, and yet the AFL constraint is satisfied:
$$\sum_i x_i = 2y - \frac{\epsilon}{2} < 2y.$$