Extending Set of Points

For a given set of two-dimensional points $$$S$$$, let's denote its extension $$$E(S)$$$ as the result of the following algorithm:

Create another set of two-dimensional points $$$R$$$, which is initially equal to $$$S$$$. Then, while there exist four numbers $$$x_1$$$, $$$y_1$$$, $$$x_2$$$ and $$$y_2$$$ such that $$$(x_1, y_1) \in R$$$, $$$(x_1, y_2) \in R$$$, $$$(x_2, y_1) \in R$$$ and $$$(x_2, y_2) \notin R$$$, add $$$(x_2, y_2)$$$ to $$$R$$$. When it is impossible to find such four integers, let $$$R$$$ be the result of the algorithm.

Now for the problem itself. You are given a set of two-dimensional points $$$S$$$, which is initially empty. You have to process two types of queries: add some point to $$$S$$$, or remove some point from it. After each query you have to compute the size of $$$E(S)$$$.