 So, now what we need to show is that suppose C of omega is a subset of A, if C of omega is a subset of A, then A is common knowledge. That is what we need to show. So, we now show that if C of omega is a subset of A, then A is common knowledge is omega. Now, see what we are what we want to show is that if C of omega is a subset of A, then A is common knowledge in omega. That is the claim we want to show. But remember C of A is C of omega is a more subspecific set as compared to A. A is more general, C of omega is a subset. So, and one of the properties of common knowledge that we have saw that if you if something specific is common knowledge, then something more general than that is also common knowledge in the same state of the world. So, if it is common knowledge that if it is common knowledge that it is raining right now, it would also be common knowledge that the road is wet or that the humidity is high or anything something anything that is implied by the fact that it is raining right now. All of those things are also common knowledge. So, therefore, it is enough it is enough to show that C of omega itself is common knowledge. If once C of omega itself becomes common knowledge, then that then we will be done because once C of omega itself is common knowledge in omega, then any super set of C of omega will also be common knowledge super set of C of omega will also be common knowledge in omega. So, this thing that I have written here what we will actually show is in place of this what we will show is we will actually show that C of omega is common knowledge in omega. And the way to show C of omega is common knowledge in omega is something related to what you are asking. So, what I want you to now reflect on is what do you think is going to be this set take any player i let i be any player. What is this? What are the states of the world in which player i knows C of omega the connected component corresponding to some state of the world omega. So, eventually we are going to show this is common knowledge. You can even guess what this would probably turn out to be this would turn out to be C of omega itself because then it will become independent of player i and then it will also end up being common also. But we can be more intelligent than that. So, let us actually show this. So, let us look at C of so fix a player i. Now, if you look at this set f i of omega take any omega dash take omega dash belonging to f i of omega. So, omega is this fixed state of the world that we are considering and omega dash is another state of the world that player i cannot distinguish from omega which means what omega dash has to therefore be in C of omega itself has to be right. So, which means that you take C of omega C of omega is always going to be contained in in this. Let me write it like this f i of omega dash where omega dash ranges over C of omega. So, you take all states of the world in C of omega. So, all states of the world that are in this particular connected component take f i for those states of the world they are all going to be in C of omega itself because player i cannot distinguish between anything any element any of the elements in f i of omega dash. So, there is always going to be an edge between all those impact that subset of vertices or impact forms a clique in the graph. So, this is necessarily going to be this is a subset. So, what is so in fact I do not need to write this. So, let us let us write it like this I will write this more neatly. So, take any omega dash in C of omega and omega double dash in f i of omega dash take any omega dash in C of omega and omega double dash in f i of omega dash. Now, player i cannot distinguish between omega dash and omega double dash. So, therefore, omega double dash is also in C of omega which means in particular C of omega is contained in this union sorry C of omega contains this union all of these are contained in C of omega. But there is something that is trivially this is this is almost total logical here because because what this in particular can see f i of omega dash contains omega dash itself. So, this is union of omega dash singleton's omega dash where omega dash belongs to C of omega and what is this left hand side equal to this is this is by definition equal to C of omega. So, what this means is if you take the f i is corresponding to various points in C of omega for any player i take the f i is corresponding to various points in C of omega take the union that becomes C of omega. So, which means C of omega is always the union of elements of partitions of a player and in this is in fact true for every player take any player C of omega is structured in such a way that it forms it is formed by taking the union of his the elements of his partition some elements of his partition. So, and the way C of omega arranges itself is that somehow these the unions of the elements of his partition the becomes C of omega and this is true for every player. So, you C of omega can be chopped in various ways and they somehow become the elements of partitions for each player. Now, the interesting thing here is that the what does this imply this implies that for one is that you know this needs a whole lot of coincidence on that on with the partitions in the absence of the partitions nicely aligning with each other right you will not it may the only set that will turn out to be common knowledge is what is why itself why will turn out to be because why is always the union of the elements of partitions. So, if you want something smaller something more specific to end up being common knowledge then it has to be that you know the partitions of the players have to agree in a you know have to you know align in a certain in a nice way. So, this means that C of omega is equal to union of Fi of omega dash over all omega dash in C of omega and this is true for if this is true for all players i in n. Now, if this is true for all players i in n then we are almost done. So, firstly C of omega is the union of elements of a partition which means what is going to be Ki of C of omega it is going to be C of omega itself. So, this is going to be C of omega and this is true for all i for every player. So, which means therefore that naturally therefore, if you take Ki 1 Ki 2 dot dot dot Ki r of C of omega that is always going to be C of omega means omega belongs to Ki 1 of Ki 2 of dot dot dot C of omega which means C of omega is common knowledge in this is the main thing we wanted to show. Now, when C of now that C of omega is common knowledge in omega it since a is a superset of C of omega a is also common knowledge in omega. So, this is how the sets that are common knowledge get structured and this also has implications on a very beautiful result regarding agreement between players. How can about whether when can two players agree on a certain fact because after all agreement is based on knowledge and whether players can agree or not depends on how they are how they are what kind of information they are getting. And so, the kind of the way the partitions get created and the way the partitions align with each other depend decide whether players can actually agree or agree or not. So, and now agree means you can again I do not mean agree in a kind of emotional or social way, but agree in a technical way. So, for this I need to define for you what agreement and so on means but essentially that is it is essentially something like what is something like what is commonly known as consensus and so on ok ok.