How does it imply that?

The definition you gave was symmetric. If I misread it, my apologies.

Why? If knowledge means "justified true belief", then for agent 1 to know that agent 2 know E, agent 1 must also know E, and vice-versa. This doesn't prove the claim that you say I am making, but goes most of the way towards proving it.

True, but it's impossible to go the rest of the way. If you see a dog and I see both you and the dog through a one-way mirror, then I know that you know that there's a dog there but you don't know that I know that there is a dog.

I am having trouble matching up your notation with the notation I'm used to.

There are two operations, which I am used to calling P and K. They also have a number attached to them.

P takes sets to bigger sets or else to themselves. P1({w}) is what A1 thinks is possible when w is true. P1(S) for any set S is what A1 might think is possible given that something in S is true.

K takes sets to smaller sets or else to themselves. K1(S) is the set of possible states of the world where A1 knows that S is true.

What I think Aumann is saying is that, if A1 knows E, and knows that A2 knows E, then for every state x in P1(E), for every event D such that x is in P2(D) , P2(D) is a subset of E. Saying this allows Aumann to go on and show that A1 and A2 can iteratively rule out possibilities until they converge on believing the same thing.

That seems to translate to the statement:

Whenever E, A1 knows that A2 knows that E.

which is stronger than just:

In the current state w, A1 knows that A2 knows that E.

Unless your P is my K in which case it translates to "E is the whole space" because all x are in K(the whole space).

This requires knowing more than what we mean when we say "A1 knows that A2 knows E".

Aumann's theorem is based on common knowledge, which is the very strong statement that A1 knows E, and A2 knows that, and A1 knows that, and so on.

However it is easy to see where this can come from. For instance, if I say "I think that the sky is blue" then it's essentially common knowledge that I said "I think that the sky is blue"

Is that the source of your confusion?

For instance, if the situation is that P1({w,v,u})={w,v}, P2({w,v,u})={w,u}, then P1 and P2 can use their common knowledge to conclude w, and thus agree.

You have P1 and P2 taking big things to small things which means that they are K.

But if you consider conditions where P1, P2, and E contain more than 3 different states between them, you can find situations that have multiple possible solutions, which the agents cannot choose between; and so cannot converge.

However they will be able to agree that it is one of those states. Moreover neither of them will have any greater information than that it's one of those states. Argument occurs when I believe "A, not B" and you believe "B, not A", not if we both believe "A or B".