I am trying to learn the theory of the Surreal numbers and I am therefore going over all the theorems and trying to prove them for myself. 


I am struggling to complete the proof of $x1 = x$.


I have the following. Assume x is a surreal number. 
Then 
$x1 = \{X\_L1 + x0 - X\_L0, \ X\_R1 + xØ - X\_RØ | X\_L1 + xØ - X\_LØ, \ X\_R1 + x0 - X\_R0 \} = 
\{X\_L, X\_R | X\_R, X\_L\}$


Why is this the same as x?. I would argue that this is the same as $\{X\_R | X\_L\}$, but this is not a surreal number (per definition). 
Could someone explain to me how you go from $\{X\_L, X\_R | X\_R, X\_L\} \equiv \{X\_L | X\_R\}=x$ 



Surreal Numbers, Proving $x1=x$
If you look at the [Wikipedia entry for surreal multiplication](https://en.wikipedia.org/wiki/Surreal_number#Multiplication), you find



> 
> The recursive formula for multiplication contains arithmetic expressions involving the operands and their left and right sets, such as the expression $$X\_{R}y+xY\_{R}-X\_{R}Y\_{R}$$ that appears in the left set of the product of x and y. This is to be understood as the set of surreal numbers resulting from choosing one number from each set that appears in the expression and evaluating the expression on these numbers. (In each individual evaluation of the expression, only one number is chosen from each set, and is substituted in each place where that set appears in the expression.)
> 
> 
> 


In your case, since $1=\{0\mid\ \}$, you have that $1\_R=\emptyset$, the empty set. Thus, any part of the term involving $1\_R$ will have no elements and therefore contribute no elements to that side of the final surreal number value for the product. 


For example, the term $X\_R1+x\emptyset−X\_R\emptyset$ in your expression adds no elements to the final value. But in your calculation, you seem to have indicated that you think it adds $X\_R1$; this is wrong and this is exactly where your calculation makes a mistake. 


Perhaps you thought that in the expression $X\_R1+x\emptyset-X\_R\emptyset$, the latter two terms just evaluate to $0$ and therefore cancel out, giving $X\_R1$. But that is not right. The correct meaning of this term is: for every element $r\in X\_R$ and every $y\in\emptyset$, we contribute the surreal numbers $r1+xy-ry$ to that part of the product surreal value. But since there are no such $y$, we actually end up contributing nothing to the product from this term, even though the first part of the expression $X\_R1$, if it had appeared alone, would have contributed $r1$.


The situation is something like the Cartesian product $X\times\emptyset$, which is empty even when $X$ is not empty, simply because elements $x\in X$ have no partner $y\in\emptyset$ with which to form a pair $(x,y)$. 


