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 $\{X_L, X_R | X_R, X_L\} \equiv \{X_L | X_R\}=x$