I read this question earlier: Algorithm to multiply nimbers. In this question, a rule about Fermat 2-powers is taken for granted. We define $F_n:=2^{2^n}$ as the $n$th Fermat 2-power. We also use $\otimes$ to represent nim-multiplication, and $\oplus$ to represent nim-addition (nim-addition is equal to bitwise exclusive or). We then define nim-multiplication recursively as follows:
\begin{equation}\label{eq:rule} x\otimes y=\text{mex}\{(x\otimes b)\oplus(a\otimes y)\oplus(a\otimes b):0\leq a<x,0\leq b<y\},\tag{1} \end{equation}
Let $x\leq F_n$. The rule then says $$ F_n\otimes x= \begin{cases} F_n\cdot x&\text{ if $x\neq F_n$}\\ \frac32 F_n&\text{ if $x=F_n$}. \end{cases} $$
My question is, how can we prove this rule? I don't really see how Fermat 2-powers are at all relevant to the definition given in display (\ref{eq:rule}), and I cannot find a proof of this on Google, which makes me suspicious that a proof is very complicated.
