I have read only one proof online that shows that the Sprague-Grundy value of a position in an impartial game, denoted by say $g(x,y)$, is equal to $g_1(x) \otimes g_2(y)$, where $\otimes$ denotes Nim-multiplication (https://www.stat.berkeley.edu/~mlugo/stat155-f11/tartan.pdf). However, I was wondering if it can be proven as follows:
Let $b = g_1(x) \otimes g_2(y)$.
If it can be proven that (1) for all integers $a < b$, $a$ is a follower of the larger game $G$ and (2) no follower/move of $G$ has an SG-value equal to $b$. This shows that the Minimum-Excludant of $G$ is $b$.
Does anyone know if this proof is available or how to approach it?