# Proof of Tartan's Theorem?

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?