On page $65$ of Combinatorial Game Theory by Siegel, under the section of Dominated and Reversible Options, there is this part which I do not understand:
Consider $G^{L_1R_1L}-G$. By assumption $G^{L_1R_1}\leq G$. So we have $G^{L_1R_1L}\triangleleft G^{L_1R_1} \leq G$, where $\triangleleft$ denotes being less than or confused with. This shows that Right must have a winning move on $G^{L_1R_1L}-G$.
I don't get why Right must have a winning move. Sure, if $G^{L_1R_1L}\leq G^{L_1R_1} \leq G$ then by transitivity $G^{L_1R_1L}-G\leq 0$ and thus Right has a winning move. But what if $G^{L_1R_1L}$ is confused with $G^{L_1R_1}$ $?$