In the proof of Jacobi's Four Square Theorem
From the equation $$\prod_{n>1}\left(1-x^{n}\right)^{6}=\frac{1}{4}\left[\sum_{m\equiv n\ (\text{ mod }2)}(2 m+1)(2 n+1) x^{\dfrac {m^{2}+n^{2}+m+n}{2}}-\sum_{m \not\equiv n(\text{ mod }2)}(2 m+1)(2 n+1)x^{\dfrac {m^{2}+n^{2}+m+n}{2}}\right]$$ We substituted $r=\frac{1}{2}(m+n) $ and $s=\frac{1}{2}(m-n) $ in the first sum and we substituted $r=\frac{1}{2}(m-n-1) $ and $s=\frac{1}{2}(m+n+1) $ in the second sum to get $$\prod_{n \geqslant 1}\left(1-x^{n}\right)^{6}=\frac{1}{2} \sum_{r, s=-\infty}^{\infty}\left((2 r+1)^{2}-(2 s)^{2}\right) x^{r^{2}+s^{2}+r}$$
Can we substitute like this ? I mean we are using same variable but their representation is different ?