Let $C$ be a commutative ring, $(E_i)_{1\leq i\leq n}$ a family of graded $C$-modules such that $E_i\cong\bigoplus_{\lambda\in\Delta}M_{i\lambda}$, for each $1\leq i\leq n$. For $\gamma\in\Delta$, define $$S_\gamma:=\left\{x\in\prod_{i=1}^n\bigcup_{\lambda\in\Delta}M_{i\lambda}\ \big|\ \sum_{i=1}^n\text{deg}(x_i)=\gamma \right\}.$$ Write $(\bigotimes_{i=1}^nE_i)_\gamma:=\text{span}_\mathbb{Z}\{\otimes_{i=1}^nx_i\ |\ x\in S_\gamma\}$. I want to show that $((\bigotimes_{i=1}^nE_i)_\gamma)_{\gamma\in\Delta}$ is a grading of $\bigotimes_{i=1}^nE_i$.
Let $\phi:\bigoplus_{\gamma\in\Delta}(\bigotimes_{i=1}^nE_i)_\gamma\rightarrow\bigotimes_{i=1}^nE_i$ be the canonical mapping. Clearly $\phi$ is surjective. I am not sure how to prove injectivity.
Let $w\in\text{ker}(\phi)$. Then $$0=\sum_{\gamma\in\Delta}w_\gamma=\sum_{\gamma\in\Delta}\sum_{x\in S_\gamma}\xi^\gamma_x(\otimes_{i=1}^nx_i).$$ Where to go from here?