= Solution
Put $N=\langle E'\rangle$ and $P=\langle E''\rangle$, the <submodules> generated by the two parts of the <basis of a module>. Every finite expression in the <basis of a module> splits into its $E'$ terms and its $E''$ terms. Thus $M=N+P$.
If $m\in N\cap P$, there are finite expressions
$$
m=\sum_{e\in H'}a_ee=\sum_{e\in H''}b_ee,
\qquad H'\subseteq E',\quad H''\subseteq E''.
$$
Subtracting gives a finite relation among distinct members of $E$, because $E'\cap E''=\varnothing$. Their <linear independence in a module> forces every coefficient to vanish, so $m=0$. Hence
$$
\boxed{M=\langle E'\rangle\oplus\langle E''\rangle}.
$$
This proves the <basis partition gives a module direct sum> lemma, also when one part is empty.
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