Let $M=K\oplus K'=L\oplus L'$. Prove $K\subset H\leq M$ implies $H=K\oplus (H\cap K')$

Question

Let $M=K\oplus K'=L\oplus L'$. Prove $K\subset H\leq M$ implies $H=K\oplus (H\cap K')$

My attempt:

If $K\subset H\leq M$ then $H=K+ (H\cap K')=(K+H)\cap (K+K')$

Then$~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~=H\cap(K+K')$

$~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~=H\cap M=H$

But I think something is wrong.

Answer 1

For instance, $A+(B \cap C) = (A+B) \cap (A+C)$ only when $B$ or $C$ contains $A$, in which case the identity can be written as $$A+(B \cap C) = \begin{cases} B \cap (A+C) & \textrm{if } A \subseteq B, \\ (A+B) \cap C & \textrm{if } A \subseteq C. \end{cases}$$ We say that the lattice of submodules of a module over a ring forms a modular lattice. Can you take it from here? Also, what role plays $L$ and $L'$ in this problem?