I need to prove that $X \subseteq Y \oplus (Z \cap X)$ if $Y \subseteq X$ and $V=Y \oplus Z$ and $X,Y,Z$ are subspaces of $V$.
We could say that because $V=Y \oplus Z$ then for $x \in X$, $y \in Y$ and $z \in Z$ we have $x=y+z$. But $Z=(Z \cap X) \cup Q$ where $Q \cap X=\{0\}$. Then I'm tempted to say that because of that $X \subseteq Y\oplus (Z \cap X)$ but I feel I haven't really proved this.
Howe can I prove this?
Every element of $V$ can be uniquely written in the form $y + z$, where $y \in Y$ and $z \in Z.$
You need to show that for $x \in X$, the element $z$ in this decomposition $x = y + z$ comes from $Z \cap X$. But this is because $$z = x-y \in X,$$ since we assumed $Y \subseteq X.$