There is this Corollary that says:
Let $U_1$ and $U_2$ be subspaces of vector space $V$, then $U_1+U_2$ is also a subspace of $V$.
I looked for proof and i found this:
$$ U_1=(x_1, y_1) $$
$$ U_2=(x_2, y_2) $$
$$ x=(x_1+x_2) x_1\in U_1, x_2\in U_2. $$
$$ y=(y_1+y_2) y_1\in U_1, y_2\in U_2 $$
$$ x+y=x_1+x_2+y_1+y_2=(x_1+y_1)+(x_2+y_2) \in U_1+U_2 $$
I know that subspaces are closed under addition but I can't relate it to what i found. Any explanation?