Prove that W_1 and W_2 are independent: proof needs checking

I have a good deal of work done for this, but could use a second opinion (and could use some info on whether I'm missing anything).

Suppose that $\displaystyle V$ and $\displaystyle W$ are vector spaces over a field $\displaystyle F$.

Let $\displaystyle W_1$ and $\displaystyle W_2$ be subspaces of a finite-dimensional vector space $\displaystyle V$.

Prove that $\displaystyle W_1$ and $\displaystyle W_2$ are independent if and only if $\displaystyle W_1\cap W_2=\{0\}$.

Here is my work so far (well, it's how my Professor showed us during his office hours, so he probably made it incomplete).

$\displaystyle \forall w_1\in W_1$ and $\displaystyle \forall w_2\in W_2$ such that $\displaystyle w_1+w_2=0$, then $\displaystyle w_1=w_2=0$

If we suppose that $\displaystyle W_1,W_2$ are independent, this implies that $\displaystyle W_1\cap W_2=\{0\}$.

Suppose that $\displaystyle w\in W_1\cap W_2$,

Clearly, $\displaystyle w+(-w)=0$ (We can let $\displaystyle w_1=w$ and $\displaystyle w_2=-w$)

$\displaystyle \Rightarrow w=0$

Using the above, we now need to prove that $\displaystyle W_1\cap W_2=\{0\}\Rightarrow W_1,W_2$ are independent.

Suppose $\displaystyle w_1\in W_1$ and $\displaystyle w_2\in W_2$ and that $\displaystyle w_1+w_2=0$.

$\displaystyle \Rightarrow w_1=-w_2$ OR $\displaystyle w_2=-w_1$

$\displaystyle \Rightarrow w_1\in W_2$ OR $\displaystyle w_2\in W_1$

These imply that $\displaystyle w_1=w_2=0$, and that they are independent.

As such, $\displaystyle W_1$ and $\displaystyle W_2$ are independent.

-----

Tell me if my answer needs any clean-up or reordering.