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 and are vector spaces over a field .

Let and be subspaces of a finite-dimensional vector space .

Prove that and are independent if and only if .

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

and such that , then

If we suppose that are independent, this implies that .

Suppose that ,

Clearly, (We can let and )

Using the above, we now need to prove that are independent.

Suppose and and that .

OR

OR

These imply that , and that they are independent.

As such, and are independent.

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