Show that If $U,W$ are subspaces of $V$ and the $\cap =\{0\}$ and $a,b$ are basis of $U,W$ resp. then disj. union of $a,b$ is a base for $U\oplus W$

Question

This is the proof for the Statement above

First a short remark on the proof of $(b)$ we actually just know that $<B>=U+ W$, we get the $\oplus$ because of our precondition that $U\cap W=\{0\}$

The Problem I have is the $\dot\cup$ sign in $(b)$.I am not sure what the message is which this sign is supposed to convey.

Does it mean the intersection of $B_U$ and $B_W$ is the empty set? If so how do we know it?

Or does it mean that there might be some elements in $B_U$ and in $B_W$ which we have to cancel out in one of the sets otherwise we don't get the base we want (i.e it is necessary).