The Dimension Theorem says $$ \dim(U+W) = \dim(U) + \dim(W) - \dim(U \cap W) $$
The proof of this theorem uses the bases of $U$, $W$, and $U\cap W$.
Is it possible to prove this theorem with just spanning lists of the three vector spaces or the spanning list of $U \cap W$ and linearly independent lists of $U$ and $W$ instead of using bases?